A quilt is made by joining pieces of cloth together, often in different shapes and sizes. To decorate or cover it, we need to know the length around it and the space it covers. In this chapter, we learn to find the perimeter and area of shapes and use them in real-life situations like measuring quilts, floors, and gardens.
Perimeter
Preetha and Adrit’s grandmother made a beautiful quilt cover from old clothes and now wants to decorate it with lace along its border, using two different colours. To know how much lace of each colour she needs, we must remember that the length around a shape is called its perimeter. Here, we learn how to find the perimeter of different shapes and use it in real-life situations like decorating and designing.
Meaning: The perimeter is the total distance around a shape. Imagine you are walking along the edge of a garden, a playground, or a quilt — the total distance you walk is its perimeter.
How to Find Perimeter:
Remember:
Write units for perimeter (cm, m, km).
The perimeter tells us the length around and not about the inside.
Try yourself:
What is the perimeter?
A.The total distance around a shape
B.The weight of a shape
C.The area inside a shape
D.The height of a shape
View Solution
Now that we know how to measure the length around a shape, let’s think about how much space a shape covers inside.
Preetha and Adrit’s grandmother is making a rug with square patches. Look at the picture of the rug – can you count how many patches she has used? Each patch takes up some space, and all the patches together cover the whole rug.
Preetha and Adrit also tried covering their table using different shapes:
Preetha used triangles and circles.
Adrit used squares and rectangles.
They found that triangle, square and rectangle shapes cover the top of the table without gaps and overlaps. Circle shape leaves gaps.
TWENTY triangles cover Table 1.
EIGHT squares cover Table 3.
SIX rectangles cover Table 4
The region covered by the triangles, squares or rectangles is called the area of the table. To find the area of a region, we usually fill it with shapes that tile well — like squares, rectangles, and triangles. Circles do not tile perfectly because they leave spaces in between. Let us learn about area in detail:
Area
Meaning:
The area of a shape is the amount of surface it covers inside its boundary. If you spread tiles, patches, or paper inside a shape until it’s filled, the space covered is the area.
How to Find Area:
Remember:
Write units for area (cm², m², km²).
The area tells us the space inside a shape, not the length around it.
Let us look at the following examples:
1. Preetha is playing with tiles. She covers her desk with different shapes, as shown below. Look at the different tiles on her desk and answer how many of the following shapes will cover the desk. (a) Green triangles __________ (b) Red triangles __________ (c) Blue squares __________
Ans: (a) Green triangles = 1 (b) Red triangles = 4 (c) Blue squares = 1
2. Compare the areas of the two gardens given below on the square grid. Share your observations.
Area of Garden A = _____ cm square Area of Garden B = _____ cm square
Ans: Area of Garden A = 12 cm² Area of Garden B = 12 cm²
3. Is the area of shape (a) less than the area of shape (b) given below? Discuss.
Preetha and Adrit’s grandmother is making another square patchwork. She arranges the patches as shown below. Can you guess how many patches she will need? How did you find it?
Did you notice that 6 is the length of one side and 4 is the length of the non-equal side of the rectangle?
Ans: Yes, the area of shape (a) is less than the area of shape (b) because shape (b) covers more square units on the grid.
For the square patchwork, there are 6 rows with 4 patches in each row. Total patches = (6×4) square cm=24 square cm Here, 6 is the length and 4 is the breadth of the rectangle, showing that:
Area of a Rectangle = Length × Breadth
Similarly, we can also find the perimeter of the rectangular shape.
Have you ever wondered what would happen if all the sides of a rectangle were equal, that is, the case of a square?
Let us think about a square whose sides are 5 units long.
Conclusion:
In this chapter, we learned how to find the perimeter (the length around a shape) and the area (the space inside a shape). We saw how to measure them for squares, rectangles, and other shapes using unit squares, counting patches, and formulas. We also learned that some shapes can cover a surface without leaving gaps, and that two shapes can have the same area but look different. Knowing how to find perimeter and area helps us in real life, like decorating quilts, measuring gardens, or tiling floors.
Symmetry is all around us — in nature, in the objects we use, and even in the letters we write. It makes things look balanced and beautiful. From the wings of a butterfly to the patterns in rangoli, symmetry helps create harmony in shapes and designs. In this chapter, we will learn what symmetry is, explore different types of lines of symmetry, and discover how to create symmetrical shapes and patterns through fun activities.
Symmetry
Meaning: Symmetry means that when we divide a shape into two halves, one half is the mirror image of the other.
Let us look at the types of Lines of Symmetry using the Alphabet Symmetry Example:
Prem and Manu want to paste ‘Happy Birthday’ cutouts on a wall for Lali’s birthday. While preparing cutouts of letters, they observe that some letters can be cut out in an easy way. They remember that they learnt about reflection symmetry and lines of symmetry in Grade 4.
Let us recall:
Reflection Symmetry: A shape has reflection symmetry if one half is the mirror image of the other half. Imagine folding it along a line — both sides match exactly.
Line of Symmetry: The line that divides a shape into two equal, mirror-image halves is called the line of symmetry.
Vertical Line of Symmetry – The line goes up and down.
Horizontal Line of Symmetry – The line goes left to right.
Some shapes have more than one line of symmetry.
They used their knowledge of lines of symmetry to make the cutouts. The letter A has a vertical line of symmetry. So, to cut out the letter ‘A’:
Similarly, for: The letter H has two lines of symmetry.
Let us try one more example:
Which of the following alphabet cutouts can be made by just drawing half (1/2 ) or quarter (1/4 ) of the letter? You can do it by drawing lines of symmetry on the letters.
Which of the letters have a horizontal line of symmetry? _________________ Which of the letters has a vertical line of symmetry? ____________________ Which letters have both vertical and horizontal lines of symmetry?________
Ans: Letters that can be made by drawing half (1/2) or quarter (1/4) of the letter: E, X, T, O (because they have at least one line of symmetry to fold along)
Letters with a horizontal line of symmetry: E, X, O
Letters with a vertical line of symmetry: X, T, O, V
Letters with both vertical and horizontal lines of symmetry: X, O
So far, we have seen shapes that can be divided into equal mirror-image halves using lines of symmetry. But what if instead of folding them, we turn them? This brings us to our next concept, known as the Rotational Symmetry.
Rotational Symmetry
Meaning: A shape has rotational symmetry if it looks the same after being turned (rotated) around its centre by a certain angle (less than a full turn).
Order of Rotational Symmetry: The number of times a shape matches itself during one complete turn (360°).
Let Us Make a Windmill Firki
To see rotational symmetry in action, let’s make a windmill firki! When the firki spins around its centre, its blades look the same in many positions during one full turn.
Lali makes firkis for her friends. Follow the steps given below to make your firki:
1. Take a square paper. 2. Fold the paper in half diagonally to make two triangles. 3. Open and fold it the other way to make two more triangles. 4. Open it again. You will see an ‘X’ shape on the paper. 5. Use scissors to cut along the four lines of the ‘X’. Stop cutting about halfway to the centre. 6. Take one corner of each triangle and fold it gently towards the centre of the paper. Do not press it flat. 7. Fold every other corner towards the centre. 8. Push a pin through the folded corners and the centre of the paper. 9. Push the pin through a stick or straw
Observe the dot in the firki. Does the firki look the same after 1/4, 1/2, 3/4, and a full turn?
Ans: Yes, the firki looks the same after 1/4, 1/2, 3/4, and a full turn because it has rotational symmetry of order 4.
Shapes with Both Reflection and Rotational Symmetry
Now that we know about reflection symmetry (folding to get mirror images) and rotational symmetry (turning to get the same shape), let us look at shapes that have both.
Some shapes, like a square or certain patterns in rangoli, can be folded along lines of symmetry and still look the same when turned. For example, a square has 4 lines of symmetry and also rotational symmetry of order 4.
Let us explore more examples having both reflection and rotational symmetry:
1. Find symmetry in the digits:
Which digit(s) have reflection symmetry? ___________________________ Which digit(s) have rotational symmetry? ___________________________ Which digit(s) have both rotational and reflection symmetries? ________
Now, let us look at the following numbers: ||, |00| Do these have (a) rotational symmetry, (b) reflection symmetry or (c) both symmetries? Give examples of 2-, 3-, and 4-digit numbers which have rotational symmetry, reflection symmetry, or both.
Ans:
Which digit(s) have reflection symmetry? 0, 1, 8 Which digit(s) have rotational symmetry? 0, 1, 8 Which digit(s) have both rotational and reflection symmetries? 0, 1, 8
For || and |00|: Both have both symmetries. Examples:
(a) Does the design have rotational symmetry? Yes/No. (b) Try to change the design by adding some shape(s) so that the new design looks the same after a 12 turn. Draw the new design in your notebook. (c) Now try to modify or add more shapes so that the new design looks the same after 14 turns. Draw the new design in your notebook. (d) Do the new designs have reflection symmetry? If yes, draw the lines of symmetry
Ans:
(a) No. The design does not look the same after turning it ½ turn (180°) or ¼ turn (90°) because the right side has an extra circle that breaks the balance.
(b) Add another identical circle on the left side, opposite to the existing one. Now the design will look the same after a ½ turn.
(c) To get ¼ turn symmetry, you will need to place identical shapes in all four directions (top, bottom, left, and right) so that after turning 90°, the design remains unchanged. This means adding circles and triangles in the missing positions.
(d) 1. The modified design with ½ turn symmetry will have one vertical line of symmetry. 2. The modified design with ¼ turn symmetry will have two lines of symmetry — vertical and horizontal (and possibly diagonal if shapes are perfectly aligned).
Have you ever lined up your pencils in neat rows or arranged your books in equal stacks? That’s an example of an array –a way to organise objects so we can count them quickly.
Arrays help us understand how multiplication and division are connected.
Multiplication tells us the total when we know the number in each group, while division helps us find out how many are in each group or how many groups there are. ?
What is Multiplication?
Multiplication is a quick way to add the same number many times.
It helps us find the total when we have equal groups.
We use the sign “×” (times) for multiplication. The numbers we multiply are called factors, and the answer is the product.
We use multiplication in daily life—like counting apples in baskets, chairs in rows, or boxes in a stack.
For example, if there are 4 baskets and each basket has 2 apples, we can multiply 2 × 4 = 8 to find the total apples, instead of adding 2 + 2 + 2 + 2.
What is Division?
Division is a way to split a number into equal parts or groups.
It’s like sharing something so everyone gets the same amount.
We use the sign “÷” (divide) or “/” for division. The number we divide is called the dividend, the number we divide by is the divisor, and the answer is the quotient.
We use division in daily life – like sharing chocolates, splitting money, or making equal teams.
For example, if there are 122 apples and 5 friends, we can divide 122 ÷ 5 = 24 to find that each friend gets 24 apples and 2 are left.
Q: Observe the following array of coconuts. Write two division facts using the given multiplication fact.
Sol: From the given multiplication fact 5 × 7 = 35, the two division facts are:
35 ÷ 7 = 5 → 35 split into 7 groups has 5 in each group.
35 ÷ 5 = 7 → 35 split into 5 groups has 7 in each group.
Example: Write the appropriate multiplication fact for the array shown below. Write two division facts that follow from the multiplication fact.
Sol:
Connection between Multiplication and Division
Multiplication and division are related.
In Multiplication, we have two numbers (factors) that make a product.
In Division, we take the product and divide it by one factor to find the other.
Patterns in Division
When we divide by 10, 100, 1000, or multiples of these, a clear pattern appears in the answers.
10 → answer is 10 times smaller.
100 → answer is 100 times smaller.
1000 → answer is 1000 times smaller.
In whole numbers, if the number ends with the same number of zeros as the divisor, those zeros can be removed.
Finding the Patterns in Division:
Sol:
Place Value
Dividing by 10 moves the digits 1 place to the right in the place-value chart.
Dividing by 100 moves the digits 2 places to the right.
Dividing by 1000 moves the digits 3 places to the right.
In decimal numbers, the decimal point moves left the same number of places as the zeros in the divisor.
Finding Place Values:
Sol:
What is happening to the quotients in each case? Discuss.
Dividing by 10, 100, or 1000 makes the number 10, 100, or 1000 times smaller.
This is why the digits move to the right in the place-value chart.
If both numbers change by the same factor, the answer stays the same.
What patterns do you notice here?
Digits move right when dividing by 10, 100, or 1000.
More zeros in the divisor → more places moved.
Some answers are round numbers like 10, 100, or 1000.
Application Problems
We use division to share things equally, so everyone gets the same.
Division helps to make equal groups when we arrange items.
It is used to find the value for one item, one hour, or one day.
We use division in daily life for money, time, distance, work, and other things.
Knowing division helps us solve word problems easily.
Distance Problem:
Q: Sabina cycles 160 km in 20 days and the same distance each day. How many kilometres does she cycle each day?
Sol: Sabina cycles 160 km in 20 days.
To find how many kilometres she cycles each day:
160÷20=8 km
Thus, she cycles 8 km each day.
Mental Strategies for DivisionWe can divide faster in our mind by using some simple tricks:
Break big numbers into smaller, easy parts, then add the results.
Use numbers that are close and easy to divide, then adjust the answer.
Halve the number twice if dividing by 4, and three times if dividing by 8.
Use multiplication tables to see how many groups fit in the number.
Estimate a close answer first to check if your result makes sense.
Try some strategies:
1.
Sol:
2.
Sol:
Partial-Quotients Division & Estimation
Sometimes we can solve big division problems in easy steps. This is called partial-quotients division.
Take away big groups of the divisor again and again.
Each time, write how many groups you took away.
Add all the groups to get the answer.
Bigger groups mean fewer steps.
First, make a guess (estimate) to help choose group sizes and check your answer.
Susie’s Farm in Kerala
Susie and Sunitha have a large coconut farm, and they harvested 1,117 coconuts in April. They sold 582 coconuts equally to 6 regular customers. How many coconuts did each customer get?
They sold 582 ÷ 6 coconuts to each customer. Susie and Sunitha both gave two different methods:Each customer gets 97 coconuts.
Do you think Sunitha’s method is better? Discuss which one you would prefer and why.
Sol: I think Sunitha’s method is better because it is faster. She takes away a big number first, so there are fewer steps. Susie’s way works, but it takes more time because she takes away small numbers many times.
Each bag can hold 25 coconuts. How many bags would be needed to pack 97 coconuts?
Sol: 3 bags will hold 75 coconuts. They will need another bag to fill the remaining coconuts. So, each person will get 4 bags.
They pack the remaining coconuts for drying and extracting oil. They can pack 25 coconuts in each bag. How many bags will they need to pack the remaining coconuts?
Sol: The number of coconuts left after selling 582 coconuts is 1117 – 582 = 535. The number of bags needed is 535 ÷ 25.
They need 21 full bags and 1 more bag to pack the 10 remaining coconuts, that is, 22 bags.
Remainders & the Division Relationship
Sometimes, the divisor (D) does not completely divide the dividend (N) and leaves a remainder (R)
Words to know:
Dividend = the big number we’re dividing.
Divisor = how many in each group.
Quotient = the number of full groups.
Remainder = what’s left over.
The rule is: N = (D × Q) + R
The remainder is always smaller than the divisor.
A remainder of 0 means no leftovers.
A non-zero remainder means there are leftovers.
Identify the remainder, if any. Check if N = D × Q + R:
1. 902 ÷ 16
Sol:
Now, after solving the above, we have:
N= 902
D= 16
Q= 56
R= 6
Checking if N = D × Q + R :
902 = 16 x 56 + 6
902 = 896 + 6
902 = 902
Hence, proved that both sides are equal.
Kalpavruksha Coconut Oil
In a particular year, Susie and Sunitha used 4376 coconuts for extracting coconut oil. They can extract 1 litre of oil from 8 coconuts. What quantity of oil were they able to extract?
Sol: They would get 4376 ÷ 8 litres of coconut oil.They extracted 547 L of oil in the year.
Division Using Place Value
Start with the biggest place value (the leftmost digit) when dividing a number.
Check if the divisor fits into the number or part of it.
Regroup if the part you’re dividing is smaller than the divisor.
Place zeros in the quotient if needed when the number doesn’t divide evenly.
Continue dividing by moving to smaller place values (tens, hundreds, ones).
After dividing, check the answer by multiplying the quotient with the divisor.
Candle Distribution Problem:
Sunitha’s mother has 62 candies to be distributed equally among 5 children. How many candies would each child get? She shows the following way of doing division using place value.
Sol: 136 = 1 Hundred + 3 Tens + 6 Ones. 1 Hundred ÷ 6 → not possible without regrouping into Tens Regroup 1 Hundred into 10 Tens. Total 13 Tens. Continue dividing.
Extended Word Problems
Some problems use both multiplication and division.
First, you might need to multiply to find a total.
Then, divide to share that total equally or find how many groups there are.
Do the steps in the correct order to get the right answer.
Solved Example:
Naina bought 5 kg of ice cream as a birthday treat for her 23 friends. 400 g of ice cream was left after everyone had an equal share. How much ice cream did each of her friends eat?
Sol: Naina bought 5 kg of ice cream. That means she had 5000 grams of ice cream. (1 kg = 1000 grams)
After the party, 400 grams were left. So, the ice cream that was eaten =5000−400=4600 grams.
She had 23 friends. Everyone got the same amount.
To find each friend’s share:4600÷23=200 grams.
Thus, each friend ate 200 grams of ice cream.
Data Handling via Division
Division helps you find missing numbers in tables or charts.
It can also help to check totals and see if they add up correctly.
Sometimes, you need to divide to find the value for one item or unit.
Vegetable Market:
Munshi Lal has a big farm in Bihar. Every Saturday, he sells the vegetables from his farm at Sundar Sabzi Mandi. Munshi ji maintains a detailed record of the quantity of vegetables he sends to the Mandi and the cost of each vegetable.
The following table shows his record book on one Saturday. His naughty grandson has erased some numbers from his record book. Help Munshi Lal complete the table.
Sol:
Radish: Cost of 1 kg = ₹26 Quantity = 78 kg26×78=2028 Total amount = ₹2028
Potato: Total amount = ₹2240 Cost of 1 kg = ₹202240÷20=112 Quantity = 112 kg
Cabbage: Cost of 1 kg = ₹32 Quantity = 56 kg32×56=1792 Total amount = ₹1792
Green peas: Total amount = ₹3125 Quantity = 125 kg3125÷125=25 Cost of 1 kg = ₹25
Total money earned:2028+2240+1792+3125=9185 Total = ₹9185
Mathematical Statements
When you see a math statement, decide if it’s always, sometimes, or never true.
Think carefully about why the statement is right or wrong, and explain your reasoning.
Solved Example:
Complete the following statements such that they are true. (a) 7 × 6 = ____ + 17 (b) 87 + 6 = ____ × 31 (c) 63 + ____ = 74 – 4 (d) ____ ÷ 9 = 16 ÷ 2
Ever bought fruits at a shop and saw the shopkeeper use a weighing scale? That’s measuring weight.
Ever filled a water bottle or a milk jug? That’s measuring capacity.
We use units like kilograms (kg), grams (g), litres (l), and millilitres (ml) to measure.
Sometimes we need to change units, add, subtract, multiply, or divide to solve problems.
These skills help in shopping, cooking, sports, farming, and many other daily activities.
Check! Check!
Anu has recorded the weights of the items in her house. Check if she has recorded them correctly by putting a tick against them if they look correct.
Sol:
Explanation:
1. This is too light because iron almirahs are very heavy and can weigh many kilograms.
2. This sounds correct because beds are heavy but not too heavy to move with help. 3. This is correct because we often buy rice in 5 kg bags. 4. This is too light because sofas are big and heavy, not as light as a few grams. 5. This is correct because buckets can weigh this much, especially if they are big or filled with something. 6. This is correct because a full water bottle can weigh around this much. 7. This is too light because refrigerators are very heavy and can weigh many kilograms.
Understanding Weighing Scale
A weighing scale tells us how heavy something is. We can see these in shops, kitchens, bathrooms, and even in school labs.
Scales can look different, but all of them do the same job – they measure weight.
The number ‘0’ on the scale means nothing is on it, so there is no weight.
Before you measure anything, look at the scale to see if it is at 0.
If the scale is not at 0 before you start, the weight will not be correct.
Some scales have a small knob or button that can move the pointer back to 0. This is called “zeroing” the scale.
When the scale is at 0, it measures only the weight of the thing you put on it.
Reading Weight
Look at the biggest numbers first. They tell you the weight in kilograms or grams.
See where the pointer is pointing. Check if it’s between two big numbers.
Count the small lines between the big numbers – each small line is usually 100 g or 50 g, depending on the scale.
Start from the smaller big number and add the small lines until you reach the pointer.
Write the weight in kilograms and grams, or only grams if it’s a small weight.
Always check from the zero mark to be sure the scale is correct before reading.
Read the ScalesRead the scales. Write the correct weight in the space given below.
Sol:
(a)
Needle between 0 and 1 kg.
Each small line = 100 g.
Needle at 600 g.
(b)
Between 1 kg and 2 kg.
Needle at 1 kg 800 g.
(c)
Between 2 kg and 3 kg.
Exactly halfway → 2 kg 500 g.
(d)
Between 2 kg and 3 kg.
Needle at 2 kg 600 g.
(e)
Between 0 g and 250 g.
Needle at 150 g.
(f)
Between 500 g and 750 g.
Needle at 650 g.
Units of Weight and their Relationship
Weight can be measured in kilograms (kg) and grams (g).
1 kilogram = 1,000 grams. This means a kilogram is a bigger unit and grams are smaller units.
If we have a weight in kilograms and want to know it in grams, we multiply by 1,000. Example: 3 kg = 3 × 1,000 = 3,000 g.
If we have a weight in grams and want to know it in kilograms, we divide by 1,000. Example: 2,500 g = 2,500 ÷ 1,000 = 2.5 kg.
Kilograms are used for heavier things like a bag of rice or a person’s weight.
Grams are used for lighter things like a chocolate bar or a pencil.
Match the weights
Match the bags that have the same weight. You can use the double number line given below.
Sol:
Step 1: Weighing Balance 1
It has weights in kilograms:
5 kg, 10 kg, 3 kg, 6 kg, 25 kg, 30 kg
Step 2: Convert them into grams
5 kg = 5 × 1000 = 5000 g
10 kg = 10 × 1000 = 10,000 g
3 kg = 3 × 1000 = 3000 g
6 kg = 6 × 1000 = 6000 g
25 kg = 25 × 1000 = 25,000 g
30 kg = 30 × 1000 = 30,000 g
Step 3: Match with Weighing Balance 2
5 kg ↔ 5000 g
10 kg ↔ 10,000 g
3 kg ↔ 3000 g
6 kg ↔ 6000 g
25 kg ↔ 25,000 g
30 kg ↔ 30,000 g
Step 4: Understanding the Double Number Line
Top line shows kilograms
Bottom line shows grams
To go from kilograms to grams, multiply by 1000
To go from grams to kilograms, divide by 1000
Examples from the number line:
1 kg = 1000 g
3 kg = 3000 g
8 kg = 8000 g
20 kg = 20,000 g
30 kg = 30,000 g
Weigh the Cake
Shrenu is baking cakes for her shop. She needs 3 kg 500 g of flour. Her kitchen scale measures only in grams. What should her kitchen scale show for 3 kg 500 g of flour?
Sol:
What would be 2 kg 250 g of flour in grams?
Sol:
Comparison between Different Weights
1. Harpreet’s family planned a picnic over the weekend. Her mother and father packed different food items to take along. The following is the list of fruits they carried.
Among the fruits they carried, which one has the (a) highest weight? __________ (b) least weight? __________ (c) Arrange the items in descending order of their weight.__________ __________ __________ __________
Sol: (a) Highest weight? Watermelon – 3 kg (b) least weight? Apples – 1 kg 250 g (c) Arrange the items in descending order of their weight. Watermelon, Mangoes, Pineapple, Apples
2. Compare the weights using <, =, > signs.
Sol:
Milligram
Milligrams (mg) are tiny units for very light things like medicine or gold.
1 gram = 1,000 mg.
To change g into mg → multiply by 1,000.
To change mg into g → divide by 1,000.
It helps in precise measurements.
Solved Example:
If a sugar sachet weighs 5g, how much will it be in milligrams?
Sol: We know that 1 g = 1000 mg
Thus, 5 g = 5 x 1000 = 5000 mg
Tonnes
Tonnes (t) are big units for very heavy things like trucks, elephants, or huge containers of goods.
1 tonne = 1,000 kilograms.
To change tonnes into kilograms → multiply by 1,000.
To change kilograms into tonnes → divide by 1,000.
It helps in measuring very heavy weights.
King’s Weight
In a kingdom, the king donates wheat grains equal to 10 times his weight on his birthday.
(a) If he donates 800 kg of wheat grain this birthday, what is his current weight? _______ kg. (b) If he had donated 780 kg of wheat grain on his last birthday, what was his weight last year? _______ kg. (c) How much weight did he gain in a year until this birthday?_______ kg
Sol: (a) 80 kg (b) 78 kg (c) (80 – 78) kg = 2 kg
From Tiny to Big
Always remember these weight conversions – they are the key to changing between milligrams, grams, kilograms, quintals, and tonnes!
Addition and Subtraction of Weights
We measure weight in kilograms (kg) and grams (g).
1 kilogram = 1,000 grams.
When adding or subtracting weights, first look at the kilograms and grams separately.
Step 1: Add or subtract the kilograms.
Step 2: Add or subtract the grams.
If the grams are 1,000 or more, change 1,000 g into 1 kg and add it to the kilograms.
Sometimes it’s easier to change the whole weight into grams first, do the calculation, and then change the answer back to kilograms and grams.
This method is very useful in real life — for example, when adding the weight of vegetables in a market or measuring ingredients in cooking.
Example: 2 kg 500 g + 1 kg 750 g → Add kilograms: 2 + 1 = 3 kg → Add grams: 500 + 750 = 1,250 g → Change 1,000 g into 1 kg and add to kilograms → Total = 4 kg 250 g
The Grocery Store
Rathna went to the local grocery store and bought several items.
She bought 2 kg 500 g rice for daily use and 1 kg 750 g additional rice for the upcoming Pongal festival. How much total rice did she buy?
Sol: We have three different waysto add these weights:
Mental Method (breaking into parts):
Add kilograms and grams separately.
2 kg + 1 kg = 3 kg
500 g + 750 g = 1250 g = 1 kg 250 g
So, 3 kg + 1 kg 250 g = 4 kg 250 g
Column Method:
Write kg and g in columns and add them.
Carry over 1000 g as 1 kg.
Answer is 4 kg 250 g.
Convert to grams:
2 kg 500 g = 2500 g
1 kg 750 g = 1750 g
Add: 2500 g + 1750 g = 4250 g
Convert back: 4250 g = 4 kg 250 g
Multiplication and Division of Weights
Weighing is not just about adding and subtracting — sometimes we need to multiply or divide weights.
Multiplication with weights
Used when we have many items of the same weight.
We multiply the weight of one item by the number of items to get the total weight.
Example: If one packet of flour weighs 2 kg and we buy 5 packets, 2 kg × 5 = 10 kg total.
Division with weights
Used when we know the total weight and need to find the weight of one item.
We divide the total weight by the number of items.
Example: If 12 kg of rice is packed equally into 4 bags, 12 kg ÷ 4 = 3 kg per bag.
Why it’s useful:
Helps when buying in bulk or sharing food equally.
Useful for finding the cost of items (price per kg × weight) or finding weight from cost (total price ÷ price per kg).
Always remember:
You can multiply or divide kilograms and grams directly.
Or convert everything to grams first, do the calculation, then change it back to kg and g.
Solved Example
1. A farmer weighs a sack of potatoes and finds it to be 10 kg 500 g.If the farmer has 4 such potato sacks, what is the total weight of all the sacks?
Sol: 4 × 10 kg 500 g = 4 ×10 kg and 4 × 500 g = 40 kg + 2000 g = 40 kg + 2 kg = 42 kg.
2. A box of nuts weighing 4 kg 800 g is equally distributed into 4 smaller boxes. What is the weight of each small box in grams?
Sol: 4 kg ÷ 4 = 1 kg 800 g ÷ 4 = 200 g So, 4 kg 800 g ÷ 4 = 1 kg 200 g
Measuring Capacity
Capacity refers to the amount of liquid a container can hold.
We measure it using litres (l) and millilitres (ml). 1 litre = 1,000 millilitres. Bigger containers (like buckets, bottles) are measured in litres. Smaller amounts (like a spoon of medicine, a small packet) are measured in millilitres.
Example:
At home, when we prepare tea, we use water and milk.
If we want to make 2 cups of tea, we don’t need 1 litre of water because 1 litre is too much.
500 ml of water is usually enough for 2 cups.
This shows we choose the right unit (ml or l) depending on the quantity.
Big to Small, Small to Big
Write the total capacity of the following containers in each blank.
Sol: Observe the beakers:
Big beaker = 1 litre (1 l = 1000 ml)
Medium beaker = 500 ml
Small beaker = 100 ml
Add the capacities in each set.
Top-left set: 1 l + 500 ml + 100 ml = 1 l 600 ml
Top-right set: 1 l + 500 ml = 1 l 500 ml
Bottom-left set: 100 ml + 100 ml + 500 ml = 700 ml (or 0 l 700 ml)
Bottom-right set: 1 l + 1 l + 100 ml = 2 l 100 ml
Different Units but Same Measure
Litres and millilitres are connected – whenever litres feel too big, we can change them into millilitres. Example: 2 litres = 2,000 ml.
Containers are marked in ml or l – like a milk vessel or water bottle. Even if the vessel is marked in ml, you can still measure litres.
Example:
Shrivastava’s family takes 12,000 ml, which means 12 litres.
Rao’s family takes 25,000 ml, which means 25 litres.
Why we learn this – it helps us when: Buying milk, petrol, or juice. Measuring medicines in ml. Cooking recipes where exact liquid amounts matter.
Measuring Milk
Khayal chacha delivers fresh cow milk to homes. Bhalerao’s family orders 2l of milk every day. This family has a vessel marked in ml only. What mark will you see in the vessel corresponding to 2l?
Khayal chacha delivers the following amounts of milk each week to different families.
Dev’s family needs 1 l milk every day. On Sunday, they need 500 ml more.Quantity of milk they need on Sunday = 1 l + 500 ml = 1,000 ml + 500 ml = 1,500 ml
Sol:
1. Bhalerao’s vessel mark for 2 litres Rule: 1 l = 1000 ml 2 l = 2 × 1000 = 2000 ml So the mark you will see is 2000 ml.
2. Fill the litres–millilitres blanks (Use 1 l = 1000 ml)
1 l → 1000 ml 2 l → 2000 ml 6 l → 6000 ml 8 l → 8000 ml 12 l → 12,000 ml 14 l → 14,000 ml 20 l → 20,000 ml 25 l → 25,000 ml
So the filled rows are:
3. Weekly delivery table (convert each using 1 l = 1000 ml)
4. Dev’s family Needs 1 l every day, and on Sunday, 500 ml more. 1 l + 500 ml = 1000 ml + 500 ml = 1500 ml
Comparing Capacity
It means finding out which container can hold more liquid and which one can hold less.
How do we compare? 1. If the liquids are given in the same unit (both in litres or both in millilitres), just look at the numbers. Example: 3 l vs 5 l → 5 l is more. 2. Example: 1 l vs 750 ml → change 1 l into 1,000 ml → now 1,000 ml is more than 750 ml.
Why do we convert? 1. Because it’s like comparing apples and oranges. 2. To compare properly, we need both in the same form (like both in ml or both in l).
Real-life examples: 1. A water bottle (1 l) vs a juice packet (250 ml) → bottle holds more. 2. A bucket (15 l) vs a mug (1 l) → bucket holds more. 3. Milkman gives 2 l to one family and 1,500 ml to another → convert 2 l = 2,000 ml → 2,000 ml is more.
Easy tip to remember: 1. Bigger number in the same unit = bigger capacity. 2. Always bring litres and millilitres to the same unit before deciding.
Let Us Compare
Kiran owns a petrol pump. She records the details of the sales of petrol in a day.
(a) How much more fuel is bought for buses than for trucks? (b) What is the total quantity of fuel filled from the petrol pump on that day?
Sol: First, find the total fuel for each vehicle type (No. of vehicles × fuel in each):
(a) How much more for buses than trucks? Buses: 1,800 litres Trucks: 1,500 litres Difference: 1,800 − 1,500 = 300 litres more for buses. (b) Total fuel filled that day Add all the totals: 1,500 + 1,800 + 500 + 96 + 125 = 4,021 litres.
Addition of Capacity
Capacity means the amount of liquid a container can hold.
To add, write litres (l) and millilitres (ml) separately.
Add ml and l column-wise.
If millilitres go over 1,000, regroup them into litres (because 1,000 ml = 1 l).
Example:
2 l 500 ml + 3 l 750 ml
= (2 + 3) l and (500 + 750) ml
= 5 l and 1,250 ml
1,250 ml = 1 l 250 ml
Final = 6 l 250 ml
Subtraction of Capacity
Write litres (l) and millilitres (ml) separately.
Subtract ml and l column-wise.
If ml in the top number is smaller, borrow 1 litre = 1,000 ml.
Example:
5 l 200 ml – 3 l 750 ml
= (5 – 3) l and (200 – 750) ml
Since 200 is smaller, borrow 1 l = 1,000 ml
200 + 1,000 = 1,200 ml
Now subtract: (4 – 3) l and (1,200 – 750) ml
Final = 1 l 450 ml
Let us solve
Sam and Tina fill petrol in their bikes. Tina bought 2 l 500 ml of petrol. Sam bought 2 l 800 ml more petrol than Tina. How much petrol didSam buy?
Sol: Sam found the quantity of petrol by adding like quantities.
2 l 500 ml + 2 l 800 ml = 2 l + 2 l and 500 ml + 800 ml = 4 l and 1,300 ml = 4 l and 1 l and 300 ml = 5 l 300 ml
Tina converted the quantities into ml, that is, 2,500 ml and 2,800 ml.
Total quantity of petrol bought by Sam = 2,500 ml + 2,800 ml = 5,300 ml =5 l 300 ml.
After refuelling, Sam found his fuel gauge reading 9 l.
How much fuel did his bike have before refuelling? The quantity of fuel Sam’s bike had before refuelling is–Sam’s bike had 3l 700ml of fuel before refuelling.
Patterns are everywhere – from woven mats to floor tiles!
Shapes fit together to make beautiful designs without gaps or overlaps.
In this chapter, you’ll learn to make patterns by weaving and tiling.
You’ll explore how triangles, squares, and other shapes create repeating patterns.
Get ready to create shapes and solve fun geometry puzzles!
Patterns
A pattern is a repeated arrangement of shapes, colours, numbers, or objects in a particular order.
Patterns can be: Shapes: Like repeating circles or squares. Numbers: Like 2, 4, 6, 8 (numbers increase by 2 each time). Colours: Like red, blue, red, blue (alternating colours).
How Patterns Work: Patterns repeat in a specific order. Once you know the rule (like “1 over, 1 under”), you can continue the pattern. Example: In a pattern of colours, “red, blue, red, blue,” you can predict that the next colour will be red.
Creating Patterns: You can create your patterns by picking a shape, number, or colour and repeating it in a certain order. Example: “circle, square, circle, square…” is a simple pattern.
Patterns in Everyday Life: You can find patterns everywhere, like in designs on clothes, tiles, or nature, such as the stripes on a zebra or the petals of a flower.
Why Are Patterns Important? Recognising and creating patterns helps us organise information, solve problems, and understand the world around us.
Weaving Mats
You may have seen woven baskets of different kinds. If you look closely, you will notice different weaving patterns on each basket.
We will try weaving some mats with paper strips.
You will need —A coloured paper (30 cm long and 20 cm wide) and eight paper strips of two different colours (3 cm wide and longer than 20 cm). (a) Take a coloured paper 30 cm long and 20 cm wide. (b) Fold the coloured paper in half along the longer side. (c) Draw vertical lines at equal distances from the closed end and cut slits, leaving a gap of 3 cm at the top. (d) Carefully unfold the paper. There will be no cuts in the paper at the top and the bottom. (e) Now cut 8 paper strips of 3 cm width in 2 colours and of length slightly longer than 20 cm. (f) Take one colour strip and weave it across the slits, going 1 under and 1 over, and again 1 under and 1 over. Repeat it for the first row. (g) Take one more strip of another colour and weave it across the slits, going 1 over and 1 under, and again 1 over and 1 under. Repeat it for the second row. (h) Weave all the strips in the same alternating pattern. Neatly fold any extra strip ends behind the mat. Your mat is ready!
Different Types of Shapes
The following table summarises different types of shapes:
Tiling and Tessellation
Tiling: Using shapes (like tiles) to cover a surface without gaps or overlaps.
Regular Shapes: Shapes with equal sides, like squares, triangles, and pentagons.
Pentagons: A regular pentagon has 5 equal sides.
When we place 3 pentagons around a point, there’s a space left.
Pentagons cannot tessellate because they don’t fit together perfectly—they leave gaps.
Tessellation happens when shapes fit together without any gaps or overlaps, like squares or triangles.
Find Out
Can regular triangles fit together at a point without any gap? How many of them fit together? Do you see that regular triangles fit around a point as shown here?
Regular triangles, when fitted around a point, leave no gaps, and there is no overlap.
Triangles with all equal sides are also called equilateral triangles. Therefore, equilateral triangles tessellate.
Sol:
Yes! Equilateral triangles (triangles with all equal sides) fit together perfectly around a point without any gaps or overlaps.
Exactly 6 equilateral triangles fit together around a point. When placed correctly, they create a tessellating pattern with no empty spaces.
Can five squares fit together around a point without any gaps or overlaps? Why or why not?
Sol:
Yes, squares (4-sided shapes) also fit together perfectly around a point without gaps or overlaps. You will need 4 squares to fit together around a point.
No, 5 squares cannot fit together at a point because there isn’t enough space to align them without leaving gaps.
Can regular hexagons (6-sided shapes with equal sides) fit together around a point without any gaps or overlaps? Try and see (a sample hexagon is given at the end of the book). How many fit together at a point?
Sol: Yes, regular hexagons (6-sided shapes) fit together perfectly around a point, with 3 hexagons fitting together at each point without any gaps or overlaps.
Here is a tessellating pattern with more than one shape. What shapes have been used in this pattern? _____________, _____________.
Sol: Triangles, Regular Hexagon
A regular octagon means a shape with eight equal sides. Do regular octagons fit together without any gaps or overlaps? Try drawing the same and check
Sol: Regular octagons do not tessellate.
Because:
The angles inside an octagon are too big to fit together perfectly at the corners.
When you place octagons next to each other, they leave gaps because their angles don’t match up to fill the space completely.
A rhombus is a shape with all equal sides.
What shapes are coming together at the marked points? Are the same set of shapes coming together at these points?
Sol:
At the marked points, we can see rhombuses and triangles coming together.
Yes, the same set of shapes (rhombuses and triangles) is repeating at each marked point. This means the pattern is consistent and repeating.
Rhombuses have equal sides, and when divided into four triangles, the angles of the triangles match up perfectly with each other. This helps the shapes fit together neatly without any gaps.
The equal sides and angles of rhombuses and triangles allow them to fill a surface by repeating without leaving any empty spaces. This is why tessellation works — the shapes fit together perfectly and create a pattern.
Types of Triangles
Equilateral Triangle
All three sides are equal.
All three angles are the same and are always 60°.
Example: A perfect triangle with all sides the same length.
Isosceles Triangle
Two sides are equal, and the third side is different.
The two angles opposite the equal sides are also equal.
Example: A triangle with two sides of the same length and one different.
Scalene Triangle
All three sides are different lengths.
All three angles are also different.
Example: A triangle with no equal sides or angles.
Quadrilaterals
Quadrilaterals whose opposite sides are equal are called parallelograms. What types of angles do quadrilaterals A and B have? Which angles are equal in each of the above parallelograms?
In parallelogram A, opposite angles are equal.
In parallelogram B, all angles are equal and are right angles. Such a parallelogram is called a rectangle.
A rectangle is a special type of parallelogram.
Tangram
A Tangram is a puzzle made of 7 flat pieces (called tans) that can be rearranged to form different shapes and figures.
Observing Angles and Sides When you look at the Tangram pieces, you can see how the angles of each piece fit together to form new shapes. The triangles have different angles, while the square and parallelogram have right angles. The pieces may have equal sides or unequal sides, depending on their shape.
Using Tangram Pieces to Make Figures You can rearrange the 7 pieces to make different shapes. For example: a bird or a fish.
Tangrams help develop spatial thinking, which means understanding how shapes fit together and how they can be rearranged.
Why is Tangram Fun? Tangram puzzles are fun because you can create many different things using the same 7 pieces. It also helps you practice problem-solving and creativity as you figure out how to arrange the pieces into new shapes.
Observe and Answer
Look at the tangram set given. Cut out all the shapes. Name them.
(a) How are they the same or different from each other? (b) What do you notice about the angles of each of the shapes? (c) What do you notice about the sides of each of the shapes? Now, use some or all of the pieces of your tangram set to make the following shapes. There may be more than one way to do it.
Sol:
A standard Tangram has 7 pieces:
2 Large right-angled isosceles triangles
1 Medium right-angled isosceles triangle
2 Small right-angled isosceles triangles
1 Square
1 Parallelogram
(a) Same:
All the shapes are polygons.
Many are right-angled isosceles triangles (triangles with one right angle and two equal sides).
Altogether, they fit perfectly into a large square.
Different:
Some shapes are triangles, one is a square, and one is a parallelogram.
Their sizes vary (small, medium, and large triangles).
The parallelogram is the only piece without right angles.
(b)
Triangles: Each has a right angle (90°) and two 45° angles.
Square: All four angles are right angles (90°).
Parallelogram: Opposite angles are equal, not all right angles.
(c)
Triangles: Two equal sides (isosceles).
Square: All four sides equal.
Parallelogram: Opposite sides equal and parallel.
Using Tangram Pieces to Make Given Shapes
Parallelogram (top left):
Can be made using the parallelogram piece itself OR by combining triangles.
Square (top right):
Can be made using the square piece directly, or by joining two small triangles.
Triangle (bottom left):
Can be made using any of the triangle pieces (small, medium, or large).
Trapezium (bottom right):
Can be made by combining two or more triangle pieces.
Kites
A kite is a four-sided shape with two pairs of adjacent sides that are equal in length.
The adjacent sides are the sides that are next to each other.
The diagonals of a kite cross each other at right angles (90°), and one diagonal bisects the other.
The angles between the unequal sides of the kite are equal.
You can make a kite shape by folding a square piece of paper diagonally and bringing the opposite corners together.
Kites are also seen in real life, like the flying kites or diamond shapes.
The name “kite” comes from the kite shape seen in flying kites!
Make your own kite shape.
(a) Start with a square piece of paper. (b) Take one corner of the paper and fold it towards the opposite corner, creating a sharp crease along the diagonal. (c) Open and fold the corner A inwards, aligning the edge with the crease you just made. (d) Repeat on the other side, folding the other corner B inwards to align with the crease at the centre. You have a kite shape! What shapes do you see in the kite
Sol: The kite shape is a quadrilateral made of two triangles and a central quadrilateral (square/diamond) formed by the folds.
Circles
A circle is a shape with no straight sides, and every point on the circle is equidistant from the centre.
The distance from the centre of the circle to any point on the circle is called the radius.
The longest distance across the circle, passing through the centre, is called the diameter. The diameter is twice the length of the radius.
The circumference is the distance around the edge of the circle.
Circles are round and have no corners or edges.
Every circle has 360° angles, meaning the entire angle around the circle adds up to 360 degrees.
Pi (π) is a special number used to calculate the circumference and area of a circle.
Play with Circles
(a) Draw a circle with a compass and mark its centre. (b) Draw its diameter. Mark the endpoints of the diameter. (c) Draw another diameter of the circle and mark the endpoints. (d) Now join the four points.What shape is formed? Check the sides of the quadrilateral and the angles obtained. Try with a different pair of diameters. What do you notice about the shape that is formed? Is it possible to create a 4-sided shape other than a rectangle through this process?
Sol: What Shape is Formed?
When you join the four points, you get a quadrilateral ACBD.
This quadrilateral is a rectangle (in some cases, a square if the diameters are perpendicular and equal divisions).
Checking the Sides:
Opposite sides are equal (AB = CD and AC = BD).
The figure is symmetric about both diameters.
Checking the Angles:
Each interior angle is 90°.
So the quadrilateral is a rectangle (and sometimes a square if the diameters are perpendicular).
Trying with Different Pairs of Diameters:
No matter which diameters you draw, the figure formed is always a rectangle (or a square).
This is because diameters always pass through the centre, and opposite points on a circle are always symmetric.
Can a 4-sided Shape Other Than a Rectangle Be Formed? No.
Any quadrilateral formed by joining the endpoints of two diameters of a circle will always be a rectangle (or a square as a special case).
It cannot be any other quadrilateral like a trapezium, parallelogram (non-rectangle), or rhombus.
Cube
A cube is a 3D shape with 6 square faces (flat surfaces).
All the faces of a cube are equal in size and shape, and all angles are 90°.
A cube has 12 edges (the sides where two faces meet) and 8 vertices (corners where edges meet).
The length, width, and height of a cube are all the same, making it a regular shape.
A cube is like a box—you can think of dice as an example of a cube.
Surface Area of a cube is the area of all 6 faces added together.
Volume of a cube is found by multiplying the length, width, and height (since all are equal, it’s simply side³).
Cube Connections
Here are some big, solid cube frames. How many small cubes have been removed from each cube?
Sol: (a) First set
The cube is 4 × 4 × 4.
Total small cubes = 64.
From each face, 1 cube is removed from the centre.
There are 6 faces, so 6 cubes are removed.
(b) Second set
The cube is 6 × 6 × 6.
Total small cubes = 216.
From each face, 4 cubes are removed (a 2 × 2 square in the centre).
6 faces × 4 cubes = 24 cubes removed.
(c) Third set
The cube is 8 × 8 × 8.
Total small cubes = 512.
From each face, 16 cubes are removed (a 4 × 4 square in the centre).
6 faces × 16 cubes = 96 cubes removed.
Icosahedron and Dodecahedron
An icosahedron is a 3D shape with 20 triangular faces. All its faces are equilateral triangles, meaning all sides are equal.
A dodecahedron is a 3D shape with 12 pentagonal faces. Each of its faces is a regular pentagon, meaning all sides and angles are equal.
Both icosahedron and dodecahedron are types of Platonic solids, which means their faces are made up of identical regular polygons.
An icosahedron has 30 edges and 12 vertices.
A dodecahedron has 30 edges and 20 vertices.
These shapes can be made by folding a 2D net into a 3D solid, and they have a special symmetry in geometry.
Let us find out
Use the nets provided at the end of the book to make icosahedron and dodecahedron models.
1. What shapes do you see in an icosahedron and a dodecahedron? Icosahedron: …….. Dodecahedron: ……..
2. Do all the faces look the same? Icosahedron: …….. Dodecahedron: ……..
3. How many faces meet at a vertex (point)? Icosahedron: …….. Dodecahedron: ……..
4. Do the same number of faces meet at each vertex? Icosahedron: …….. Dodecahedron: ……..
5. How many edges do you see? Icosahedron: …….. Dodecahedron: ……..
Sol:
1. Icosahedron: Triangles Dodecahedron: Pentagons
2. Icosahedron: Yes Dodecahedron: Yes
3. Icosahedron: 5 Dodecahedron: 3
4. Icosahedron: Yes Dodecahedron: Yes
5. Icosahedron: 30 Dodecahedron: 30
How did you count them such that you do not miss out any edge or count an edge twice? Sol: I counted by looking at one face at a time and then checking how many new edges appeared as I moved around the shape. I also remembered that every edge belongs to exactly two faces, so I did not count the same edge twice.
Can you think of any other solid shapes that have faces that look the same? Sol: Yes. A cube has 6 square faces that are all the same. A tetrahedron has 4 triangular faces that are all the same. An octahedron has 8 triangular faces that are all the same.
Do the same number of faces meet at each common vertex? Sol: Yes. In both the icosahedron and the dodecahedron, the same number of faces meet at every vertex. That is why they are called regular solids.
You can also build some 3-D shapes using straws or ice-cream sticks and clay or play-dough. Which shapes did you make? Sol: I made a cube, a triangular prism, and a pyramid using straws and clay.
This chapter will help you learn and understand multiplication in many different ways. There are stories, examples, and many ways to multiply numbers which will make your calculations easier and faster. You will see how multiplication helps in our daily life, especially on a dairy farm!
Multiplication is a method of adding the same number over and over again. It helps us find out how much we have in total when we have many groups of the same size.
What is Multiplication?
Suppose you have 3 bags, and each bag has 4 apples. If you want to know how many apples you have in total, you can add:
4 + 4 + 4 = 12
But instead of adding many times, we can use multiplication:
3 bags × 4 apples in each bag = 12 apples
So, multiplication makes our work much quicker and easier!
The Parts of Multiplication
The number of groups: 3 bags
The number in each group: 4 apples
The answer (total): 12 apples
We write it as:
3 × 4 = 12
‘×’ means ‘times’ or ‘groups of’
Order of Numbers in Multiplication
Daljeet Kaur arranges butter packets different ways:
What pattern do you notice?
3 × 2 = 6
2 × 3 = 6
Pattern: Changing the order of numbers does not change the answer.
Think 20 × 5 = 100, then subtract 2 × 5 =10, so 100-10=90.
Therefore, All ways are correct! Multiplication can be done in many ways to make calculation easier.
Doubling and Halving
Butter packets are arranged in the following ways. Let us find some strategies to calculate the total number of packets.
Key Trick:
Halving one number and doubling the other does not change the product.
This method is extremely useful when multiplying with numbers like 4, 8, 5, 25, or 50.
Nearest Multiple
When a number is close to a multiple that is easy to multiply with, use that:
Example:
Here, 20 is close to 19, and multiplying with 20 is alot easier, so we multiply 4 x 20 instead of 4 x 19, because multiplying with 19 is very hard. Now, see the figure, when we multiply 4 with 20, we actually added 1 more column of 4 squares. So, we get extra 4 squares, we need to subtract these squares after multiplying 4 x 20 to get the result of 4 x 19.
This method is useful if one of the numbers is close to a multiple of 10, like 19, 99, etc.
Waste and Composting
A family makes 35kg waste in one month. How much waste will the family produce in a year?
Quantity of kitchen waste in 1 month is 35 kg.
Quantity of kitchen waste in 12 months is 12 × 35 kg
Lets calculate this number
Different Ways to Multiply:
1. Nida’s solution
Lets understand what Nida did:
Nida split the numbers in 2 parts: 35 = 30 + 5, 12 = 10 + 2 and then she multiplied these numbers.
This method splits the numbers into parts, multiplies each pair, and then adds everything together. It’s based on the distributive property:
Lets see how it works
(10+2)×(30+5)=(10×30)+(10×5)+(2×30)+(2×5)
= (10×30)+(10×5)+(2×30)+(2×5)
=300+50+60+10=420
2. Kanti’s solution
Lets understand what Kanti did:
10. Multiplication Using Place Value
When we multiply large numbers, we break them into parts using their place value.
Example:
574 × 125
574 × 100
574 × 20
574 × 5 Add up the answers.
This is called the expanded form method, or Mili’s father’s method.
11. Practice Problems
You get to practice multiplying using different methods:
Using breaking into parts (expanded form method)
Using doubling and halving
Using nearest multiple
Changing the order of numbers
Example Problems:
Multiply: 78 × 4, 83 × 9, 86 × 3, 94 × 5, etc.
Solve word problems based on multiplication
12. Dairy Cooperative ExamplesHow Many Cows?
If 1 villager has 4 cows, and there are 268 villagers: Total cows = 268 × 4 = 1,072
How Much Milk?
453 Gir cows, each cows gives 13 litres: Total milk = 453 × 13 = 5,889 litres
How Much Ghee Sold?
1kg ghee = ₹574
125kg produced = 574 × 125 = ₹71,750
13. Patterns in Multiplication
Multiplying by 10s and 100s adds 0s or two 0s to the product.
Multiplying in any order gives the same product (commutative).
Multiplying in parts, you can break any big number to smaller parts, multiply, and add.
14. The King’s Reward Story
3 ministers get different numbers of coins with different rules.
You must multiply and see which gives the most coins.
(Double, triple, or five times every day for 7 days)
Moral:
Multiplication grows very fast when we multiply every day!
15. Multiplication Patterns and Tricks
Look at how numbers and products change in patterns.
E.g., 11 × 11 = 121, 111 × 111 = 12,321
You can predict the next numbers using observed patterns.
16. Practice Section
Match multiplication problems to their answers
Find different ways to solve a multiplication
Some numbers are easier to multiply if broken down cleverly
In this chapter, we will learn about measuring how far or how near things are. We will also learn about different units we use to measure length and how they are related to each other.Let Us Find
Identify the appropriate units for measuring each of the following.
We use different units to measure different lengths. For example, we use metre (m) or centimetre (cm).
Height of India Gate: This is a big structure, so we use metre (m).
Length of a handkerchief: This is small, so we use centimetre (cm). cm)
Depth of a well: This can be deep, so we use metre (m).
Length of a mobile phone: This is small, so we use centimetre (cm). (
Length of an elephant’s trunk: This is quite long, so we use metre (m).
Distance between two buttons on a shirt: This is very small, so we use centimetre (cm).
Different Units but Same Measure
Sometimes, the same length can be written using different units. Let’s see how Shikha and Sonu are measuring saris and stoles.
We can use a double number line to understand this better. Remember, 1 metre (m) = 100 centimetres (cm).
Let’s match the measures:
Let Us Compare
Let’s compare different lengths using the signs < (less than), > (greater than), or = (equal to).
Remember: 1 m = 100 cm
Comparing lengths of rods:
(a) 456 cm ____ 5 m Convert 5 m to cm: 5 m = 5 x 100 cm = 500 cm. So, 456 cm < 500 cm.
(b) 55 cm 200 cm ____ 200 cm 54 cm Left side: 55 cm + 200 cm = 255 cm. Right side: 200 cm + 54 cm = 254 cm. So, 255 cm > 254 cm.
(c) 6 m 5 cm ___ 6 m 50 cm Here, metres are the same (6 m). Compare the centimetres: 5 cm and 50 cm. So, 6 m 5 cm < 6 m 50 cm.
(d) 2 m 150 cm ___ 3 m 50 cm Convert 150 cm to metres: 150 cm = 1 m 50 cm. So, 2 m 150 cm = 2 m + 1 m 50 cm = 3 m 50 cm. So, 3 m 50 cm = 3 m 50 cm.
(e) 238 cm ____ 138 cm 1 m Convert 1 m to cm: 1 m = 100 cm. Right side: 138 cm + 100 cm = 238 cm. So, 238 cm = 238 cm.
World’s tallest statue comparison:
Statue of Unity, India: 182 m
Spring Temple Buddha, China: 128 m
Statue of Guanyin of Nanshan, China: 108 m
Statue of Liberty, USA: 93 m
The Motherland Calls, Russia: 91 m
Christ the Redeemer, Brazil: 38 m
Measuring Long Distances
For very long distances, we use special tools and units. We often use kilometre (km) for long distances.
Here are some ways to measure long distances:
• Long Tape: A very long measuring tape.
• Rangefinder: A device that measures distance to a target.
• Laser Distance Meter: Uses a laser to measure distance.
• Rodometer (measuring wheel): A wheel that measures distance as it rolls.
Try yourself:
What unit do we often use for long distances?
A.Meter
B.Centimetre
C.Mile
D.Kilometre
View SolutionKilometre Race
Sheena and Jennifer are organising a 3-km race. Let’s help them!
Remember: 3 km = 3 x 1000 m = 3000 m
1. Water stations are to be arranged after every 500 m. How many water stations must be set up? At what positions from the starting point will these water stations be placed?
Total distance = 3000 m.
Water stations every 500 m.
Number of stations = 3000 m ÷ 500 m = 6 stations.
Positions: 500 m, 1000 m, 1500 m, 2000 m, 2500 m, 3000 m (finish line).
2. Children need to stand at an interval of 300 mto direct the runners. How many children are needed? At what positions from the starting point will the children be standing?
Total distance = 3000 m.
Children every 300 m.
Number of children = 3000 m ÷ 300 m = 10 children.
Positions: 300 m, 600 m, 900 m, 1200 m, 1500 m, 1800 m, 2100 m, 2400 m, 2700 m, 3000 m.
3. Red and blue flags are to be placed alternately at every 50 m. How many red and blue flags are needed till the finish line?
Total distance = 3000 m.
Flags every 50 m.
Number of flag positions = 3000 m ÷ 50 m = 60 positions.
Since flags are placed alternately (red, blue, red, blue…), half will be red and half will be blue.
Number of red flags = 60 ÷ 2 = 30 red flags.
Number of blue flags = 60 ÷ 2 = 30 blue flags.
The World of Small Things
How do we measure very small things like a sprout, small screws, or a nail?
Small Things
We use a smaller unit called millimetre (mm).
Look at your scale. You will see small marks between 1 cm and 2 cm. If you count them, there are ten small marks.
Each of these small parts is called 1 millimetre (mm).
So, 1 centimetre (cm) = 10 millimetres (mm).
These small marks help us measure very small lengths accurately.
Relationships between Different Units
Let’s summarise the relationships between different units of length:
• 10 mm = 1 cm (Millimetre to Centimetre)
• 100 cm = 1 m (Centimetre to Metre)
• 1,000 m = 1 km (Metre to Kilometre)
Try yourself:
How many millimeters are in one centimeter?
A.15 mm
B.20 mm
C.10 mm
D.5 mm
View SolutionAdding and Subtracting Lengths
We can add and subtract lengths just like we add and subtract numbers. We need to be careful with units.
Example 1: Saji’s walk
Saji walked 3 km 450 m in the morning and 4 km 650 m in the evening. How much did he walk in total?
Method 1: Adding km and m separately
Add kilometres: 3 km + 4 km = 7 km.
Add metres: 450 m + 650 m = 1100 m.
Convert 1100 m to km and m: 1100 m = 1000 m + 100 m = 1 km100 m.
Add the converted metres to kilometres: 7 km + 1 km 100 m = 8 km 100 m.
Method 2: Convert everything to metres and then add
3 km 450 m = 3 x 1000 m + 450 m = 3000 m + 450 m = 3450 m.
4 km650 m = 4 x 1000 m + 650 m = 4000 m + 650 m = 4650 m.
Total distance = 3450 m + 4650 m = 8100 m.
Convert 8100 m to km and m: 8100 m = 8000 m + 100 m = 8 km 100 m.
Example 2: Cable left
Electricians need 63 m of cable. They used 16 m 75 cm in the first room. What is the length of cable left?
Method 1: Convert everything to cm and then subtract
Total cable needed: 63 m = 63 x 100 cm = 6300 cm.
Cable used: 16 m 75 cm = 16 x 100 cm + 75 cm = 1600 cm + 75 cm = 1675 cm.
Cable left = 6300 cm – 1675 cm = 4625 cm.
Convert 4625 cm to m and cm: 4625 cm = 4600 cm + 25 cm = 46 m 25 cm.
Method 2: Subtracting m and cm separately (with borrowing)
Multiplying and Dividing Lengths
We can also multiply and divide lengths.
Example 1: Cloth for shirts
We need 1 m 80 cm cloth for one shirt. How much cloth is needed for 20 children?
Method 1: Multiply m and cm separately
Multiply metres: 20 x 1 m = 20 m.
Multiply centimetres: 20 x 80 cm = 1600 cm.
Convert 1600 cm to metres: 1600 cm = 16 x 100 cm = 16 m.
Add the metres: 20 m + 16 m = 36 m.
Method 2: Convert everything to cm and then multiply
1 m 80 cm = 100 cm + 80 cm = 180 cm.
Total cloth = 20 x 180 cm = 3600 cm.
Convert 3600 cm to metres: 3600 cm = 36 x 100 cm = 36 m.
Example 2: Cost of cloth
A shop sells cloth for ₹100 for 5 m. How much money is needed to buy 1 m cloth?
Cost of 5 m cloth = ₹100.
Cost of 1 m cloth = ₹100 ÷ 5 = ₹20.
Example 3: Embroidery thread
Anita needs a 1 m long thread to embroider a 50 cm sari. How much thread would she need for a 5 m sari border?
For 50 cm sari, thread needed = 1 m.
5 m sari = 500 cm sari.
Number of 50 cm sections in 500 cm = 500 cm ÷ 50 cm = 10 sections.
Thread needed = 10 x 1 m = 10 m.
If a 1 m long thread costs ₹50, how much money will be needed to buy 10 m thread?
Cost of 1 m thread = ₹50.
Cost of 10 m thread = 10 x ₹50 = ₹500.
Example 4: Road laying work
A road 12 km 600 m long is being laid. Workers complete the work in 6 days, laying an equal length each day. How much road-laying work is done each day?
Total length of road = 12 km 600 m.
Number of days = 6.
Convert total length to metres: 12 km 600 m = 12 x 1000 m + 600 m = 12000 m + 600 m = 12600 m.
Road laid each day = 12600 m ÷ 6 = 2100 m.
Convert 2100 m to km and m: 2100 m = 2000 m + 100 m = 2 km 100 m.
Try yourself:
How much cloth is needed for 20 children if one shirt requires 1 m 80 cm?
A.20 m
B.36 m
C.10 m
D.30 m
View SolutionLet Us Estimate
Estimating means making a good guess about a measurement without actually measuring it. We use things we already know as a reference.
Height of a single-storeyed house: You can estimate it to be around 3-4 metres.
Height of an electric pole: You can estimate it to be around 8-10 metres.
The height of the tallest building in your neighbourhood. What did you use as a reference to estimate the height?
Example: If you know a 2-storey house is about 6 metres tall, you can use that to guess the height of a taller building.
The height of the tallest tree in your neighbourhood. What did you use as a reference?
Example: You can compare it to the height of a building or a known object.
The depth of a well or swimming pool in your neighbourhood. How did you find out?
Example: You can drop a stone and listen for the splash, or compare it to your own height.
Let Us Explore (Feet and Inches)
In daily life, we also use other units for measuring length, especially for height. These are feet (ft) and inches (in).
Look at your ruler. Some rulers also have inches marked on them.
Namaste, little travellers! In this chapter, we are going to learn some super interesting things about numbers and how they help us in our daily lives, especially when we are travelling! We will explore how to make sums equal, understand how fuel works for vehicles, and even learn about big numbers and how to add and subtract them. So, get ready for an exciting journey with numbers!
Making Sums Equal
You have two groups of numbers. The two groups have numbers in them, and each group has a total. But the totals are not the same.
Your goal: Swap some numbers between the groups so that both groups add up to the same total.
You should try to do this by moving as few numbers as possible!
Let’s look at the groups one by one:
(a)
Group 1: 1, 2, 7, 9
Group 2: 3, 4, 5, 9
(b)
Group 1: 5, 7, 12, 15
Group 2: 9, 11, 13, 14
(c)
Group 1: 11, 15, 19, 23
Group 2: 13, 17, 21, 25
(d)
Group 1: 77, 78, 79, 80
Group 2: 81, 82, 83, 84
2. Add Up Each Group
Use your fingers or a calculator if needed!
(a)
Group 1: 1 + 2 + 7 + 9 = 19
Group 2: 3 + 4 + 5 + 9 = 21
Difference is 2.
(b)
Group 1: 5 + 7 + 12 + 15 = 39
Group 2: 9 + 11 + 13 + 14 = 47
Difference is 8.
(c)
Group 1: 11 + 15 + 19 + 23 = 68
Group 2: 13 + 17 + 21 + 25 = 76
Difference is 8.
(d)
Group 1: 77 + 78 + 79 + 80 = 314
Group 2: 81 + 82 + 83 + 84 = 330
Difference is 16.
3. Swap Numbers to Make Sums
Your goal is to swap numbers between the groups so both add up to the same.
How to Swap:
The best swap is when the difference between the two numbers you trade is half the difference between the group sums!
Example: If groups differ by 8, try to swap numbers whose difference is 4.
Let’s Try: (a) Difference is 2. Swap numbers with a difference of 1.
Swap 2 (Group 1) and 3 (Group 2):
New Group 1: 1, 3, 7, 9 = 20
New Group 2: 2, 4, 5, 9 = 20
Now both groups have 20!
(b) Difference is 8. Swap numbers with a difference of 4.
Swap 7 (Group 1) and 11 (Group 2):
New Group 1: 5, 11, 12, 15 = 43
New Group 2: 7, 9, 13, 14 = 43
Now both groups have 43!
(c) Difference is 8. Swap numbers with a difference of 4.
Swap 15 (Group 1) and 17 (Group 2):
New Group 1: 11, 17, 19, 23 = 70
New Group 2: 13, 15, 21, 25 = 70
Now both groups have 70!
(d) Difference is 16. Swap numbers with a difference of 8.
Swap 80 (Group 1) and 84 (Group 2):
New Group 1: 77, 78, 79, 84 = 318
New Group 2: 81, 82, 83, 80 = 326
Still not equal, try another pair: swap 78 (Group 1) and 86 (not present). Since numbers go up by 1, try swapping 79 (Group 1) and 87 (not present). Instead, swap two pairs:
Swap 77, 78 (Group 1) with 83, 84 (Group 2)
New Group 1: 79, 80, 83, 84 = 326
New Group 2: 77, 78, 81, 82 = 318
Now, both are 326 and 318. This puzzle needs two swaps, or adjust till both sums match.
The main idea here is to understand that when you swapnumbers, the sums of the groups change. You need to find the right swap to make them equal! This is like a fun puzzle where you use addition and a little bit of trial and error.
Fuel Arithmetic
Have you ever wondered how cars, bikes, and buses run? They need fuel! Just like we need food to get energy, vehicles need fuel to move. Different vehicles need different amounts of fuel.
Motorbikes: Usually need a small amount, like 5 to 15 litres.
Cars: Need a bit more, around 15 to 50 litres.
Lorries and Trucks: These are big vehicles, so they need a lot more fuel, from 150 to 500 litres.
Trains: Imagine how big a train is! They need a huge amount, like 5,000 litres!
Remember: Fuel is a limited resource, which means it will not last forever. So, we must save fuel! Saving fuel also helps to keep our air clean and reduce pollution. That’s why electric vehicles are becoming popular, as they don’t use natural fuel and help keep our environment healthy.
Let’s solve some problems related to fuel:
A lorry has 28 litres of fuel in its tank. An additional 75 litres is filled. What is the total quantity of fuel in the lorry? To find the total quantity, we need to add the fuel already in the tank and the additional fuel filled.Total quantity of fuel = 28 litres + 75 litresDo you remember how to add two numbers using place value? Let’s do it step-by-step:Explanation:
Step 1: Add the Ones place. 8 Ones + 5 Ones = 13 Ones. We know that 10 Ones make 1 Ten. So, 13 Ones is 1 Ten and 3 Ones. Write down 3 in the Ones place and carry over 1 (Ten) to the Tens place.
Step 2: Add the Tens place. 1 (carried over) + 2 Tens + 7 Tens = 10 Tens. We know that 10 Tens make 1 Hundred. So, 10 Tens is 1 Hundred and 0 Tens. Write down 0 in the Tens place and carry over 1 (Hundred) to the Hundreds place.
Step 3: Add the Hundreds place. 1 (carried over) + 0 Hundreds = 1 Hundred. Write down 1 in the Hundreds place.
So, the total quantity of fuel in the lorry is 103 litres.
Try yourself:
What do vehicles need to move?
A.Food
B.Fuel
C.Water
D.Air
View Solution
Relationship Between Addition and Subtraction
Addition and subtraction are like best friends! They are closely related. If you know an addition fact, you can easily find two subtraction facts from it. And if you know a subtraction fact, you can find an addition fact!
Let’s look at some examples to understand this better:
1. Find the relationship between the numbers in the given statements and fill in the blanks appropriately.
(a) If 46 + 21 = 67, then,
67 – 21 = 46
67 – 46 = 21
Think: If you add 46 and 21 to get 67, then if you take away 21 from 67, you will get 46. And if you take away 46 from 67, you will get 21. It’s like undoing the addition!
(b) If 198 – 98 = 100, then,
100 + 98 = 198
198 – 100 = 98
Think
: If you subtract 98 from 198 to get 100, then if you add 100 and 98, you will get 198. Also, if you take away 100 from 198, you will get 98.
(c) If 189 + 98 = 287, then,
287 – 98 = 189
287 – 189 = 98
(d) If 872 – 672 = 200, then,
200 + 672 = 872
872 – 200 = 672
2. In each of the following, write the subtraction and addition sentences that follow from the given sentence.
(a) If 78 + 164 = 242, then,
242 – 78 = 164
242 – 164 = 78
(b) If 462 + 839 = 1301, then,
1301 – 462 = 839
1301 – 839 = 462
(c) If 921 – 137 = 784, then,
784 + 137 = 921
921 – 784 = 137
(d) If 824 – 234 = 590, then,
590 + 234 = 824
824 – 590 = 234
More Fuel ArithmeticFuel Problems with Subtraction
Let’s go back to our fuel problems, but this time, we will use subtraction!
A minibus has 18 litres of fuel left. After refuelling, the fuel meter indicates 65 litres. How much fuel has been filled in the fuel tank of the minibus?
To find out how much fuel was filled, we need to subtract the fuel that was already there from the total fuel after refuelling.
Quantity of fuel filled = 65 litres – 18 litres
Do you remember how to subtract numbers using place value? Let’s do it step-by-step:
Explanation:
Step 1: Subtract the Ones place. We cannot subtract 8 from 5. So, we need to regroup from the Tens place. Borrow 1 Ten (which is 10 Ones) from the 6 Tens. Now, we have 5 Tens left in the Tens place, and 5 10 = 15 Ones in the Ones place. Now, 15 Ones – 8 Ones = 7 Ones. Write 7 in the Ones place.
Step 2: Subtract the Tens place. Now we have 5 Tens (because we borrowed 1 Ten) – 1 Ten = 4 Tens. Write 4 in the Tens place.
So, 47 litres of fuel has been filled in the fuel tank of the minibus.
Check your answer: Is 18 + 47 = 65? Yes, it is! So our answer is correct.
Sums of Consecutive Numbers
What are consecutive numbers?
They are numbers that follow each other in order, one after another, without skipping any number. It’s like counting! For example:
1, 2, 3, 4, 5 are consecutive numbers.
29, 30, 31, 32 are consecutive numbers.
512, 513 are consecutive numbers.
2023, 2024, 2025 are consecutive numbers.
Now, let’s look at adding these consecutive numbers and see what interesting patterns we find!
In each of the boxes above, state whether the sums are even or odd. Explain why this is happening.
Sum of 2 consecutive numbers: The sums are 3, 5, 7, 9. All these are odd numbers.
Why? When you add an odd number and an even number, the result is always odd. In any pair of consecutive numbers, one will be odd and the other will be even (e.g., 1 and 2, 2 and 3, 3 and 4). So, the sum is always odd.
Sum of 3 consecutive numbers: The sums are 6, 9, 12, 15. These sums are sometimes even and sometimes odd.
Why? If you have three consecutive numbers, you will either have:
Odd + Even + Odd = Even (e.g., 123 = 6, 345 = 12)
Even + Odd + Even = Odd (e.g., 234 = 9, 456 = 15)
Sum of 4 consecutive numbers: The sums are 10, 14, 18, 22. All these are even numbers.
Why? When you add four consecutive numbers, you will always have two odd numbers and two even numbers. The sum of two odd numbers is always even, and the sum of two even numbers is always even. So, even + even = even. That’s why the sum of four consecutive numbers is always even.
What is the difference between two successive sums in each box? Is it the same throughout?
Sum of 2 consecutive numbers:
5 – 3 = 2
7 – 5 = 2
9 – 7 = 2
Yes, the difference is 2 and it is the same throughout.
Sum of 3 consecutive numbers:
9 – 6 = 3
12 – 9 = 3
15 – 12 = 3
Yes, the difference is 3 and it is the same throughout.
Sum of 4 consecutive numbers:
14 – 10 = 4
18 – 14 = 4
22 – 18 = 4
Yes, the difference is 4 and it is the same throughout.
What will be the difference between two successive sums for —
(a) 5 consecutive numbers: Based on the pattern we saw (2 for 2 numbers, 3 for 3 numbers, 4 for 4 numbers), the difference between two successive sums for 5 consecutive numbers will be 5.
(b) 6 consecutive numbers: Following the same pattern, the difference between two successive sums for 6 consecutive numbers will be 6.
For 3 consecutive numbers:
1 + 2 + 3 = 6 (Here, the middle number is 2, and 2 x 3 = 6)
2 + 3 + 4 = 9 (Here, the middle number is 3, and 3 x 3 = 9)
3 + 4 + 5 = 12 (Here, the middle number is 4, and 4 x 3 = 12)
Pattern: For 3 consecutive numbers, the sum is the middle number multiplied by 3.
For 4 consecutive numbers:
1 + 2 + 3 + 4 = 10 (The numbers are 1, 2, 3, 4. The sum of the first and last number is 1 + 4 = 5. The sum of the two middle numbers is 2 + 3 = 5. And 5 + 5 = 10. Also, notice that the sum is 2 times the sum of the middle two numbers divided by 2, or (first + last) * (number of terms / 2) = (1 + 4) * (4/2) = 5 * 2 = 10. Or, (first + last) * 2 = 10. Another way to think is that the average of the two middle numbers is 2.5, and 2.5 * 4 = 10.)
Pattern: For 4 consecutive numbers, the sum is 2 times the sum of the first and last number (or 2 times the sum of the two middle numbers).
For 5 consecutive numbers:
1 + 2 + 3 + 4 + 5 = 15 (Middle number is 3, and 3 x 5 = 15)
2 + 3 + 4 + 5 + 6 = 20 (Middle number is 4, and 4 x 5 = 20)
3 + 4 + 5 + 6 + 7 = 25 (Middle number is 5, and 5 x 5 = 25)
Pattern: For 5 consecutive numbers, the sum is the middle number multiplied by 5.
Now, use your understanding to find the following sums without adding the numbers directly:
(a) 67 + 68 + 69: This is a sum of 3 consecutive numbers. The middle number is 68. Sum = 68 x 3 = 204
(b) 24 + 25 + 26 + 27: This is a sum of 4 consecutive numbers. The sum of the first and last number is 24 + 27 = 51. Sum = 51 x 2 = 102
(c) 48 + 49 + 50 + 51 + 52: This is a sum of 5 consecutive numbers. The middle number is 50. Sum = 50 x 5 = 250
(d) 237 + 238 + 239 + 240 + 241 + 242: This is a sum of 6 consecutive numbers. For 6 consecutive numbers, you can add the first and last number, and multiply by 3 (because there are 3 pairs). First + Last = 237 + 242 = 479. Sum = 479 x 3 = 1437
Quick Tip:
For an odd number of consecutive numbers, the sum is the middle number multiplied by the count of numbers. For an even number of consecutive numbers, the sum is the (first number + last number) multiplied by (count of numbers / 2).
Try yourself:What do sums of consecutive numbers refer to?A.Dividing numbers by their totalB.Subtracting numbers from each otherC.Multiplying numbers togetherD.Adding numbers in a sequenceView Solution
The Longest Land Route — Adding Large Numbers
Imagine travelling really, really far! The longest road trip you can take on Earth is between a place called Talon in Russia and Sagres in Portugal. This road is super long, about 15,150 kilometres!
Now, let’s talk about India. In 2019, we had a very long road called the North–South Corridor. It started from Srinagar in Jammu and Kashmir and went all the way down to Kanniyakumari in Tamil Nadu. How long was it?
Your textbook tells us about a place on this corridor that was 1,855 km from Srinagar and 1,862 km from Kanniyakumari.
To find the total length of the North–South Corridor in 2019, we need to add these two distances:
Total length = 1,855 km + 1,862 km
Do you remember how to add large numbers? It’s just like adding smaller numbers, but with more digits! We use place values like Thousands, Hundreds, Tens, and Ones.
Explanation:
Step 1: Add the Ones place. 5 Ones + 2 Ones = 7 Ones. Write 7 in the Ones place.
Step 2: Add the Tens place. 5 Tens + 6 Tens = 11 Tens. We regroup 10 Tens as 1 Hundred. So, we have 1 Ten left. Write 1 in the Tens place and carry over 1 to the Hundreds place.
Step 3: Add the Hundreds place. 1 (carried over) + 8 Hundreds + 8 Hundreds = 17 Hundreds. We regroup 10 Hundreds as 1 Thousand. So, we have 7 Hundreds left. Write 7 in the Hundreds place and carry over 1 to the Thousands place.
Step 4: Add the Thousands place. 1 (carried over) + 1 Thousand + 1 Thousand = 3 Thousands. Write 3 in the Thousands place.
So, the total length of the North–South Corridor was 3,717 km in 2019.
Now, let us try finding the sum of 5-digit numbers.
Mahesh and his family decided to drive from Srinagar to Kanniyakumari. This is a very long journey! He spent ₹21,880 on fuel and toll tax and ₹38,900 on other expenses during this trip. How much did he spend in total?
To find the total money spent, we need to add these two amounts:
Total spent = ₹21,880 + ₹38,900
Adding larger numbers is the same as adding smaller numbers. Just make sure to align the digits properly: Ones below Ones, Tens below Tens, and so on.
Explanation:
Step 1: Add the Ones place. 0 + 0 = 0. Write 0.
Step 2: Add the Tens place. 8 + 0 = 8. Write 8.
Step 3: Add the Hundreds place. 8 + 9 = 17. Write 7, carry over 1.
Step 4: Add the Thousands place. 1 (carried over) + 1 + 8 = 10. Write 0, carry over 1.
Step 5: Add the Ten Thousands place. 1 (carried over) + 2 + 3 = 6. Write 6.
So, Mahesh spent a total of ₹60,780 on his journey.
Important Tip: When adding large numbers, if you keep the digits aligned (Ones below Ones, Tens below Tens, etc.), you don’t always need to write the place value labels (TTh, Th, H, T, O) at the top. You can mentally keep track of the positions of the digits as you add.
Just like we add large numbers, we can also subtract them! It’s very useful in real life, especially when you want to find out how much is left or the difference between two big numbers.
Important Places for Travel
A bus stand or bus station is where passengers get on a bus.
A railway station is where people get on trains.
A port is where people get on ships.
Your textbook talks about the ports of Mumbai and Chennai, which are very important in India. Ships travelling from Mumbai to Chennai pass by another important port called Cochin Port. You can spot these places on the map of India!
Distance Calculation
The total distance of the sea route from Mumbai to Chennai is 2,700 km. A ship starting from Mumbai first reaches the Cochin port, travelling 1,083 km by sea. How much more distance does it have to travel to reach the Chennai port?
To find the remaining distance, we need to subtract the distance already travelled from the total distance:
Remaining distance = 2,700 km – 1,083 km
Subtraction Steps Using Place Value
Let’s do it step-by-step:
Explanation
Step 1: Subtract the Ones place. We cannot subtract 3 from 0. We need to regroup. There are no Tens to borrow from, so we go to the Hundreds place. Borrow 1 Hundred (which is 10 Tens) from the 7 Hundreds. Now, we have 6 Hundreds left, and 10 Tens in the Tens place. Now, borrow 1 Ten (which is 10 Ones) from the 10 Tens. Now, we have 9 Tens left, and 0 + 10 = 10 Ones in the Ones place. Now, 10 Ones – 3 Ones = 7 Ones. Write 7 in the Ones place.
Step 2: Subtract the Tens place. Now we have 9 Tens (because we borrowed 1 Ten) – 8 Tens = 1 Ten. Write 1 in the Tens place.
Step 3: Subtract the Hundreds place. Now we have 6 Hundreds (because we borrowed 1 Hundred) – 0 Hundreds = 6 Hundreds. Write 6 in the Hundreds place.
Step 4: Subtract the Thousands place. 2 Thousands – 1 Thousand = 1 Thousand. Write 1 in the Thousands place.
So, the ship has to travel 1,617 km more to reach Chennai.
Check if the solution is correct: Add the answer to the number subtracted: 1,617 + 1,083 = 2,700. Yes, it is correct!
Longest Land Route
As you learned earlier, the longest land route is 15,150 km between Talon (Russia) and Sagres (Portugal). The longest highway in Africa is 10,228 km long, connecting the cities of Cairo, in Egypt and Cape Town, in South Africa.
How much longer is the land route between Talon and Sagres compared to the highway between Cairo and Cape Town?
To find out how much longer one route is than the other, we need to find the difference between their lengths. This means we subtract the smaller length from the larger length.
Difference = 15,150 km – 10,228 km
Subtraction Steps for Route Lengths
Explanation
Step 1: Subtract the Ones place. We cannot subtract 8 from 0. Regroup from the Tens place. 5 Tens becomes 4 Tens, and 0 Ones becomes 10 Ones. 10 Ones – 8 Ones = 2 Ones. Write 2.
Step 5: Subtract the Ten Thousands place. 1 Ten Thousand – 1 Ten Thousand = 0 Ten Thousands. Write 0 (or leave it blank if it’s the first digit).
So, the land route connecting Talon and Sagres is 4,922 km longer than the road connecting Cairo and Cape Town.
Check if the answer is correct: Add the answer to the number subtracted: 4,922 + 10,228 = 15,150. Yes, it is correct!
Try yourself:
What is the first step in subtracting large numbers?
A.Multiply the numbers
B.Line up the digits
C.Ignore the smallest number
D.Add the numbers
View Solution
Mental Subtraction
Like addition, here too we can try not to write the positions of the digits and align the numbers appropriately. You can mentally keep track of the position of the digits.
For example:
Here, you mentally subtract 4 from 3 (regroup), then 5 from 0 (regroup), then 1 from 5.
Quick Sums and Differences
Sometimes, we need to find missing numbers in sums or differences quickly. Let’s look at some fun ways to do this!
Sukanta’s Challenge
Sukanta likes the numbers 10, 100, 1,000, and 10,000. He wants to figure out what number he should add to a given number such that the sum is 100 or 1,000. Let’s help him fill in the blanks!
32 _______ = 100
To find the missing number, you can think: “How much more do I need to add to 32 to reach 100?” This is a subtraction problem: 100 – 32 = 68.
Piku’s Method
Sukanta’s friend Piku shows him an interesting way to solve these problems:
Then, add 1 to the sum:
This method is a bit tricky. Let’s understand it better. Piku is trying to make the number easier to work with by getting it close to 100. If you add 67 to 32, you get 99. Then, adding 1 more makes it 100. So the missing number = 68.
Trying Piku’s Method
Do you think this method will always work? Let’s try this method for the number 59:
59 _______ = 100
Using Piku’s method:
Then, add 1 to the sum:
So, the missing number is = 100
Piku’s Method Explained
This method works by finding the difference to the next multiple of 10, then to 100. For example, for 32 to reach 100: 32 + 8 = 40, then 40 + 60 = 100. So 8 +60 = 68.
Piku’s method is a variation of this, where you try to reach 99 first, and then add 1.
Units Digit is 0
Will this method work if the units digit is 0? What do you think?
Let’s try with numbers ending in 0:
(a) 180 ________ = 1,000 If we use Piku’s method, we would try to reach 999. But 180 already ends in 0. It’s easier to just subtract: 1,000 – 180 = 820. So, 180 + 820 = 1,000.
The simplest method is always to subtract the given number from the target sum. For example, to find the missing number in A ? = B, you just calculate B – A = ?.
Namita’s Challenge
Namita likes the number 9. She wants to subtract 9 or 99 from any number. Let’s find a way to quickly subtract 9 or 99 from any number.
Now, use the above solutions to find answers to the following problems. Do not calculate again.
Namita wonders if she can get 9 or 99 as the answer to any subtraction problem. Find a way to get the desired answer.
(a) 32 – ________ = 9 We know that 32 – 23 = 9. So the missing number is 23. (Think: 32 – 9 = 23)
(b) 56 – ________ = 9 We know that 56 – 47 = 9. So the missing number is 47. (Think: 56 – 9 = 47)
(c) 877 – ________ = 99 We know that 877 – 778 = 99. So the missing number is 778. (Think: 877 – 99 = 778)
(d) 666 – ________ = 99 We know that 666 – 567 = 99. So the missing number is 567. (Think: 666 – 99 = 567)
Even and Odd Numbers
Even Numbers:
These are numbers that can be divided into two equal groups without anything left over. They always end in 0, 2, 4, 6, or 8.
Examples: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20…
Odd Numbers:
These are numbers that cannot be divided into two equal groups. When you try to divide them by 2, there’s always 1 left over. They always end in 1, 3, 5, 7, or 9.
Examples: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19…
1. Circle the numbers that are even.
Let’s go through the list and identify the even numbers:
(a) 297 – Ends in 7, so it’s Odd.
(b) 498 – Ends in 8, so it’s Even.
(c) 724 – Ends in 4, so it’s Even.
(d) 100 – Ends in 0, so it’s Even.
(e) 199 – Ends in 9, so it’s Odd.
(f) 789 – Ends in 9, so it’s Odd.
(g) 49 – Ends in 9, so it’s Odd.
(h) 6,893 – Ends in 3, so it’s Odd.
(i) 846 – Ends in 6, so it’s Even.
(j) 111 – Ends in 1, so it’s Odd.
(k) 222 – Ends in 2, so it’s Even.
(l) 1,023 – Ends in 3, so it’s Odd.
So, the even numbers are: 498, 724, 100, 846, 222.
2. Observe the given arrangement.
This figure shows a paired arrangement for numbers. This means arranging items in pairs to see if any are left over.
Paired arrangement for 18: If you arrange 18 items in pairs, you will have 9 pairs, and nothing will be left over. This shows 18 is an even number.
Paired arrangement for 23: If you arrange 23 items in pairs, you will have 11 pairs, and 1 item will be left over. This shows 23 is an odd number.
Add 2 to 18. What changes or does not change in the arrangement?
If you add 2 to 18, you get 20. When you arrange 20 items in pairs, you will still have no items left over. So, adding 2 to an even number keeps it an even number. The arrangement will still be perfectly paired.
Add 2 to 23. What changes or does not change in the arrangement?
If you add 2 to 23, you get 25. When you arrange 25 items in pairs, you will still have 1 item left over. So, adding 2 to an odd number keeps it an odd number. The arrangement will still have one item left over.
What do you notice about the sums in each of the following cases?
Do you think it will be true for all pairs of such numbers? Explain your observations. You may use the paired arrangement to explain your thinking.
(a) 12 and 6 are a pair of even numbers.
Choose 5 such pairs of even numbers. Add the numbers in each of the pairs.
12 + 6 = 18 (Even)
4 + 8 = 12 (Even)
10 + 2 = 12 (Even)
20 + 14 = 34 (Even)
30 + 10 = 40 (Even)
Observation: When you add two even numbers, the sum is always an even number.
Explanation: An even number means there are no items left over when paired. If you combine two groups that both have no items left over, the combined group will also have no items left over. So, Even + Even = Even.
(b) 13 and 9 are a pair of odd numbers.
Choose 5 such pairs of odd numbers. Add the numbers in each of the pairs.
13 + 9 = 22 (Even)
5 + 3 = 8 (Even)
7 + 1 = 8 (Even)
11 + 15 = 26 (Even)
21 + 23 = 44 (Even)
Observation: When you add two odd numbers, the sum is always an even number.
Explanation: An odd number means there is 1 item left over when paired. If you combine two groups that each have 1 item left over, those two leftover items can form a new pair. So, Odd + Odd = Even.
(c) 7 and 12 are a pair of odd and even numbers.
Choose 5 such pairs of odd and even numbers. Add the numbers in each of the pairs.
7 + 12 = 19 (Odd)
3 + 6 = 9 (Odd)
9 + 2 = 11 (Odd)
15 + 10 = 25 (Odd)
21 + 4 = 25 (Odd)
Observation: When you add an odd number and an even number, the sum is always an odd number.
Explanation: An odd number has 1 item left over when paired, and an even number has no items left over. If you combine these two groups, the 1 leftover item from the odd number will still be left over in the combined group. So, Odd + Even = Odd.
Have you ever noticed how things move around you? Like a fan spinning, or a door opening? All these movements involve ‘turns’, and these turns help us understand ‘angles’. Let’s explore this fun topic together!
What is a Turn?
Can you recognise the child in the picture who has made a full turn? Who has made a half turn? How do you know?
The child in the picture that has made a full turn is “C” and the child in the picture who has made a half turn is “A”
Lets see how we know this.
Turns
When you spin all the way around and come back to where you started, that’s a full turn. Think about the minute hand of a clock. When it goes all the way around from 12 and comes back to 12, it has made a full turn.
Example: A giant wheel makes a full turn when it comes back to its starting position.
If you spin halfway around, so you are facing the opposite direction, that’s a half turn.
Example: If Reema takes two half turns in the same direction, it’s like she has made a full turn.
If you spin just a little bit, like turning to face the side, that’s a quarter turn. A quarter turn is one-fourth of a full turn.
Example: If Reema takes 2 quarter turns in the same direction, it’s like she has made a half turn. If she takes 4 quarter turns in the same direction, it’s like she has made a full turn.
Ashutosh and Sahana are making circles, each having one foot fixed and rotating at one spot on the ground.
Ashutosh made a complete circle by making a full turn. Sahana is making a half-moon shape with a half turn.
Everyday Objects and Turns
Taps: When you open or close a tap, you turn it.
Doors with hinges: Doors swing open and close, making turns.
Scissors: When you open and close scissors, they make turns.
Clothes clip: When you open a clothes clip, it makes a turn.
Think about the maximum turn these objects can make. Some can make a quarter turn, some a half turn, and some even a full turn!
Try yourself:
What is a full turn?
A.Turning halfway around
B.Spinning all the way around
C.Turning to the side
D.Making a half-moon shape
View Solution
Different Types of Angles
Angles are formed when two lines or objects meet at a point and turn. Let’s learn about different types of angles using the example of Pragya and her straws.
Pragya joined a green and yellow straw with paper clips. She holds the green straw steady and turns the yellow straw around. Observe different turns of the yellow straw:
Right Angle
When the yellow straw makes a quarter (1/4) of a full turn, it looks like a right angle. A right angle is like the corner of a square or a book. It looks like an ‘L’ shape.
Acute Angle
If the yellow straw makes less than a quarter turn, it looks like an acute angle. Acute angles are smaller than a right angle. Think of a sharp point, like the tip of a pencil.
Obtuse Angle
When the yellow straw makes more than a quarter turn but less than a half turn, it looks like an obtuse angle. Obtuse angles are bigger than a right angle but smaller than a straight angle.
Straight Angle
If the yellow straw makes two quarter turns (which is the same as a half turn), it forms a straight angle. A straight angle looks like a straight line.
Angles arise in situations that involve a turn.
Angle Measuring Tool
Measuring Turns and Angles
Wouldn’t it be great if we could measure turns and angles? Well, we can! Let’s make our own tool to measure turns.
(a) Making Your Tool
Take a tracing paper and cut out a circle from it.
Fold the circle carefully to make 8 equal parts. Imagine folding it in half, then in half again, and then in half one more time.
Attach a straw to the very center of the circle.
Mark a starting point on the edge of the circle.
Now you have your very own angle measuring tool!
(b) Using Your Tool
Let’s try using your new tool to understand turns:
Show a 1/8 turn: Move the straw from the starting point to the first fold line. That’s a 1/8 turn.
Show a 2/8 turn (which is 1/4 turn): Move the straw to the second fold line. This is a quarter turn, or a right angle!
Show a 3/8 turn: Move the straw to the third fold line.
Show a 4/8 turn (which is 1/2 turn): Move the straw to the fourth fold line. This is a half turn, or a straight angle!
Keep turning the straw by 5/8, 6/8, 7/8, and 8/8. When you reach 8/8, you have completed a full turn!
Making a Permanent Tool
You can cut out parts of your folded circle (like 1/8 part and 2/8 parts and paste them on a thicker paper or board. These can be used as handy angle measuring tools.
Try yourself:
What is an angle measuring tool used for?
A.Drawing
B.Cooking
C.Measuring angles
D.Writing
View Solution
Which Direction?
When things turn, they turn in a certain direction. There are two main directions of turning:
Clockwise Movement
Think about how the hands of a clock move. They move from 12 to 1, then to 2, and so on. This direction of movement is called clockwise movement.
Anti-clockwise Movement
The opposite of clockwise movement is called anti-clockwise movement. If you move your hand from 12 to 11, then to 10, that would be anti-clockwise.
The creatures below have made a quarter turn once. Tick the direction in which they have moved.
Fun with Turns
1. The children in a class are playing a game in which the teacher tells them the direction in which they should rotate. Complete the table by filling the direction the children will face on completing the given turns. The starting direction is given in the table.
2. Padma is facing the toy shop. What place will she face if she takes a half turn clockwise?
She will face the ice cream side
What other way can she turn to face the same place?
She can turn a half turn anti clockwise to face the same side.
Tamanna is a student of Grade 5. She has two chocolates of different sizes. She says that of chocolate 1 is bigger than of the chocolate 2.
Chocolate 1 Chocolate 2
Is that correct?
We can see that chocolate 2 is alot bigger than chocolate 1, they are of different sizes, so we cant really compare them both
Let’s learn about fractions to understand this!
When we compare fractions, we need to make sure the chocolates (wholes) are the same size.
If Tamanna’s chocolates are different sizes, then of a bigger chocolate can indeed be larger than of a smaller chocolate!
To compare two fractions of two wholes, the wholes from which the fractions are made must be the same.
Playing with a Grid
Let’s understand fractions by shading grids. This activity helps us visualise what fractions look like:
Grid A:
Total squares = 48
Shade 1/8 in red (shade 1 square out of 8)
This means we divide the grid into 8 equal parts and colour 1 part
Understanding Equivalent Fractions Is equal to ?
In this picture we can see that, 2 parts out of 6 are shaded.
Now, lets divide the same picture in three equal parts
In this picture, we can see that, 1 part out of 3 is shaded.
So, when we look at both the pictures showing the same shaded area, we can see that = . These are called equivalent fractions.
Equivalent fractions are fractions that represent the same part of a whole but are written differently.
Fun with Fraction Kit
Using a fraction kit helps us understand how fractions work. A fraction kit contains strips divided into equal parts.
Activity 1: Making a Whole
How many 1/5 pieces do you need to make a whole? Answer: 5 pieces
How many 1/8 pieces do you need to make a whole? Answer: 8 pieces
How many 1/3 pieces do you need to make a whole? Answer: 3 pieces
Activity 2: Combining Different Pieces
One piece of 1/2 and two pieces of 1/4 make a whole
This shows us: 1/2 + 1/4 + 1/4 = 1 whole
Also: 1/2 = 2/4 (equivalent fractions)
Understanding the Pattern: When a 1/2 piece is broken into 2 equal parts, each part becomes 1/4. So 2 pieces of 1/4 equal 1/2.
Lets see some more combinations:
1/3 + 1/3 + 1/3 = 1 whole
1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1 whole
1/4 + 1/4 = 1/2
Try yourself:
What do you need to make a whole using 1/5 pieces?
A.3 pieces
B.5 pieces
C.8 pieces
D.2 pieces
View Solution
Making Equivalent Fractions
What are Equivalent Fractions?
Equivalent fractions are different fractions that show the same part of a whole.
Example: , , , all represent the same amount even though they look different.
Sameer shaded one-third () of some shapes.
When he split the shapes into smaller equal parts by drawing lines, he found fractions like:
, , – all exactly cover the same shaded area as 1/3.
This means = = =
These are equivalent fractions.
How to check if two fractions are equivalent?
If you multiply or divide both the top number (numerator) and the bottom number (denominator) of one fraction by the same number, you get an equivalent fraction.
For example:
In short, to get equivalent fractions, multiply or divide numerator and denominator by the same number.
Look at figures showing shaded.
Lets find equivalent fractions of using this:
Multiply numerator and denominator by 2
Multiply numerator and denominator by 3:
Multiply numerator and denominator by 4:
These all represent the same part of the whole as .
When fractions have the same denominator (bottom number), we compare the numerators (top numbers). This is like comparing pieces of the same size.
Comparing Fractions—Same Denominator
Fractions have two parts: numerator (top number) and denominator (bottom number).
When fractions have the same denominator, it means the whole is divided into the same number of equal parts.
To compare such fractions, we only need to look at the numerators.
Example: Sevi and Shami sharing a chikki
The whole chikki is divided into 3 equal parts (denominator is 3).
Sevi ate 1 part → 13
Shami ate 2 parts → 23
Since the denominator is the same (3), compare numerators:
Numerator 1 (Sevi) and Numerator 2 (Shami)
2 is greater than 1, so Shami ate more chikki than Sevi.
Comparing Fractions – Same Numerator
What does it mean to compare fractions with the same numerator?
When fractions have the same numerator (same number of parts), but different denominators (different total parts), the bigger fraction is the one with the smaller denominator.
This is because smaller parts make each piece bigger.
Example: Comparing who ate more paratha
Suppose:
You ate 13 paratha yesterday evening.
Your friend ate 14 paratha yesterday evening.
Both have the numerator 1 (one piece), but denominators are different (3 and 4).
Since 3 is smaller than 4, 13 is bigger than 14.
So, you ate more paratha than your friend.
Rule: When numerators are the same, the fraction with the smaller denominator is bigger.
Try yourself:
What do we compare when fractions have the same numerator?
A.Values
B.Denominators
C.Sizes
D.Numerators
View Solution
Fractions Greater Than 1
We know, a fraction tells us how many parts of a whole we have. For example, if a paratha (a soft flatbread) is cut into equal parts:
12 (one-half) means the paratha is cut into 2 equal parts, and you have one of those parts.
14(one-fourth or a quarter) means the paratha is cut into 4 equal parts, and you have one of those parts.
Sometimes, you can eat more than one whole paratha. This happens when you eat many pieces of paratha, so the total amount is more than one whole. This is called a “fraction greater than 1.”
Let’s see this with some examples.
Example 1: Eating Halves (12’s)
Imagine Raman’s father cuts parathas into halves (2 equal parts each).
If Maa took 5 pieces of 12 paratha, how many whole parathas did she eat?
Since 2 halves make 1 whole paratha, 5 halves are:
5×12=52=212 (which means 2 whole parathas and half of another one)
So, Maa ate 2 and a half parathas.
Using a Number Line to Understand
If you draw a line and divide the space between 0 and 1 into 2 equal parts (because halves),
Then each part is 12.
Moving 5 steps of 12 along the line gets you to 2 and 12.
Example 2: Radhika’s halves
Radhika took 6 pieces of 12 paratha.
Number Line method
62=3 parathas
So, she ate 3 whole parathas.
Dadiji took 7 pieces of 12 paratha = 72=312 parathas
Raman ate 6 pieces of 12 paratha = 3 parathas
Dadaji ate 7 pieces of 12 paratha = 3 and 12parathas
Baba ate 5 pieces of 12 paratha = 2 and 12 parathas
To find how many parathas were made on that day, add all these fractions.
Now, some day, all parathas were cut into 4 pieces each (fourths).
Dadaji took 9 pieces of 14 paratha. How many parathas did he eat
Since 4 fourths make 1 whole paratha, 9 pieces are:
2 + 14
= 214
So, he ate 2 and one-fourth parathas.
Example 4: Sharing Pizzas
The family ordered 2 pizzas, and each pizza is cut into 3 equal slices.
Total slices = 2 pizzas × 3 slices = 6 slices
6 family members need 1 slice each.
Dadiji and Dadaji gave their slices to Raman.
Maa and Baba gave theirs to Radhika.
That means:
Raman gets his slice + 2 slices from Dadiji and Dadaji = 3 slices
Radhika gets her slice + 2 slices from Maa and Baba = 3 slices
Each slice is:
13 (one slice out of 3)
So,
Raman’s total pizza =
Radhika’s total pizza =
When Raman gave 1 slice of pizza to Radhika
Raman eats 2 slices = 2×13=23 pizza
Radhika had 3 slices, now gets 1 more slice from Raman
Comparing Fractions With Reference to 1
Let us compare some more fractions. Between Sevi and Shami can you tell who ate less?
of a bigger chocolate can indeed be larger than 
of a smaller chocolate!
Example: , , , all represent the same amount even though they look different.
Look at figures showing shaded.
