Mathematics for Class 8 ( Ganita Prakash ) Worksheet

Multiple Choice Questions (MCQs)

Q1. Simplify the ratio 56 : 72.
a) 14 : 18
b) 7 : 9
c) 28 : 36
d) 8 : 9

Q2. If a : b = 2 : 3 and b : c = 3 : 5, then a : b : c = ?
a) 2 : 3 : 5
b) 2 : 3 : 9
c) 2 : 3 : 5
d) 2 : 3 : 7.5

Q3. A sum of $600 is divided in the ratio 3 : 5. The smaller share is:
a) $225
b) $200
c) $250
d) $180

Q4. If 6 pencils cost $24, the cost of 9 pencils is:
a) $28
b) $30
c) $36
d) $32

Q5. The fourth proportion of 3, 9, and 12 is:
a) 27
b) 36
c) 24
d) 18

Q6. The third proportion of 12 and 18 is:
a) 24
b) 27
c) 36
d) 30

Fill in the Blanks

Q1: The ratio of 75 cm to 2.5 m is ___ : ___.

Q2: If 4 pens cost $20, then the cost of 10 pens is ___.

Q4: The ratio of 1 hour to 45 minutes is ___ : ___.

Q5: If 7 : x = 21 : 63, then x = ___.

Q6: The third proportion of 8 and 12 is ___.

Q7: If a : b = 5 : 7, then b : a = ___ : ___.

Answer the following Questions: 

Q1. Simplify the ratio 42 : 63

Q2. Ron gets 20% more marks than John. Find the ratio of their marks.

Q3. Divide $490 in the ratio 4 : 3

Q4. A man distributes $4000 among three sons in the ratio 4 : 3 : 3. Find amount for first son.

Q5. If the ratio a : b = 2 : 3, and b : c = 3 : 4. Find the ratio a : c.

Q6. Two numbers: Five times the first = Four times the second. Find ratio.

Q7. Find the fourth proportion of 4, 9, and 12.

Q8. Find the third proportion of 16 and 36.

You can access the solutions to this unit test here.

Fill in the blanks

Q1: The distributive property of multiplication over addition is written as: a(b + c) = ______.

Q2: 12(x + 4) = ______ + ______.

Q3: (2x + 7)(x + 3) expands to ______.

Q4: The perimeter of a rectangle with length (3x + 5) and breadth (x + 2) is ______.

Q5: The total number of pencils in (x + 3) boxes, each having (2x + 4) pencils, is ______.

State True or False

Q1: The degree of a constant term is 0

Q2: A school buys (3x + 1) books, each having (x + 2) pages. A student claims the total pages are 3x² + 7x + 1.

Q3: 1 is an algebraic expression

Q4: A fruit seller sells (2x + 5) apples every day for 12 days. He says the total apples are 24x + 60.

Q5: In like terms, the numerical coefficients should also be the same

Answer the following questions

Q1: The volume of a rectangular box where length, breadth, and height are 2a, 4b, 8crespectively.
Q2: Carry out the multiplication of the expressions in each of the following pairs.
(i) ​​​​p − q, 9pq²
(ii) b² − 16, 5b

Q3: Simplify x(2x−1)+5 and find its value at x=−3
Q4:A shopkeeper sells (x + 4) pens at ₹15 each and (2x + 3) pencils at ₹5 each. Find the total money he earns.
Q5: Add: x(x − y), y(y − z), and z(z − x)

Q6: Multiply: (m² + 3n²) × (2m − n)

Q7: A fruit seller sells (2x + 5) apples every day for 12 days. Each apple costs ₹(x + 2). Find the total cost.

Q8 Simplify the expression and evaluate them as directed:4y(3y – 2) + 5(y + 3) – 12for y = -1

Q9:Add 4x(2x + 3) and 5x² – 7x + 10.

Q10: Simplify (x²−3x+2)(5x−2)−(3x²+4x−5)(2x−1)

1. Multiple Choice Questions (MCQs)

Q1: Which of the following is not a property of a square?
(a) All angles are 90°
(b) Opposite sides are parallel
(c) Only one pair of sides is equal

Q2: What will be the sum of interior angles of a polygon having 8 sides?
(a) 720°
(b) 1080°
(c) 1260°
(d) 1440°

Q3: Which quadrilateral has exactly two distinct consecutive pairs of equal sides?
(a) Kite
(b) Rhombus
(c) Trapezium
(d) Square

Q4: The sides of a quadrilateral are in the ratio of 2:5:4:1. Find out the sum of the smallest and largest angles.
(a) 120°
(b) 180°
(c) 240°
(d) 360°

Q5: If the area of a square field is 144 sq m, then find the perimeter.
(a) 24 m
(b) 36 m
(c) 48 m
(d) 60 m

Q6: If the base of a triangle is 3 cm and the height is 6 cm, then find the area.
(a) 6 sq cm
(b) 9 sq cm
(c) 12 sq cm
(d) 18 sq cm

Q7: An isosceles trapezium has:
(a) Both pairs of opposite sides parallel
(b) Non-parallel sides equal in length
(c) Diagonals equal and perpendicular
(d) All sides equal in length

Q8: In a parallelogram:
(a) Only one pair of sides is parallel
(b) Opposite sides are equal
(c) Diagonals are always equal in length
(d) All angles are 90°

Q9: If the three angles of a quadrilateral are 70°, 90° and 120°, then find the measure of the fourth angle.
(a) 100° 
(b) 75° 
(c) 80° 
(d) 60°

Q10: The measure of two adjacent angles of a parallelogram are in the ratio 2:3. Find the measure of each of the angles of a parallelogram.
(a) 72°, 108° 
(b) 54°, 112° 
(c) 68°, 99° 
(d) 86°, 114°

2. True/FalseQ1: A kite has all four sides equal.

Q2: A square is a special type of rectangle and parallelogram.

Q3: The sum of the smallest and largest angles of a quadrilateral, with sides in the ratio 2:5:4:1, is 240°. 

Q4: The perimeter of a square field, with an area of 144 sq m, is 48 m. 

Q5: The area of a triangle with a base of 3 cm and height of 6 cm is 9 sq cm. 3. Fill in the Blanks

Q1: A polygon in which all sides and all angles are equal is called a __________ polygon.

Q2: The diagonals of a rectangle are equal in length and __________ each other.

Q3: In a parallelogram, adjacent angles are __________.

Q4: The diagonals of a rhombus bisect each other at __________ degrees.

Q5: A trapezium has at least __________ pair of opposite sides parallel.

4. Very Short Answer QuestionsQ1: Can all the angles of a quadrilateral be right angles?

Q2: The sum of all angles in a quadrilateral is equal to_____ right angles.

Q3:  Name the quadrilateral whose diagonals are equal.

Q4: Each angle of a square measures ___°.

Q5: How many parallel lines are in a trapezium?

Q6: Which figure is equiangular and equilateral polygons?

Q7: It rhombus also satisfied the properties of a_______.

Q8: If the diagonals of a quadrilateral are perpendicular bisectors of each other then it is always a______.

5. Answer the following questions: 
Q1: A room has a length of 10 m, breadth of 5m and height of 8 m. Find out the area of the room.

Q2: The length of one side of a rhombus is 6.5 centimeters and its altitude is 10 centimeter. if the length of one side of its diagonals is 26 centimeter find the length of the other diagonal.

Q3: If three angles of a trapezium is 50°, 130° and 120°. Then find the other angle.

Q4: If two adjacent angles of a parallelogram are in the ratio 2:3 Find all the angles of the parallelogram.

Q5: if the angles of a quadrilateral are in the ratio 3:6:8:13. The largest angle is?

Q6: diagonals of a quadrilateral ABCD bisect each other. If A=45°. Then B=?

Q7: The angles of a quadrilateral are x°, x+5°, x+10°, x+25°. Then find the value of x.

Q8: ABCD is a trapezium such that AB || CD, ∠A : ∠D = 2 : 1, ∠B : ∠C = 7 : 5, find the angles of the trapezium.

Q9: ABCD is a parallelogram where m∠A = (2x + 50°) and m∠C = (3x + 40°).
(i) Find the value of x.
(ii) Find the measure of each angle.

Q10: In the below figure, ABCD is a rectangle. Its diagonals meet at O. Find x, if OA = 3x + 1 and OB = 2x + 4.

You can access the solutions to this unit test here.

Q1: In the Roman numeral system, what number does MCMXLIV represent?
a) 1944
b) 1444
c) 1949
d) 1494
Answer: a) 1944

Q2: If the base of a number system is 7, what is the 4th landmark number after 1?
a) 28
b) 49
c) 343
d) 7
Answer: b) 49

Q3: Symbolis representation for which number?

a) 60
b) 360
c) 3600
d) any multiple of 60

Answer: a) 60

Q4: In a base-8 system, what is the value of the number 345 (base 8) in base-10?
a) 229
b) 228
c) 230
d) 231
Answer: a) 229

Q5: In a base-5 system, the third landmark number after 1 is:
a) 15
b) 25
c) 20
d) 30
Answer: b) 25

2. True / False

Q1: The Bakhshali manuscript contains the earliest known example of the number zero written as a dot. 

Ans: True

Q2: Roman numerals can easily be used for multiplication and division. 

Ans: False

Q3:The Gumulgal people counted numbers in groups of 2. 

Ans: True

Q4: The Lebombo bone is believed to be younger than the Ishango bone. 

Ans: False

Q5: In the Egyptian number system, each landmark number is 10 times the previous one. 

Ans: True

3. Fill in the blanks

Q1: The ancient counting device made of a frame with rods and beads is called an __________.
Ans: abacus

Q2: In the Roman numeral system, C represents the number __________.
Ans: 100

Q3: The Roman numeral for 1000 is represented as __________.
Ans: M

Q4: The Roman numeral for 50 is represented as __________.
Ans: L

Q5: A number system in which each landmark number is obtained by multiplying the previous landmark number by a fixed number n is called a __________ system.
Ans: base-n

4. Answer the following Questions

Q1. Represent the following numbers in the Roman system.

(i) 1444

Ans: Break it down:
1000 + 400 + 40 + 4 = M + CD + XL + IV

Answer: MCDXLIV

(ii) 1867

Ans: Break it down:
1000 + 800 + 60 + 7 = M + DCCC + LX + VII

Answer: MDCCCLXVII

(iii) 2539

Ans: Break it down:
2000 + 500 + 30 + 9 = MM + D + XXX + IX

Answer: MMDXXXIX

(iv) 948

Ans: Break it down:
900 + 40 + 8 = CM + XL + VIII

Answer: CMXLVIII

Q2. Consider the extension of the Gumulgal number system beyond 6 in the same way of counting by 2s. Come up with ways of performing the different arithmetic operations (+, –, ×, ÷) for numbers occurring in this system, without using Hindu numerals. Use this to evaluate the following:

(i) (ukasar-ukasar-ukasar-urapon) + (ukasar-urapon)

Ans:Break it down:
(2 + 2 + 2 + 1) + (2 + 1) = 7 + 3 = 10

(Gumulgal): ukasar-ukasar-ukasar-ukasar-ukasar
(= ukasar repeated 5 times → 2×5 = 10)

(ii) (ukasar-ukasar-ukasar-ukasar-urapon) − (ukasar-ukasar-urapon)

Ans: Break it down:
(2+2+2+2+1) − (2+2+1) = 9 − 5 = 4

(Gumulgal): ukasar-ukasar
(= 2 + 2 = 4)

(iii) (ukasar-ukasar-ukasar) × (ukasar-urapon)

Ans: Break it down:
(2+2+2) × (2+1) = 6 × 3 = 18

(Gumulgal): (ukasar repeated 9 times)
= 9×2 = 18 → write ukasar nine times:

ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar

(iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)

Break it down:
(2+2+2+2+2+2) ÷ (2+2) = 12 ÷ 4 = 3

(Gumulgal): ukasar-urapon
(= 3)

Q3: Represent the following numbers in the Egyptian system:

(i) 54321

(ii) 8888

​(iii) 26005

​Ans:

(i) 

​(ii) 

​(iii) 

Q4: Express the number 87 in this base-5 symbolic system.

Ans: Start grouping with the largest landmark number smaller than 87, which is 5² = 25.
We get:

87 = 25 + 25 + 25 + 5 + 5 + 1 + 1

Using the standard symbols:

• 5⁰ = 1 → ▲
• 5¹ = 5 → ■
• 5² = 25 → ⬡

So the number 87 in the new system is:

⬡ ⬡ ⬡ ■ ■ ▲ ▲

Q5: Add : XLVIII + XXXVI

Ans: Step 1: Write all symbols together:
X + L + V + I + I + I + X + X + X + V + I

Step 2: Group and simplify:

• I + I + I + I = IV → but keep as V when grouped
• V + V = X
• X + X + X + X = XL

So the final result is LXXXIV.

Q6: Write Mesopotamian symbol representation for each number.

(i) 58

(ii) 214

(iii) 305

(iv) 499

(v) 7,281

Ans: 

(i) 58

​(ii) 214

​(iii) 305

Q7: Convert the decimal number 150 to its base-7 representation. Show your working and write the answer using digits from 0 to 6.

Ans:  To convert decimal 150 to base-7 :

Write the remainders in reverse order: 3 0 3

Q8: Express the number 999 in this new system.

Ans: Largest landmark number smaller than 999 is 5⁴ = 625.
We get:

999 = 625 + 125 + 125 + 125 + (–1) + … wait, no, careful:

Step-by-step:

999 – 625 = 374
Largest ≤ 374 is 125:
374 – 125 = 249

Another 125:
249 – 125 = 124
Largest ≤ 124 is 25:
124 – 25 = 99

Another 25:
99 – 25 = 74

Another 25:
74 – 25 = 49

Another 25:
49 – 25 = 24

Largest ≤ 24 is 5:
24 – 5 = 19

Another 5:
19 – 5 = 14

Another 5:
14 – 5 = 9

Another 5:
9 – 5 = 4

Largest ≤ 4 is 1:
4 – 1 = 3

Another 1:
3 – 1 = 2

Another 1:
2 – 1 = 1

Another 1:
1 – 1 = 0

So:
625 (5⁴) + 125 (5³) × 2 + 25 (5²) × 4 + 5 (5¹) × 4 + 1 (5⁰) × 4

Symbols:
~ ○ ○ ⬡ ⬡ ⬡ ⬡ ■ ■ ■ ■ ▲ ▲ ▲ ▲

Q9. Add the following numerals that are in the base-5 system that we created:

Remember that in this system, 5 times a landmark number gives the next one!

Ans: Let’s convert this to numerals

First numeral (left side)

This is: 1 circle, 2 hexagons, 1 square, 2 triangles

Value:

• 1 × 125 = 125
• 2 × 25 = 50
• 1 × 5 = 5
• 2 × 1 = 2

Sum: 125 + 50 + 5 + 2 = 182

Second numeral (right side):

Value:

• 3 × 125 = 375
• 1 × 25 = 25
• 2 × 5 = 10
• 2 × 1 = 2

Sum: 375 + 25 + 10 + 2 = 412

Q10: How would the Mesopotamians have written 20, 50, 100?

Ans: The Mesopotamian (Babylonian) system was a base-60 positional system using symbols for 1 (⟐) and 10 (⟐). Numbers were grouped into powers of 60, with a placeholder for zero in later periods. 
Assuming the simplified notation from Section 3.4:

• 20: 20 = 20 × 1 = ⟐⟐ (two 10s).
• 50: 50 = 5 × 10 = ⟐⟐⟐⟐⟐ (five 10s).
• 100: 100 = 1 × 60 + 40 × 1 = ⟐,⟐⟐⟐⟐ (one 60 and four 10s). 

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Multiple Choice Questions

Q1: What is the base of the exponent 6⁹?
(a) 6
(b) 2
(c) 9
(d) None

Q2: Find the missing number 

(a) 2
(b) −5
(c) 1
(d) None

Q3: Find the value of  (5²)²
(a) 125
(b) 625
(c) 25
(d) 0

Q4: Find the value of x, when 2^x=4⁴
(a) x=6
(b) x=2
(c) x=8
(d) x=−5

Q5: Find the value of (2¹¹+6²−5¹)⁰= ?
(a) 0
(b) −1
(c) 1
(d) None

State true or false

Q1:  (10⁰+12⁰)(16⁰+12⁰)=8²

Q2: (3⁴)²=3⁸

Q3: (5²)³=100000

Q4: Among 2⁷,3²,4², and 6³, 6³ is the greatest.

Q5: 625 can be expressed as 4⁵.

Answer the following Questions

Q1: Follow the pattern and complete

Q2: If 2^x × 5^x=1000 then x=?

Q3: Find 3³+ 4³ + 5³ and give the answers in cube

​Q4: Find the missing number x in  5²+x²=13²

​Q5: Simplify in exponent form (3⁴× 3²)÷ 3⁻⁴

​Q6: Expand
(a) 1526.26
(b) 8379
Using exponents

Q7: Express the following number as a product of powers of prime factors.
(a) 1225
(b) 3600

​Q8: Express the following large no’s in its scientific notation.
(a) 491200000
(b) 301000000

​Q9: Express the following in usual form
(a) 3.02 ×10⁻⁶
(b) 5.8 × 10¹²

Q10: Prove that 

You can access the solutions to this unit test here.

Multiple Choice QuestionsQ1: Cube root of the expression 125²
(a) 5
(b) 25
(c) 10
(d) 125

Q2: Which of the following is not a perfect square number?

(a) 1156
(b) 4657
(c) 4624
(d) 7056

Q3: Which of these is not a perfect cube but a perfect square of a number 
(a) 729
(b) 2197
(c) -1331
(d) 169

Q4: Which of these is a perfect cube
(a) 216
(b) 392
(c) 8640
(d) 243

Q5: √0.09 is
(a) 0.3
(b) 0.03
(c) 0.9
(d) 0.33

Q6: The sum of first n odd natural numbers is
(a) n²
(b) 2n
(c) n²+1
(d) n²−1

Q7: The one’s digit of the cube of 53 is:

(a) 9
(b) 3
(c) 7
(d)  1

Q8: The volume of the cube is 5832m³ , the side is
(a) 18m
(b) 16 m
(c) 28 m
(d) 19m

Q9:A perfect square can never have the following digit in its ones place
(a) 8
(b) 4
(c) 0
(d) 1

Q10: Which of the following expressions is not a perfect cube
(a) 27 x 125 x 64
(b) 1331 x 125 x 8
(c) 15 x 8 x 25 x 9
(d) None of these

True and FalseQ1: 648 is not a perfect cube.

Q2: 999 is a perfect cube.

Q3: The square of a number is positive, so the cube of that number will also be positive.

Q4: 125×8×27 is a perfect cube.

Q5: For an integer p, p³ is always greater than p²_.

Q6: 8³ = 5.12

Fill in the blanks

Q1: The square of an even number is _____

Q2: The least number by which 72 be multiplied to make it a perfect cube is __________.

Q3: √4096 is ____

Q4: If 8x³=216, then x ix ____

Q5: The cube of .5 is _____

Q6: There are _________ perfect cubes between 1 and 1000.

Q7: The digit at the ones place of 23³ is ____

Q8: The cube of the even natural number is ____

Find the cube roots by prime factorization method

Q1: 15625

Q2: 2744

Q3: 125/2197

Q4: 5832

Q5: 64000

Find the square roots by prime factorization method

Q1: 144

Q2: 3600

Q3: 2025

Q4: 81/256

Q5: 1024

Q6: 3844

Answer the following questions

Q1: Is 256 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube.
Q2: What could be the possible ‘one’s’ digits of the square root of each of the following numbers?
(i) 1801
(ii) 856
(iii) 1008001
(iv) 6577525
Q3: Three numbers are in the ratio 1:2:3 and the sum of their cubes is 4500. Find the numbers.

Q4: Find the smallest number by which following number must be divided to get a perfect square. Also, find the square root of the perfect square so obtained.
(i)600
(ii)2904

Q5: Find the smallest number by which following number must be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained.
(i) 1008
(ii) 1280
(iii) 1875

​You can access the solutions to this Unit Test here.