Class 8 Mathematics · Part 2 · Chapter 1

1. Fractions in Disguise

Solutions to all Figure it Out and in-text questions. CBSE Class 8 Mathematics, NCERT Ganita Prakash Part 2.

1.1 Fractions as Percentages

1Surya’s deep orange colour is 34 red. What percentage of the colour was made using yellow?

Yellow = 1 − 34 = 14 = 25100 = 25%

2Complete Method 3 by filling the boxes (fifths of 100%).

15 = 20%, 25 = 40%, 35 = 60%, 45 = 80%.

So the savings for the canvas (25) = 40%

Figure it Out (Page 3)

1Express the following fractions as percentages.

(i) 35 × 100 = 60%

(ii) 714 = 12 = 50%

(iii) 920 × 100 = 45%

(iv) 72150 × 100 = 48%

(v) 13 × 100 = 3313%

(vi) 511 × 100 ≈ 45.45%

2Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are white?

1525 × 100 = 60%. Answer: (iv) 60%

3In a school, 15 of the 80 students come by walking. What percentage come by walking?

1580 × 100 = 18.75%

4Match the percentage of the race each runner has approximately completed.

A ≈ 38%, B ≈ 55%, C ≈ 72%, D ≈ 93%

A is about one-third of the way, B a little past half, C about three-fourths, and D is very close to the finish. 20% and 84% are not used.

5Write ‘>’, ‘<’ or ‘=’.

(i) 50% > 5%

(ii) 510 = 50%

(iii) 311 < 61% (311 is about 27%)

(iv) 30% < 13 (13 is about 33%)

6If we want the same denominator, why choose 100? Why not 10, 50, 1000 or 43?

Our number system is base 10, so 100 fits easily with decimals (31% = 0.31).

100 is large enough to show detail and small enough to grasp easily. Per 10 is too rough for many purposes.

1.2 Percentage of Some Quantity

7Madhu ate 120 g of biscuits with 25% sugar and Madhav ate 95 g with 35% sugar. Who consumed more sugar?

Madhu: 25% of 120 = 14 × 120 = 30 g.

Madhav: 35% of 95 = 0.35 × 95 = 33.25 g.

Madhav ate more sugar.

8Calculate mentally the indicated percentages of the values in the table. Also find 40% and 15%.

%	100	200	50	80	10	35	287
25%	25	50	12.5	20	2.5	8.75	71.75
10%	10	20	5	8	1	3.5	28.7
20%	20	40	10	16	2	7	57.4
5%	5	10	2.5	4	0.5	1.75	14.35
40%	40	80	20	32	4	14	114.8
15%	15	30	7.5	12	1.5	5.25	43.05

Methods: 10% = divide by 10. 20% = double of 10%. 5% = half of 10%. 25% = 20% + 5% (or a quarter). 40% = double of 20%. 15% = 10% + 5%.

9How would you mentally calculate 75%, 90%, 70%, 55% of a value?

75% = 50% + 25% (or 3 quarters). 90% = 100% − 10%. 70% = 50% + 20%. 55% = 50% + 5%.

Example for 80: 75% = 60, 90% = 72, 70% = 56, 55% = 44.

10To find 10% of a quantity, what decimal value should be multiplied? Complete the table.

Multiply by 0.1.

Per cent	50%	100%	25%	75%	10%	1%	5%	43%
Fraction	12	1	14	34	110	1100	120	43100
Decimal	0.5	1.0	0.25	0.75	0.1	0.01	0.05	0.43

11Kishanlal’s target is ₹5000. Find the percentage of target on Day 5 (₹7800) and Day 6 (₹9550). Find sales on Day 7 (150%) and Day 8 (210%).

Day 5: 78005000 × 100 = 156%

Day 6: 95505000 × 100 = 191%

Day 7: 150% of 5000 = 1.5 × 5000 = ₹7,500

Day 8: 210% of 5000 = 2.1 × 5000 = ₹10,500

12Complete the table of percent, fraction and decimal.

Percent	90%	110%	200%	250%	15%	173%	358%	28.9%	305%
Fraction	910	1110	2	52	320	173100	17950	2891000	6120
Decimal	0.9	1.1	2.0	2.5	0.15	1.73	3.58	0.289	3.05

On the number line, mark each decimal value: for example 2.5 is halfway between 200% and 300%, and 3.58 is a little before 400%.

Figure it Out (Page 12)

1Find the missing numbers.

(ii) The bar has 10 equal parts, so one part = 10%. The total is 90 and ? covers 6 parts: 6 × 9 = 54

(iii) The bar has 4 equal parts, so one part = 25%. The total is 140 and ? covers 3 parts: 3 × 35 = 105

2Find the value of the following and draw their bar models.

(i) 25% of 160 = 40

(ii) 16% of 250 = 40

(iii) 62% of 360 = 223.2

(iv) 140% of 40 = 56

(v) 1% of 1 hour = 1% of 60 min = 0.6 min = 36 seconds

(vi) 7% of 10 kg = 0.7 kg = 700 g

Bar model: Draw a bar for the whole (100%). Divide it into equal parts (for example 4 parts of 25%) and shade the required part.

3Surya made 60 ml of deep orange paint. Red made up 34. How much red paint did he use?

34 × 60 = 45 ml

4Write ‘>’, ‘<’ or ‘=’.

(i) 50% of 510 < 50% of 515

(ii) 37% of 148 < 73% of 148

(iii) 29% of 43 < 92% of 110

(iv) 30% of 40 (= 12) < 40% of 50 (= 20)

(v) 45% of 200 (= 90) > 10% of 490 (= 49)

(vi) 30% of 80 (= 24) > 24% of 64 (= 15.36)

5Fill in the blanks appropriately.

(i) 30% of k is 70. 60% of k is 140, 90% of k is 210, 120% of k is 280

(ii) 100% of m is 215. 10% of m is 21.5, 1% of m is 2.15, 6% of m is 12.9

(iii) 90% of n is 270. 9% of n is 27, 18% of n is 54, 100% of n is 300

(iv) Sample: 25% of p is 40; find 50%, 75% and 100% of p (80, 120, 160).

6Fill in the blanks.

(i) 3 is 1% of 300.

(ii) 1.6 is 40% of 4.

(iii) 40 is 80% of 50.

7Is 10% of a day longer than 1% of a week?

10% of 24 hours = 2.4 hours. 1% of 168 hours = 1.68 hours.

So yes, 10% of a day is longer.

8The bull is given n + 1 units and eats n units. Represent these as percentages. What do you observe?

Day 1: 12 = 50%, Day 2: 23 ≈ 66.7%, Day 3: 34 = 75%, Day 4: 80%, …, Day 99: 99100 = 99%.

The percentage keeps increasing and comes closer and closer to 100%, but never reaches it.

9Workers take 18 days to pick berries in 20% of the plantation. How many days for the whole plantation?

20% → 18 days, so 100% → 18 × 5 = 90 days

We assume the rate of work stays the same. If the number of workers or the speed changes, the time will change.

10Warm up : play : cool down = 10% : 80% : 10% for 90 minutes. How long is each activity?

Warm up = 9 min, play = 72 min, cool down = 9 min

1190% of the world’s population lives in the Northern Hemisphere. Find the approximate number.

World population ≈ 8.2 billion. 90% of 8.2 billion ≈ 7.4 billion (about 738 crore).

Use the latest world population figure for the current year.

12Halwa for 4 people has Rava 40%, Sugar 40%, Ghee 20%. (i) For 8 people, what is the proportion? (ii) If the total weight is 2 kg, find each.

(i) The proportion stays the same: 40%, 40%, 20%. Only the quantities double.

(ii) Rava = 800 g, Sugar = 800 g, Ghee = 400 g

1.3 Using Percentages

13KYC: Complete the table of percentages for DEF and Zacni. Which has a larger share of badam? Which uses a smaller proportion of food chemicals?

Product	Sugar	Milk solids	Badam powder	Food chemicals
DEF (150 g)	66%	20%	8%	6%
Zacni (400 g)	68%	16%	10%	6%

Zacni has a larger share of badam (10%).

Both use the same proportion (6%) of food chemicals.

The percentages of each product add up to 100%.

14Find the profit percentage of the wholesaler and the manufacturer.

Manufacturer: CP ₹230, SP ₹253. Profit = 23. Profit % = 23230 × 100 = 10%

Wholesaler: CP ₹253, SP ₹300. Profit = 47. Profit % = 47253 × 100 ≈ 18.6%

15Shambhavi buys notebooks at ₹36 and sells with a 20% profit margin. Find the selling price. Crayon boxes are sold at ₹50 with a 25% margin. Find the cost price.

Notebook SP = 120% of 36 = ₹43.20

Crayons: 125% of CP = 50, so CP = 50 ÷ 1.25 = ₹40

16Could we have calculated the loss percentage per kg instead? Would it be the same?

Yes. Per kg: CP = ₹35, SP = ₹30, loss = ₹5. Loss % = 535 × 100 ≈ 14.28%. It is the same.

17Snehal sells strawberries at ₹80 per kg with a 12% loss. What is the cost price?

88% of CP = 80. CP = 80 ÷ 0.88 ≈ ₹90.91 per kg

18A 35% discount on a cooker with MRP ₹1800. Find the SP. If CP was ₹900, find the profit percentage.

SP = 65% of 1800 = ₹1,170

Profit = 1170 − 900 = 270. Profit % = 270900 × 100 = 30%

19Check if the calculations in the bill are correct.

3 × ₹150 = ₹450. CGST 9% of 450 = ₹40.50. SGST 9% of 450 = ₹40.50.

Total = 450 + 40.50 + 40.50 = ₹531. The bill is correct.

Figure it Out (Page 19)

1A geometry box is bought for ₹75 and sold for ₹110. Find the profit margin with respect to the cost.

Profit = 35. Profit % = 3575 × 100 ≈ 46.67%

2Materials for a chair cost ₹475. I want a 50% profit margin. At what price should I sell?

SP = 150% of 475 = ₹712.50

3A company’s revenue was ₹2.5 crore with a profit margin of 25%. What was the total expenditure?

Profit is 25% of revenue = 0.25 × 2.5 crore = 0.625 crore.

Expenditure = 2.5 − 0.625 = ₹1.875 crore

If the 25% margin is taken on cost, then cost = 2.5 ÷ 1.25 = ₹2 crore.

4A 25% discount on a shirt of ₹300. How much will Anwar pay?

75% of 300 = ₹225

5Petrol was ₹60 in 2015 and ₹100 in 2025. Find the percentage increase.

Increase = 40. Percentage increase = 4060 × 100 ≈ 66.66%. Answer: (iv) 66.66%

6Samson bought a car for ₹4,40,000 after a 15% discount. What was the original price?

85% of original = 4,40,000. Original = 4,40,000 ÷ 0.85 ≈ ₹5,17,647

71600 people voted and the winner got 500 votes. What percent did the winner get? Guess the minimum number of candidates.

Winner = 5001600 × 100 = 31.25%

The other candidates got 1100 votes, and each must have fewer than 500. Two others cannot share 1100 this way (one would get at least 550). Three others can (for example 400, 400, 300).

So at least 4 candidates stood for the election.

8Rice was ₹38 in 2024 and ₹42 in 2025. What is the rate of inflation?

Increase = 4. Inflation = 438 × 100 ≈ 10.5%

9A number increased by 20% becomes 90. What is the number?

120% of the number = 90. Number = 90 ÷ 1.2 = 75

10Two buffaloes were sold at ₹80,000 each, one at 5% profit and the other at 10% loss. Find the overall profit or loss.

CP of first = 80000 ÷ 1.05 ≈ ₹76,190.48. CP of second = 80000 ÷ 0.9 ≈ ₹88,888.89.

Total CP ≈ ₹1,65,079.37. Total SP = ₹1,60,000.

Overall loss ≈ ₹5,079.37 (about 3.08%).

11The elephant population increased by 5% in the last decade. If it was p, the population now is:

p + 5% of p = p × 1.05. Answer: (iv) p × 1.05

12Which statements mean the same as “The demand for cameras has fallen by 85% in the last decade”?

If demand fell by 85%, 15% of the old demand remains. Answer: (iii) The demand now is 15% of the demand a decade ago.

In the textbook, the questions on page 20 are numbered 3 to 9 again (a printing slip). Here they are numbered 6 to 12.

Figure it Out (Page 22) — Growth and Compounding

1Bank of Yahapur: 10% p.a. on ₹20,000 for 2 years. Compare with and without compounding.

Without compounding: 20000 + 2 × 2000 = ₹24,000

With compounding: 20000 × 1.1 × 1.1 = ₹24,200

Compounding gives ₹200 more.

2Bank of Wahapur: 5% p.a. on ₹20,000 for 4 years. Compare.

Without compounding: 20000 + 4 × 1000 = ₹24,000

With compounding: 20000 × (1.05)⁴ ≈ ₹24,310.13

3What do you observe in the two answers above?

Without compounding, both give the same amount (₹24,000), because 10% × 2 = 5% × 4.

With compounding, the 5% for 4 years gives more, because interest is added back more times.

20Write the formula for the total interest gained in each option.

Without compounding: Interest = p × r × t = prt

With compounding: Interest = p(1 + r)^t − p

Figure it Out (Page 24)

4Jasmine invests p for 4 years at 6% p.a. without compounding. Which expression gives the total amount?

Amount = p + (p × 0.06 × 4). Answer: (vii)

5₹50,000 at 7% p.a. for 3 years. Find the interest without compounding. How much more with compounding?

Without: 50000 × 0.07 × 3 = ₹10,500

With: 50000 × (1.07)³ ≈ ₹61,252.15, so interest ≈ ₹11,252.15.

Compounding gives about ₹752.15 more

6Giridhar borrows ₹12,500 at 12% p.a. for 3 years without compounding. Raghava borrows the same at 10% p.a. compounded. Who pays more interest and by how much?

Giridhar: 12500 × 0.12 × 3 = ₹4,500.

Raghava: 12500 × (1.1)³ − 12500 = 16637.50 − 12500 = ₹4,137.50.

Giridhar pays more, by ₹362.50

7₹1000 grows at 10% p.a. How long to double with and without compounding?

Without compounding: ₹100 is added each year, so it takes 10 years

With compounding: after 7 years it is ≈ ₹1949 and after 8 years ≈ ₹2144. So it doubles in about 7½ years (during the 8th year).

Yes. Compounding multiplies every year (exponential growth). Without compounding, the same amount is added every year (linear growth).

8A city’s population of 1.5 crore rises by 3% every year. Find the population after 3 years.

1.5 crore × (1.03)³ ≈ 1,63,90,905 (about 1.64 crore).

9Bacteria increase at 2.5% per hour. Initial count 5,06,000. Find the number after 2 hours.

506000 × 1.025 × 1.025 ≈ 5,31,616

Tricky Percentages

21Would You Rather? Option A: deposit ₹100, get ₹300. Option B: deposit ₹1000, get ₹1500. Which would you choose?

Option A gain = 200% (₹200). Option B gain = 50% (₹500).

A has a bigger percentage, but B gives more money. Since we can choose only once, Option B gives the larger amount.

22Choose 20% discount or ₹50 discount for a purchase of (i) ₹180 (ii) ₹225 (iii) ₹300.

(i) 20% of 180 = 36 → choose ₹50 off

(ii) 20% of 225 = 45 → choose ₹50 off

(iii) 20% of 300 = 60 → choose 20% off

(Both are equal at ₹250.)

23Ariba has 120% of Arun’s marbles. What could Arun say?

Arun has 100120 × 100 ≈ 83.33% of Ariba’s marbles, that is, about 16.67% fewer.

Figure it Out (Page 28)

1Bengaluru’s population in 2025 is 250% of that in 2000 (50 lakh). Find the 2025 population.

2.5 × 50 lakh = 125 lakh = 1.25 crore

2World population ≈ 8.2 billion. Match each country to its approximate share.

Germany: 83 million ÷ 8.2 billion ≈ 1%

India: 1.46 billion ÷ 8.2 billion ≈ 18%

Bangladesh: 175 million ÷ 8.2 billion ≈ 2%

USA: 347 million ÷ 8.2 billion ≈ 4%

4% is not in the given options (2% is printed twice). The USA’s share is about 4.2%.

3A phone costs ₹8,250 plus 18% GST. Which gives the final price?

(v) 8250 × 1.18 and (vi) 8250 + 8250 × 0.18. Both give ₹9,735.

4Mice population changes: +5%, −2%, −3%. Which statements are true?

Population = p × 1.05 × 0.98 × 0.97 ≈ 0.998p.

True: (ii) and (vi). The population is slightly less than p.

5A 35% profit margin and then a 30% discount. Profit or loss?

SP = 1.35 × 0.70 × CP = 0.945 × CP.

So he makes a loss of 5.5%.

6What percentage of the area is occupied by region E?

The big square is 8 × 8 = 64 small squares. D and E together form a 4 × 4 square (16), cut by a diagonal. So E = 8.

E = 864 × 100 = 12.5%

7Compare 5% of 40 and 40% of 5, 25% of 12 and 12% of 25, 15% of 60 and 60% of 15. Make a general statement.

5% of 40 = 2 = 40% of 5. 25% of 12 = 3 = 12% of 25. 15% of 60 = 9 = 60% of 15.

In general, x% of y = x100 × y = xy100 = y100 × x = y% of x.

840% of students are Grade 8, and 60% of them are girls. (i) What percentage are Grade 8 girls? (ii) If 160 students go, how many are Grade 8 girls?

(i) 60% of 40% = 0.6 × 0.4 = 24%

(ii) 24% of 160 = 38.4 ≈ 38 girls

The exact answer 38.4 is not a whole number, so the given data is approximate.

9SP of 3 pencils = CP of 5 pencils. Profit or loss? Find the percentage.

SP of 1 pencil = 53 CP. Profit = 23 CP.

Profit % = 23 × 100 ≈ 66.67% profit

10Bus fares increased by 3% last year and 4% this year. Find the overall increase.

1.03 × 1.04 = 1.0712.

Overall increase = 7.12%

11Length of a rectangle is increased by 10% and the area is unchanged. By what percentage does the breadth decrease?

New breadth = 11.1 = 1011 of old breadth.

Decrease = 111 × 100 = 9111% ≈ 9.09%

12A 65 g chips packet has Potato 70%, Vegetable oil 24%, Salt 3%, Spices 3%. Find the weight of each.

Potato = 45.5 g, Vegetable oil = 15.6 g, Salt = 1.95 g, Spices = 1.95 g

13Shop A: Buy 1 get 1 free. Shop B: Buy 2 get 1 free. Shop C: Buy 3 get 1 free. Price of one item = ₹100.

(i) Effective price: A = ₹200 ÷ 2 = ₹50, B = ₹200 ÷ 3 ≈ ₹66.67, C = ₹300 ÷ 4 = ₹75. Cheapest to costliest: A, B, C.

(ii) Discount: A = 12 = 50%, B = 13 ≈ 33.33%, C = 14 = 25%.

(iii) For 4 items: A costs ₹200, B costs ₹300 (buy 2 get 1, plus 1 more), C costs ₹300. Choose Shop A.

14In a room of 100 people, 99% are left-handed. How many left-handed people must leave to bring it down to 98%?

There is 1 right-handed person. For this person to be 2% of the room, the room must have 50 people.

So 50 left-handed people must leave.

15Based on the graph on computer use, which statements are valid?

(i) Valid. The twenties have the highest bars (26% and 37%).

(ii) Valid. The female bar is shorter in every age group.

(iii) Not valid. The graph shows percentages, not the number of people.

(iv) Not valid. Only 14% of women and 25% of men in their thirties can use computers.

(v) Valid. Only 2% of women and 4% of men aged 60+ can.

(vi) Not valid. Only 26% to 37% in their twenties can.

Puzzle: Peaceful Knights

1Place 8 knights on the board so that no knight attacks another.

A knight always moves from a light square to a dark square, or from dark to light.

So if all 8 knights are placed on squares of the same colour, no knight can attack another. For example, place them on 8 dark squares of the first two rows.

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