Q: Write the following fractions as a sum of fractions and also as decimals:Ans:
Figure it Out(Page 73)
Q1: Recall that a tenth is 0.1, a hundredth is 0.01, and so on. Find the following products in tenths, hundredths and so on: (a) 6 × 4 tenths = 24 tenths (b) 7 × 0.3 (c) 9 × 5 hundredths Ans: (a) 6 × 4 tenths = 24 tenths (b) 0.3 = 3 tenths 7 × 3 tenths = 21 tenths (c) 9 × 5 hundredths = 45 hundredths Q2: Find the products: (a) 27.34 × 6 (b) 4.23 × 3.7 (c) 0.432 × 0.23 Ans: (a) 27.34 × 6 2734 × 6 = 16404 Decimal places in 27.34 = 2 Decimal places in 6 = 0 Total decimal places in the product = 2 + 0 = 2 So, 27.34 × 6 = 164.04.
(b) 4.23 × 3.7 423 × 37 = 15651 Decimal places in 4.23 = 2 Decimal places in 3.7 = 1 Total decimal places in the product = 2 + 1 = 3 So, 4.23 × 3.7 = 15.651.
(c) 0.432 × 0.23 432 × 23 = 9936 Decimal places in 0.432 = 3 Decimal places in 0.23 = 2 Total decimal places in the product = 3 + 2 = 5 So, 0.432 × 0.23 = 0.09936. Q3:Thejus needs 1.65 m of cloth for a shirt. How many metres of cloth are needed for 3 shirts? Ans: Given: Thejus needs 1.65 m of cloth for a shirt.
For 3 shirts, the total cloth needed =1.65×3
=165100×3
=495100
=4.95
Q4: Meenu bought 4 notebooks and 3 erasers. The cost of each book was ₹15.50 and each eraser was ₹2.75. How much did she spend in all?
Ans: Here cost of 1 notebook = ₹ 15.50
∴ Cost of 4 notebooks = 4×15.50
=4×1550100
=6200100
=₹ 62
and cost of 1 eraser = ₹ 2.75
∴ Cost of 3 erasers = 3×₹ 2.75
=3×275100
=825100
=₹ 8.25
∴ Total amount spent = 62+8.25=₹ 70.25
Q5: The thickness of a rupee coin is 1.45 mm. What is the total height of the cylinder formed by placing 36 rupee coins one over the other? Write the answer in centimeters. Ans: Thickness of 1 coin = 1.45 mm
Total thickness of 36 coins = 36×1.45
=36×145100
=5220100
=52.2 mm
Now 10 mm = 1 cm
1 mm=110 cm
∴ 52.2 mm=52.210=5.22 cm
Q6: The price of 1 kg of oranges is ₹56.50. What is the price of 2.250 kg of oranges? Can we write 56.50 as 56.5 and 2.250 as 2.25 and multiply? Will we get the same product? Why?
Ans: Price of 1 kg of oranges = ₹ 56.50
Price of 2.250 kg of oranges = 56.50×2.250
=5650×2250100×1000
=12712500100000
=₹ 127.125
Now 56.5×2.25=127.125
Hence, we will get the same product.
The zeroes at the end of a decimal do not change its value.
As we saw, 56.50 is the same as 56.5, and 2.250 is the same as 2.25.
Hence, the product of the two numbers will be the same.
Q7: Dwarakanath purchases notebooks at a wholesale price of ₹23.6 per piece and sells each notebook at ₹30/-. How much profit does he make if he sells 50 books in a week?
Ans: Profit per notebook = Selling price – wholesale price = 30 – 23.6 = ₹ 6.4 Total profit = Profit per notebook × No. of notebooks = 6.4 × 50 = ₹ 320
Q8: Given that 18 × 12 = 216, find the products: (a) 18 × 1.2 (b) 18 × 0.12 (c) 1.8 × 1.2 (d) 0.18 × 0.12 (e) 0.018 × 0.012 (f) 1.8 × 12 In which of the cases above is the product less than 1? Ans: (a) Here 18×12=216 …..(i)
Now 18×1.2=18×1210 [Using (i)]
=21610
=21.6 (1 decimal place)
(b) 18×0.12=18×12100 (Using (i))
=216100
=2.16 (2 decimal places)
(c) 18×1.2=1810×1210=216100 [Using (i)]
=2.16 (2 decimal places)
(d) 0.18×0.12=18100×12100=216100×100 [Using (i)]
=0.0216 (4 decimal places)
(e) 0.018 × 0.012 18 × 12 = 216 Decimal places in 0.018 = 3 Decimal places in 0.012 = 3 Total decimal places in the product = 3 + 3 = 6 So, 0.018 × 0.012 = 0.000216
(f) 1.8×12=18×1210=21610 [Using (i)]
=21.6 (1 decimal place)
When multiplying two numbers positive if both numbers are less than 1, their product will also be less than 1.
In (d) and (e) product is less than 1.
Q9: In which of the following multiplications is the product less than 1? Can you find the answer without actually doing the multiplications? (a) 7 × 0.6 (b) 0.7 × 0.6 (c) 0.7 × 6 (d) 0.07 × 0.06 Ans: (a) 7 × 0.6
Here, one number (0.6) is between 0 and 1, and the other number (7) is greater than 1.
So, the product is less than 7 but greater than 0.7.
Hence, the product is greater than 1. (b) 0.7 × 0.6 Here, both numbers, 0.7 and 0.6, are between 0 and 1. So, the product is less than both numbers. Since both numbers are less than 1, their product is less than 1. (c) 0.7 × 6 Here, one number (0.7) is between 0 and 1, and the other number (6) is greater than 1. So, the product is less than 6 but greater than 0.7. Hence, the product is greater than 1. (d) 0.07 × 0.06 Here, both numbers, 0.07 and 0.06, are between 0 and 1. So, the product is less than both numbers. Since both numbers are less than 1, their product is less than 1. Therefore, the product is less than 1 in the following cases: (b) 0.7 × 0.6 and (d) 0.07 × 0.06. Q10:Multiplying the following numbers by 10, 100 and 1000 to complete the table.
Ans:
Figure it Out(Page 83)
Q1: Find the quotient by converting the denominator into 1, 10, 100 or 1000 and verify the solution by the long division method (division by place value).
(a) 185
(b) 4154
(c) 12172
(d) 48278
Ans:
(a) Given 185
To convert the denominator 5 into 10, multiply both the numerator and Dr by 2.
18×25×2=3610=3.6
Verification
18÷5
Dividing 1 ten and 8 ones into 5 equal parts.
1<5
It means we need to regroup 1 ten as 10 ones,
i.e., 10+8=18 ones
18 ones÷5
3 ones remain.
To divide 3 ones into 5 equal parts.
Regroup the 3 ones as 30 Tenths. (Place a decimal while regrouping ones into tenths).
30 Tenths÷5=6
Then, 18÷5=3.6
(b) Given 4154
To convert the denominator 4 into 100, multiply both the Nr and Dr by 25.
415×254×25=10375100=103.75
Verification
By following the steps
415÷4=103.75
Hence verified.
(c) Given 12172
To convert the denominator 2 into 10, multiply both the Nr and Dr by 5.
1217×52×5=608510=608.5
Verification
By following the steps:
1217÷2=608.5
Hence verified.
(d) Given 48278
To convert the denominator 8 into 1000, multiply both the Nr and Dr by 125.
So, the quotient is 3.77. Q3: Evaluate the following using the information 156 × 12 = 1872. (a) 15.6 × 1.2 = __________ (b) 187.2 ÷ 1.2 = __________ (c) 18.72 ÷ 15.6 = __________ (d) 0.156 × 0.12 = __________ Ans: Given 156×12=1872 ……(i)
⇒156=187212……(ii)
⇒12=1872156……(iii)
(a) Now converting division into a fraction
15.6×1.2=156×1210×10=1872100=18.72[using (i)]
(b) Converting division into a fraction 187.2÷1.2
187.2÷1.2=187.21.2=187212=156[using (ii)]
(c) Converting division into a fraction 18.72÷15.6, we get
Is the quotient obtained in 24.6 ÷ 1.5 the same as the quotient obtained in 2.46 ÷ 0.15?
∴ Quotient = 164
Now
24.61.5=246×1015×10=24615
and
2.460.15=246×10015×100=24615
Both are the same.
Hence quotient obtained in 24.6÷1.5 is the same as the quotient obtained in 2.46÷0.15.
Q6: A 4 m long wooden block has to be cut into 5 pieces of equal length. What is the length of each piece? Ans: Here total length = 4 m
No. of pieces = 5
Length of each piece = Total lengthNo. of pieces
=45
=0.8 m
Q7: If the perimeter of a regular polygon with 12 sides is 208.8 cm, what is the length of its side? Ans: Here Perimeter = 208.8 cm
No. of sides = 12
Length Of a side = PerimeterNo. of sides
=208.812
=17.4 cm
Q8: 3 liters of watermelon juice is shared among 8 friends equally. How much watermelon juice will each get? Express the quantity of juice in millilitres. Ans: Total watermelon juice = 3 litres Number of friends = 8 Quantity of juice each will get = 3 ÷ 8 = 0.375 litres = 0.375 × 1000 = 375 ml Therefore, each friend will get 375 millilitres of watermelon juice.
Q9: A car covers 234.45 km using 12.6 litres of petrol. What is the distance travelled per litre? Ans: Given total distance = 234.45 km Total petrol = 12.6 litres
Hence, the total distance travelled per litre of petrol is 18.607 km.
Q10: 13.5 kg of flour (aata) was distributed equally among 15 students. How much flour did each student receive? Ans:
Total quantity of flour = 13.5 kg
No. of students = 15
Flour per student = 13.515=0.9 kg
Now
Each student receives = 0.9 kg.
Ans:
What pattern do you observe? Why are 2 and 5 related in this way?
Ans: The decimal values of keep getting smaller and each step adds one more decimal place. This happens because 2 and 5 are factors of 10. Since 2 × 5 = 10, powers of 2 and 5 combine to make powers of 10, and numbers with denominator 10n can be written easily as decimals. Therefore, fractions with powers of 2 or 5 in the denominator always give terminating decimals.
Figure it Out (Page 93)
Q1: A 210 gram packet of peanut chikki costs ₹70.5, while a 110 gram packet of potato chips costs ₹33.25. Which is cheaper? Ans: Peanut chikki Weight = 210 g Cost = ₹70.5 Cost per gram of peanut chikki = 70.5 ÷ 210 = 0.336 per g
Potato chips Weight = 110 g Cost = ₹33.25 Cost per gram of potato chips = 33.25 ÷ 110 = 0.302 per g Since, 0.302 < 0.335 Therefore, potato chips are cheaper. Q2: Write the decimal number at the arrow mark:
Ans:
Q3: Shyamala bought 3 kg bananas at ₹30/- per kg. She counted 35 bananas in all. She sells each banana for ₹5/-. How much profit does she make selling all the bananas?
Ans: Cost of 1 kg bananas = ₹30 Cost of 3 kg bananas = 3 × 30 = ₹90 Number of bananas bought = 35 Selling price of each banana = ₹5 Total selling price = 5 × 35 = ₹175 Profit = 175 – 90 = ₹85 Therefore, Shyamala makes a profit of ₹85 by selling all the bananas. Q4: A teacher placed textbooks that are 2.5 cm thick on a bookshelf. The teacher wanted to place 80 textbooks on the shelf. The bookshelf is 160 cm long. How many books could be placed on the shelf? Was there any space left? If yes, how much? Ans: Thickness of one textbook = 2.5 cm Length of the bookshelf = 160 cm Number of books that can be placed on the shelf = 160 ÷ 2.5 = 64 Space occupied by 64 textbooks = 64 × 2.5 = 160 cm Space left on the shelf = 160 – 160 = 0 cm Therefore, 64 textbooks could be placed on the shelf and no space was left.
Q5: Fill in the following blanks appropriately:
Ans: Here, (i) 1 km=1000 m
∴ 5.5 km=5.5×1000=5500 m
(ii) 1 m=100 cm
∴ 35 cm=35100=0.35 m
(iii) 1 cm=10 mm
∴ 14.5 cm=14.5×10=145 mm
(iv) 1 kg=1000 g
∴ 68 g=681000=0068 kg
(v) 1 m=1000 mm
∴ 9.02 m=9.02×1000=9020 mm
(vi) 1 l=1000 ml
∴ 125.5 ml=125.51000=0.1255 l
Q6: The following problem was set by Sridharacharya in his book, Patiganita. “614 is divided by 212, and 6014 is divided by 312. Tell the quotients separately.” Can you try to solve it by converting the fractions into decimals? Ans:
∴ Quotient = 17.21
Q7: Fill the boxes in at least 2 different ways: (a) ☐ × ☐ = 2.4 (b) ☐ × ☐ = 14.5 Ans: (a) ☐ × ☐ = 2.4 Two different ways: 0.6 × 4 = 2.4 1.2 × 2 = 2.4
(b) ☐ × ☐ = 14.5 Two different ways: 1.45 × 10 = 14.5 2.9 × 5 = 14.5
Q8: Find the following quotients given that 756 ÷ 36 = 21: (a) 75.6 ÷ 3.6 (b) 7.56 ÷ 0.36 (c) 756 ÷ 0.36 (d) 75.6 ÷ 360 (e) 7560 ÷ 3.6 (f) 7.56 ÷ 0.36 Ans: (a) 75.6 ÷ 3.6 Multiply both numbers by 10, 75.6 ÷ 3.6 = 756 ÷ 36 = 21.
(b) 7.56 ÷ 0.36 Multiply both numbers by 100, 7.56 ÷ 0.36 = 756 ÷ 36 = 21.
(c) 756 ÷ 0.36 Multiply both numbers by 100, 756 ÷ 0.36 = 75600 ÷ 36 Since 756 ÷ 36 = 21, 75600 ÷ 36 = 2100.
(d) 75.6 ÷ 360 Multiply both numbers by 10, 75.6 ÷ 360 = 756 ÷ 3600 Divide both by 36, 756 ÷ 3600 = 21 ÷ 100 = 0.21.
(e) 7560 ÷ 3.6 Multiply both numbers by 10, 7560 ÷ 3.6 = 75600 ÷ 36 = 2100.
(f) 7.56 ÷ 0.36 Multiply both numbers by 100, 7.56 ÷ 0.36 = 756 ÷ 36 = 21.
9. Find the missing cells if each cell represents a ÷ b:
Ans:
Q10. Using the digits 2, 4, 5, 8, and 0 fill the boxes ☐☐.☐ × ☐.☐ to get the: (a) maximum product (b) minimum product (c) product greater than 150 (d) product nearest to 100 (e) product nearest to 5 Ans: (a)Q11: Sort the following expressions in increasing order: (a) 245.05 × 0.942368 (b) 245.05 × 7.9682 (c) 245.05 ÷ 7.9682 (d) 245.05 ÷ 0.942368 (e) 245.05 (f) 7.9682 Ans: Let A = 245.05, B = 0.942368, C = 7.9682 We note that B < 1 and C > 1 (a) Now A × B = 245.05 × 0.942368 < 245.05 (∴ Multiplying a number by a value less than 1 results in a smaller number)
(b) A × C = 245.05 × 7.9682 > 245.05 (Multiplying a number by a value greater than 1 results in a larger number)
(c) A ÷ C = 2.4505 ÷ 7.9682 < 245.05 (Dividing a number by a value greater than 1 results in a smaller number)
(d) A ÷ B = 245.05 ÷ 0.942368 > 245.05 (Dividing a number by a value greater than 1 results in a larger number)
(e) Now, expression less than 245.05 ∴ 0.942368 is closer to 1 than 7.9682 is to 1. 0.942368 will result in a value closer to 245.05 than dividing by 7.9682. ∴ (c) < (a) Again, 0.942368 is closer to 1 than 7.9682 is to 1. ∴ Dividing by 0.942368 will result in a value closer to 245.05 than multiplying by 7.9682. ∴ (d) < (b) Also, (f) 7.9682 is significantly smaller than 245.05; it will be the smallest value.
(f) ∴ 7.9682 < (e) 245.05 ÷ 7.9682 Combining all, we get (f), (c), (a), (e),(d), (b).
keep getting smaller and each step adds one more decimal place. This happens because 2 and 5 are factors of 10. Since 2 × 5 = 10, powers of 2 and 5 combine to make powers of 10, and numbers with denominator 10n can be written easily as decimals. Therefore, fractions with powers of 2 or 5 in the denominator always give terminating decimals.
Q11: Sort the following expressions in increasing order: