NCERT Solutions for Class 8 Maths Chapter 8 Fractions in Disguise – step-by-step answers to every exercise and in-text question of the chapter, as per the latest NCERT syllabus for 2027. Read them online or download the free question-answer PDF.
The opening chapter of Ganita Prakash Part II shows that a percentage is nothing new — it is a fraction wearing a denominator of 100. From that one idea the chapter builds comparison of proportions, percentage increase and decrease, profit, loss, discount, GST, compound growth and depreciation.
NCERT Solutions for Class 8 Maths Chapter 8 Fractions in Disguise PDF
All the answers are on this page below. The PDF also explains every answer – why it is right and how to check it – so download it for revision.
NCERT Solutions for Class 8 Maths Chapter 8 Fractions in Disguise – Questions and Answers
All 85 questions of the chapter (textbook pages Part II, 1 – 32) are answered below, in the order of the book – in-text questions, activities and the end-of-chapter exercises. Each answer is written the way you would write it in the exam.
In-text Questions — Page 2Section 1.1 Fractions as Percentages
Q1Can you tell what percentage of the colour was made using yellow?
Answer
25%.
The mixture is made of red paint and yellow paint only, so the two shares must together make up the whole mixture.
In-text Questions — Page 3Section 1.1 Fractions as Percentages
Q1Try completing Method 3 by filling the boxes.
Method 3, page 3 — the bar is marked 0, 1⁄5, …, 5⁄5 = 1 along the top and 0% … 100% along the bottom. The four empty boxes are the ones to fill.
Answer
The boxes are 20%, 40%, 60% and 80%, and the savings for the canvas is 40% of the prize money.
Fraction of prize money
0
1/5
2/5
3/5
4/5
5/5 = 1
Percentage
0%
20%
40%
60%
80%
100%
Surya saves 2/5 of the money, and the shaded band “Savings for canvas” reaches the mark 2/5. So the savings are 40%.
Figure it Out — Pages 3–4Section 1.1 Fractions as Percentages
Q1Express the following fractions as percentages. (i) 3/5 (ii) 7/14 (iii) 9/20 (iv) 72/150 (v) 1/3 (vi) 5/11
Answer
A fraction is of one unit and a percentage is per 100 units, so multiply each fraction by 100.
Q2Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are white? (i) 10% (ii) 15% (iii) 25% (iv) 60% (v) 40% (vi) None of these
Answer
(iv) 60%.
White marbles as a fraction of all marbles = 15/25
15/25 = 15 × 4 / 25 × 4 = 60/100
= 60%
Q3In a school, 15 of the 80 students come to school by walking. What percentage of the students come by walking?
Answer
Fraction who walk = 15/80 = 3/16
Percentage = 3/16 × 100 = 300/16
= 18.75%
Q4A group of friends is participating in a long-distance run. The positions of each of them after 15 minutes are shown in the following picture. Match (among the given options) what percentage of the race each of them has approximately completed.
Page 4 — where A, B, C and D had reached after 15 minutes, with the six options printed below the track.
Answer
Reading each runner’s position as a fraction of the whole Start-to-Finish line:
Runner
Position on the line
Matching option
A
about a third of the way
38%
B
just past the halfway mark
55%
C
about three-quarters of the way
72%
D
almost at the Finish
93%
The four runners placed on the Start-to-Finish line, with the option each one matches.
The options 20% and 84% are not used — no runner is that close to the Start, and none sits between C and D.
Q5Pairs of quantities are shown below. Identify and write appropriate symbols ‘>’, ‘<’, ‘=’ in the blanks. Try to do it without calculations. (i) 50% ____ 5% (ii) 5/10 ____ 50% (iii) 3/11 _____ 61% (iv) 30% ____ 1/3
Answer
(i) 50% > 5% — 50 hundredths against 5 hundredths
(ii) 5/10 = 50% — 5/10 = 50/100
(iii) 3/11 < 61% — 3/11 is under 3/10, i.e. under 30%
(iv) 30% < 1/3 — 1/3 = 33⅓%
In-text Questions — Page 4Section 1.1 Fractions as Percentages
Q1Well, if percentages are just a particular type of fraction, why do we need them? Why can’t we just continue using fractions?
Answer
Because fractions with different denominators cannot be compared at a glance, while percentages always share the denominator 100.
Take the book’s own example — sugar makes up 9/34 of Variety 1 and 13/45 of Variety 2.
9/34 × 100 = 26.47%
13/45 × 100 = 28.88%
So Variety 2 is the more sugary one.
Q2If we want to have the same denominator, why choose 100 in particular? Why not 10, 50, 1000, or 43? Think.
Answer
Any denominator would work in principle. 100 is chosen because it balances two things a comparison needs — fine enough detail and easy mental handling.
It fits our number system. Our numerals are base ten, so 10, 100 and 1000 turn straight into decimals: 31% = 31/100 = 0.31. Moving between fraction, decimal and percentage costs nothing.
Per 10 is too coarse. 9/34 would be 2.647 “per decem” — you need decimals before you have even started.
Per 1000 is too heavy. 9/34 becomes 264.7 per mille — accurate, but hard to picture.
43 is awkward. Nothing in our number system makes 43 easy to divide by or to picture.
In-text Questions — Page 7Section 1.2 Percentage of Some Quantity — Free-hand Computations
Math Talk
Q1Try to calculate (without using pen and paper) the indicated percentages of the values shown in the table below. Write your answers in the table.
100
200
50
80
10
35
287
25%
25
10%
20%
5%
The table printed on page 7. The book has filled in only one entry — 25% of 100.
Answer
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
None of these needs written work if you use the friendly fractions behind the percentages:
10% of a number = the number ÷ 10 (shift the decimal point one place left)
20% = double the 10% value
5% = half the 10% value
25% = one quarter, i.e. halve, then halve again
So for 287: 10% is 28.7 → 20% is 57.4 → 5% is 14.35 → 25% is 57.4 + 14.35 = 71.75.
Q2How did you find these values? Discuss the methods with the class. Do you find anything interesting in the table?
100
200
50
80
10
35
287
25%
25
10%
20%
5%
The table from page 7 — the same one you have just filled in.
Answer
Several patterns run through the table:
The 20% row is exactly double the 10% row, in every column.
Q3Similarly, to find 10% of a quantity, what decimal value should be multiplied?
Answer
0.1
10% = 10/100 = 1/10 = 0.1
So 10% of 350 = 0.1 × 350 = 35
Q4Complete the following table:
Per cent
50%
100%
25%
75%
10%
1%
5%
43%
Fraction
50⁄100
Decimal
0.5
The table printed on page 8. The first column is filled in as an example.
Answer
Per cent
50%
100%
25%
75%
10%
1%
5%
43%
Fraction
50/100 = 1/2
100/100 = 1
25/100 = 1/4
75/100 = 3/4
10/100 = 1/10
1/100
5/100 = 1/20
43/100
Decimal
0.5
1.0
0.25
0.75
0.1
0.01
0.05
0.43
Q5Activity: How Close Can You Get? Make a pair. Each of you choose a number. Suppose, the numbers chosen are a and b. Share your numbers with each other. Both of you should estimate the percentage equivalent to the fraction a/b (where a < b) and announce your answers by a fixed time, say, 5 seconds. The one whose estimate is the closest wins this round. Play this for 10 rounds.
Answer
The skill this game trains is bracketing — trapping the answer between two percentages you already know, instead of dividing.
Suppose the pair chooses a = 7, b = 24.
Half of 24 is 12, and 7 < 12, so 7/24 < 50%
A quarter of 24 is 6, and 7 > 6, so 7/24 > 25%
A third of 24 is 8, and 7 < 8, so 7/24 < 33.3%
So the answer lies between 25% and 33%, nearer the top → estimate about 29%
(the exact value is 700/24 = 29.17%)
Q1On Day 7, he achieved 150% of his target. On Day 8, he achieved 210% of his target. Find the sales made on these days.
Answer
This reverses the earlier question: the percentage is known, the sales amount is not.
Day 7: 150% of 5000 = 150/100 × 5000 = 1.5 × 5000 = ₹7500
Day 8: 210% of 5000 = 210/100 × 5000 = 2.1 × 5000 = ₹10,500
Q2Complete the table below. Mark the approximate locations in the following diagram.
Percent
90%
110%
200%
250%
15%
173%
358%
28.9%
305%
Fraction
Decimal
The table printed on page 12.
Page 12 — the diagram to mark on. The solid block is 0% to 100%; the dashed blocks carry on to 400%. The arrow shows where 90% has been marked as an example.
Answer
Per cent
90%
110%
200%
250%
15%
173%
358%
28.9%
305%
Fraction
90/100 = 9/10
110/100 = 11/10
200/100 = 2
250/100 = 5/2
15/100 = 3/20
173/100
358/100 = 179/50
289/1000
305/100 = 61/20
Decimal
0.9
1.1
2.0
2.5
0.15
1.73
3.58
0.289
3.05
All nine percentages marked on the diagram. Each mark sits at its decimal value: 100% is at (1), 200% at (2), and so on.
On the diagram the marks sit at the decimal values, since 100% is the point 1:
15% just after the start | 28.9% about a third of the way to (1)
90% just before (1) | 110% just after (1)
173% between (1) and (2), nearer (2) | 200% exactly at (2)
250% halfway between (2) and (3) | 305% just after (3)
358% between (3) and (4), nearer (4)
Figure it Out — Pages 12–14Section 1.2 Percentage of Some Quantity
Math Talk
Q1Estimate first before making any computations to solve the following questions. Try different methods including mental computations. Find the missing numbers. The first problem has been worked out.The three pairs of bar models on pages 12–13. In each pair both bars stand for 100%. The arrow under the left bar marks one part; the arrow under the right bar marks some of the parts, and the number at the right end is what the whole right bar stands for.
Answer
In each pair, the left bar tells you what one part is worth as a percentage, and the right bar uses that to find a value.
Parts in the bar
One part
Shaded portion
Total
Missing value
(i)
5
20%
4 parts = 80%
75
60 (worked out)
(ii)
10
10%
6 parts = 60%
90
54
(iii)
4
25%
3 parts = 75%
140
105
(ii) One part = 100% ÷ 10 = 10%
Shaded 6 parts = 60% of 90 = 0.6 × 90 = 54
(iii) One part = 100% ÷ 4 = 25%
Shaded 3 parts = 75% of 140 = 3/4 × 140 = 105
Q2Find the value of the following and also draw their bar models. (i) 25% of 160 (ii) 16% of 250 (iii) 62% of 360 (iv) 140% of 40 (v) 1% of 1 hour (vi) 7% of 10 kg
Answer
(i) 25% of 160 = 1/4 × 160 = 40
(ii) 16% of 250 = 0.16 × 250 = 40
(iii) 62% of 360 = 0.62 × 360 = 223.2
(iv) 140% of 40 = 1.4 × 40 = 56
(v) 1% of 1 hour = 0.01 × 60 min = 0.6 min = 36 seconds
(vi) 7% of 10 kg = 0.07 × 10 kg = 0.7 kg = 700 g
Bar models for (i)–(vi). The whole bar is 100% of the quantity; the shaded part is the percentage asked for. In (iv) the bar runs past 100% because 140% is more than the whole.
Q3Surya made 60 ml of deep orange paint, how much red paint did he use if red paint made up 3/4 of the deep orange paint?
Answer
Red = 3/4 of 60 ml
= 3 × 60 / 4 = 180/4
= 45 ml
The remaining 15 ml is yellow — which matches the 25% found earlier.
Q4Pairs of quantities are shown below. Identify and write appropriate symbols ‘>’, ‘<’, ‘=’ in the boxes. Visualising or estimating can help. Compute only if necessary or for verification. (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 __ 40% of 50 (v) 45% of 200 __ 10% of 490 (vi) 30% of 80 __ 24% of 64
Answer
(i) 50% of 510 < 50% of 515 — same percentage, bigger base
(ii) 37% of 148 < 73% of 148 — same base, bigger percentage
(iii) 29% of 43 < 92% of 110 — smaller share of a smaller base
(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
Q5Fill in the blanks appropriately: (i) 30% of k is 70, 60% of k is _____, 90% of k is _____, 120% of k is ______. (ii) 100% of m is 215, 10% of m is _____, 1% of m is ______, 6% of m is ______. (iii) 90% of n is 270, 9% of n is ______, 18% of n is _____, 100% of n is ______. (iv) Make 2 more such questions and challenge your peers.
Answer
(i) 60% is twice 30% → 140
90% is three times 30% → 210
120% is four times 30% → 280
(ii) 10% is one tenth of 100% → 21.5
1% is one hundredth of 100% → 2.15
6% = 6 × 1% → 12.9
(iii) 9% is one tenth of 90% → 27
18% is twice 9% → 54
100% = 90% + 10%, and 10% = 30 → 300
(iv) Sample answer — two questions of the same kind:
40% of t is 96. Find 10% of t, 25% of t and 150% of t. (24, 60, 360)
75% of w is 45. Find 25% of w, 100% of w and 5% of w. (15, 60, 3)
Q6Fill in the blanks: (i) 3 is ____ % of 300. (ii) _____ is 40% of 4. (iii) 40 is 80% of _____.
Answer
(i) 3/300 × 100 = 1 → 3 is 1% of 300
(ii) 40% of 4 = 0.4 × 4 = 1.6
(iii) 80% of x = 40 → x = 40/0.8 = 50
Q7Is 10% of a day longer than 1% of a week? Create such questions and challenge your peers.
Answer
Yes — 10% of a day is longer.
1 day = 24 hours = 1440 minutes
10% of a day = 0.1 × 1440 = 144 minutes = 2 hours 24 min
1 week = 7 × 1440 = 10,080 minutes
1% of a week = 0.01 × 10,080 = 100.8 minutes = 1 hour 40.8 min
144 min > 100.8 min
Q8Mariam’s farm has a peculiar bull. One day she gave the bull 2 units of fodder and the bull ate 1 unit. The next day, she gave the bull 3 units of fodder and the bull ate 2 units. The day after, she gave the bull 4 units and the bull ate 3 units. This continued, and on the 99th day she gave the bull 100 units and the bull ate 99 units. Represent these quantities as percentages. This task can be distributed among the class. What do you observe?
Answer
On day n she gives (n + 1) units and the bull eats n units, so the percentage eaten is n/(n + 1) × 100.
Day
Given
Eaten
Fraction eaten
Percentage eaten
Percentage left
1
2
1
1/2
50%
50%
2
3
2
2/3
66.67%
33.33%
3
4
3
3/4
75%
25%
4
5
4
4/5
80%
20%
9
10
9
9/10
90%
10%
19
20
19
19/20
95%
5%
49
50
49
49/50
98%
2%
99
100
99
99/100
99%
1%
What we observe: the percentage eaten climbs steadily — 50%, 66.67%, 75%, 80%, … , 99% — but it never reaches 100%.
Q9Workers in a coffee plantation take 18 days to pick coffee berries in 20% of the plantation. How many days will they take to complete the picking work for the entire plantation, assuming the rate of work stays the same? Why is this assumption necessary?
Answer
20% of the plantation takes 18 days
100% is 5 times 20%
Time for the whole plantation = 5 × 18 = 90 days
Why the assumption is necessary: multiplying the days by 5 because the area is 5 times bigger only works if every additional stretch of the plantation costs the same amount of time as the first. That needs the number of workers, their hours, the density of berries and the terrain to stay the same throughout.
Q10The badminton coach has planned the training sessions such that the ratio of warm up : play : cool down is 10% : 80% : 10%. If he wants to conduct a training of 90 minutes. How long should each activity be done?
Answer
Warm up = 10% of 90 min = 0.1 × 90 = 9 minutes
Play = 80% of 90 min = 0.8 × 90 = 72 minutes
Cool down = 10% of 90 min = 9 minutes
Q11An estimated 90% of the world’s population lives in the Northern Hemisphere. Find the (approximate) number of people living in the Northern Hemisphere based on this year’s worldwide population.
Answer
The chapter itself gives the world population in 2025 as about 8.2 billion.
90% of 8.2 billion = 0.9 × 8.2
= 7.38 billion
≈ 7.4 billion people (about 738 crore)
That leaves roughly 0.8 billion — about 82 crore people — in the Southern Hemisphere.
Q12A recipe for the dish, halwa, for 4 people has the following ingredients in the given proportions — Rava: 40%, Sugar: 40%, and Ghee: 20%. (i) If you want to make halwa for 8 people, what is the proportion of each of the above ingredients? (ii) If the total weight of the ingredients is 2 kg, how much rava, sugar and ghee are present?
Answer
(i) The proportions do not change: Rava 40%, Sugar 40%, Ghee 20%.
You need twice as much of everything, but doubling all three amounts leaves each one the same share of the total.
(ii) Total = 2 kg = 2000 g.
Rava = 40% of 2000 g = 800 g
Sugar = 40% of 2000 g = 800 g
Ghee = 20% of 2000 g = 400 g
Check: 800 + 800 + 400 = 2000 g ✓
In-text Questions — Page 15Section 1.3 Using Percentages — Know Your Contents (KYC)
Q1They are at a shop to buy badam drink mix. They are looking at two products and wondering which has a larger share of badam. Can you figure it out? Which product uses a smaller proportion of food chemicals? [DEF: sugar 99 g, milk solids 30 g, badam powder 12 g, food chemicals 9 g, total weight 150 g. Zacni: sugar 272 g, milk solids 64 g, badam powder 40 g, food chemicals 24 g, total weight 400 g.]
Answer
Zacni has the larger share of badam. Neither uses a smaller proportion of food chemicals — the two are equal.
Badam in DEF = 12/150 × 100 = 8%
Badam in Zacni = 40/400 × 100 = 10%
Food chemicals in DEF = 9/150 × 100 = 6%
Food chemicals in Zacni = 24/400 × 100 = 6%
Q2Complete this table by calculating the percentages to answer the questions: [DEF: sugar 99 g, milk solids 30 g, badam powder 12 g, food chemicals 9 g, total weight 150 g. Zacni: sugar 272 g, milk solids 64 g, badam powder 40 g, food chemicals 24 g, total weight 400 g.]
Sugar
Milk Solids
Badam Powder
Food Chemicals
DEF
66%
Zacni
The table printed on page 15. DEF’s sugar entry is filled in as the worked example.
Q2Shambhavi owns a stationery shop. She procures 200 page notebooks at ₹36 per book. She sells them with a profit margin of 20%. Find the selling price.
Q4Could we have just calculated the loss percentage per kg instead? Would it be the same? [Raghu bought rice at ₹35 per kg and sold 10 kg for ₹300, a loss of ₹50 on ₹350, i.e. 14.28%.]
Answer
Yes — the loss percentage per kilogram is exactly the same, 14.28%.
Per kg: CP = ₹35, SP = 300/10 = ₹30
Loss per kg = 35 – 30 = ₹5
Loss % = 5/35 × 100 = 500/35 = 14.28% (14.2857…%)
For 10 kg: loss = ₹50 on a cost of ₹350
Loss % = 50/350 × 100 = 14.28%
In-text Questions — Page 18Section 1.3 — Profit, Loss and Discount
Q1Due to heavy rains, Snehal could not transport strawberries to Hyderabad from his farm in Panchgani. He sells some of his stock at ₹80 per kg with a 12% loss. What is the cost price?
Answer
A 12% loss means the selling price is 100% – 12% = 88% of the cost price
0.88 × CP = 80
CP = 80/0.88 = 8000/88 = 1000/11
= ₹90.91 per kg (90.909…, to the nearest paisa)
Check: 12% of 90.91 = ₹10.91, and 90.91 – 10.91 = ₹80. ✓
Q2A utensil store is offering a 35% discount on the cooker with an MRP ₹1800. What is the selling price? If the cost price was ₹900, what is the percentage profit made after the sale?
Answer
Selling price: a 35% discount leaves 65% of the MRP
SP = 0.65 × 1800 = ₹1170
Q1Check if the calculations are correct in the bill shown.
XY ELECTRICALS — SALES RECEIPT Date: 06/07/2025
Item
Qty.
Price
Amount
CFL Bulb
3
₹150.00
₹450.00
Sub Total
₹450.00
CGST 9%
₹40.50
SGST 9%
₹40.50
TOTAL
₹531.00
The bill printed on page 19.
Answer
Every figure on the bill is correct.
Line on the bill
Check
Verdict
Amount ₹450.00
3 × 150 = 450
correct
Sub Total ₹450.00
only one item
correct
CGST 9% ₹40.50
0.09 × 450 = 40.50
correct
SGST 9% ₹40.50
0.09 × 450 = 40.50
correct
TOTAL ₹531.00
450 + 40.50 + 40.50 = 531
correct
Total tax = 40.50 + 40.50 = ₹81
Tax as a percentage of the sub total = 81/450 × 100 = 18%
Q2You may share any bills you have at home with the class. Observe the different elements present in the bills. Are there any similarities or differences in these bills?
Answer
(after comparing a grocery bill, a restaurant bill and an electricity bill):
What is usually the same:
The seller’s name, the date and a bill number.
Item, quantity, rate per unit, and amount (= quantity × rate).
A sub total, then tax lines, then a grand total.
The GST shown as two equal halves, CGST and SGST.
What differs:
The GST rate is not the same for every item — for example 5% on many everyday goods, 18% on many others, and none on many fresh foods.
Some bills show a discount line before the tax, some after.
Some prices are marked “inclusive of all taxes”, so no tax line appears at all.
Figure it Out — Pages 19–20Section 1.3 — Profit, Loss, Discount and Percentage Change
Q1If a shopkeeper buys a geometry box for ₹75 and sells it for ₹110, what is his profit margin with respect to the cost?
Q2I am a carpenter and I make chairs. The cost of materials for a chair is ₹475 and I want to have a profit margin of 50%. At what price should I sell a chair?
Answer
SP = CP + 50% of CP = 150% of CP
= 1.5 × 475
= ₹712.50
Equivalently: profit = half of 475 = ₹237.50, and 475 + 237.50 = ₹712.50.
Q3The total sales of a company (also called revenue) was ₹2.5 crore last year. They had a healthy profit margin of 25%. What was the total expenditure (costs) of the company last year?
Answer
Total expenditure = ₹2 crore (taking the profit margin on the cost, as in Questions 1 and 2).
Revenue = cost + 25% of cost = 125% of the expenditure.
If the margin is taken on the revenue instead: profit = 25% of ₹2.5 crore = ₹0.625 crore, so expenditure = ₹1.875 crore.
Q4A clothing shop offers a 25% discount on all shirts. If the original price of a shirt is ₹300, how much will Anwar have to pay to buy this shirt?
Answer
He pays 100% – 25% = 75% of the marked price
= 3/4 × 300
= ₹225
Or: discount = 25% of 300 = ₹75, so he pays 300 – 75 = ₹225.
Q5The petrol price in 2015 was ₹60 and ₹100 in 2025. What is the percentage increase in the price of petrol? (i) 50% (ii) 40% (iii) 60% (iv) 66.66% (v) 140% (vi) 160.66%
Q6Samson bought a car for ₹4,40,000 after getting a 15% discount from the car dealer. What was the original price of the car? [Printed as question 3 on page 20; the book restarts its numbering there.]
Answer
After a 15% discount he pays 85% of the original price
0.85 × original = 4,40,000
original = 4,40,000 / 0.85 = 44,00,000/8.5
= ₹5,17,647 (5,17,647.06 to the nearest paisa)
Check: 15% of 5,17,647 = ₹77,647, and 5,17,647 – 77,647 = ₹4,40,000. ✓
Q71600 people voted in an election and the winner got 500 votes. What percent of the total votes did the winner get? Can you guess the minimum number of candidates who stood for the election? [Printed as question 4 on page 20.]
Votes left for everyone else = 1600 – 500 = 1100
No other candidate can have 500 or more (or they would not have lost)
With 3 candidates: 2 others share 1100 votes
→ at least one of them gets 1100 ÷ 2 = 550 > 500 — impossible
With 4 candidates: 3 others share 1100 votes
→ e.g. 400, 350, 350 — all below 500. Possible.
Q8The price of 1 kg of rice was ₹38 in 2024. It is ₹42 in 2025. What is the rate of inflation? (Inflation is the percentage increase in prices.) [Printed as question 5 on page 20.]
Q9A number increased by 20% becomes 90. What is the number? [Printed as question 6 on page 20.]
Answer
Let the number be x. Increasing it by 20% makes it 120% of x
1.2x = 90
x = 90/1.2 = 900/12
= 75
Check: 20% of 75 = 15, and 75 + 15 = 90. ✓
Q10A milkman sold two buffaloes for ₹80,000 each. On one of them, he made a profit of 5% and on the other a loss of 10%. Find his overall profit or loss. [Printed as question 7 on page 20.]
Answer
He made an overall loss of about ₹5,079, which is about 3.08%.
The two selling prices are equal, but the two cost prices are not — so work each one out first.
Total cost price = 76,190.48 + 88,888.89 = ₹1,65,079.37
Total selling price = 80,000 + 80,000 = ₹1,60,000
Loss = 1,65,079.37 – 1,60,000 = ₹5,079.37
Loss % = 5,079.37 / 1,65,079.37 × 100 = 3.08%
Q11The population of elephants in a national park increased by 5% in the last decade. If the population of the elephants last decade is p, the population now is (i) p × 0.5 (ii) p × 0.05 (iii) p × 1.5 (iv) p × 1.05 (v) p + 1.50 [Printed as question 8 on page 20.]
Answer
(iv) p × 1.05.
Population now = p + 5% of p
= p + 0.05p
= p(1 + 0.05)
= 1.05p
Q12Which of the following statement(s) mean the same as — “The demand for cameras has fallen by 85% in the last decade”? (i) The demand now is 85% of the demand a decade ago. (ii) The demand a decade ago was 85% of the demand now. (iii) The demand now is 15% of the demand a decade ago. (iv) The demand a decade ago was 15% of the demand now. (v) The demand a decade ago was 185% of the demand now. (vi) The demand now is 185% of the demand a decade ago. [Printed as question 9 on page 20.]
Answer
Only (iii) means the same thing.
Let the demand a decade ago be d.
Fallen by 85% → the fall is 0.85d
Demand now = d – 0.85d = 0.15d = 15% of d
Statement
What it says
Same?
(i)
now = 0.85d — a fall of only 15%
No
(ii)
d = 0.85 × now, so demand has risen
No
(iii)
now = 0.15d
Yes
(iv)
d = 0.15 × now, so demand has risen sharply
No
(v)
d = 1.85 × now, a fall of about 46%
No
(vi)
now = 1.85d, a rise of 85%
No
Figure it Out — Pages 22–24Section 1.3 — Growth and Compounding
Math Talk
Q1Bank of Yahapur offers an interest of 10% p.a. Compare how much one gets if they deposit ₹20,000 for a period of 2 years with compounding and without compounding annually.
Answer
Without compounding — the principal stays ₹20,000 every year
Interest each year = 10% of 20,000 = ₹2000
Interest for 2 years = 2 × 2000 = ₹4000
Amount = 20,000 + 4000 = ₹24,000
With compounding — each year’s interest joins the principal
After year 1: 20,000 × 1.1 = ₹22,000
After year 2: 22,000 × 1.1 = ₹24,200
(or 20,000 × 1.1² = 20,000 × 1.21 = 24,200)
Compounding gives ₹200 more.
Q2Bank of Wahapur offers an interest of 5% p.a. Compare how much one gets if one deposits ₹20,000 for a period of 4 years with compounding and without compounding annually.
Answer
Without compounding
Interest each year = 5% of 20,000 = ₹1000
Interest for 4 years = 4 × 1000 = ₹4000
Amount = ₹24,000
With compounding
Amount = 20,000 × (1.05)⁴
1.05² = 1.1025, so 1.05⁴ = 1.1025 × 1.1025 = 1.21550625
= 20,000 × 1.21550625 = ₹24,310.125 = ₹24,310.13 to the nearest paisa
Year
Opening amount
Interest at 5%
Closing amount
1
₹20,000
₹1000
₹21,000
2
₹21,000
₹1050
₹22,050
3
₹22,050
₹1102.50
₹23,152.50
4
₹23,152.50
₹1157.63
₹24,310.13
Compounding gives about ₹310 more.
Q3Do you observe anything interesting in the solutions of the two questions above? Share and discuss.
Answer
Without compounding both deposits give exactly the same amount, ₹24,000 — but with compounding they do not.
10% for 2 years
5% for 4 years
Total simple interest
₹4000
₹4000
Amount, no compounding
₹24,000
₹24,000
Amount, compounded
₹24,200
₹24,310.13
Without compounding the amount is p(1 + rt), and rt is the same for both:
0.10 × 2 = 0.20 and 0.05 × 4 = 0.20
With compounding the amount is p(1 + r)ᵗ, and these differ:
(1.10)² = 1.2100 but (1.05)⁴ = 1.2155
Q4Jasmine invests amount ‘p’ for 4 years at an interest of 6% p.a. Which of the following expression(s) describe the total amount she will get after 4 years when compounding is not done? (i) p × 6 × 4 (ii) p × 0.6 × 4 (iii) p × (0.6/100) × 4 (iv) p × (0.06/100) × 4 (v) p × 1.6 × 4 (vi) p × 1.06 × 4 (vii) p + (p × 0.06 × 4)
Answer
Only (vii).
Without compounding, amount = p + interest
Interest = p × r × t = p × 0.06 × 4 = 0.24p
Amount = p + 0.24p = 1.24p — which is what (vii) says
Option
What it actually computes
(i) p × 6 × 4
24p — treats 6% as the number 6
(ii) p × 0.6 × 4
2.4p — uses 60%, not 6%
(iii) p × (0.6/100) × 4
0.024p — a hundred times too small
(iv) p × (0.06/100) × 4
0.0024p — divides by 100 twice over
(v) p × 1.6 × 4
6.4p
(vi) p × 1.06 × 4
4.24p — multiplies the whole amount by 4 instead of the interest
(vii) p + (p × 0.06 × 4)
1.24p ✓
Q5The post office offers an interest of 7% p.a. How much interest would one get if one invests ₹50,000 for 3 years without compounding? How much more would one get if it was compounded?
Answer
Without compounding
Interest = p × r × t = 50,000 × 0.07 × 3
= ₹10,500
Extra from compounding = 11,252.15 – 10,500 = ₹752.15
Q6Giridhar borrows a loan of ₹12,500 at 12% per annum for 3 years without compounding and Raghava borrows the same amount for the same time period at 10% per annum, compounded annually. Who pays more interest and by how much?
Q7Consider an amount ₹1000. If this grows at 10% p.a., how long will it take to double when compounding is done vs. when compounding is not done? Is compounding an example of exponential growth and not-compounding an example of linear growth?
Answer
Without compounding: 10 years. With compounding: 8 years. And yes — compounding is exponential growth, and not-compounding is linear growth.
No compounding: interest is ₹100 every year
Amount = 1000 + 100t. Doubling needs 100t = 1000
t = 10 years
With compounding: Amount = 1000 × (1.1)ᵗ
(1.1)⁷ = 1.9487 → ₹1948.72 — not yet doubled
(1.1)⁸ = 2.1436 → ₹2143.59 — past ₹2000
So it doubles during the 8th year → 8 years on annual compounding
End of year
1
2
4
6
7
8
10
No compounding (₹)
1100
1200
1400
1600
1700
1800
2000
Compounded (₹)
1100
1210
1464.10
1771.56
1948.72
2143.59
2593.74
Q8The population of a city is rising by about 3% every year. If the current population is 1.5 crore, what is the expected population after 3 years?
Answer
A yearly percentage rise compounds, because each year’s growth is measured on the population at the start of that year.
Population after 3 years = 1.5 crore × (1.03)³
1.03² = 1.0609, so 1.03³ = 1.0609 × 1.03 = 1.092727
= 1.5 × 1.092727 crore = 1.6390905 crore
≈ 1.64 crore (about 1,63,90,905 people)
Q9In a laboratory, the number of bacteria in a certain experiment increases at the rate of 2.5% per hour. Find the number of bacteria at the end of 2 hours if the initial count is 5,06,000.
After 1 hour: 5,06,000 × 1.025 = 5,18,650
After 2 hours: 5,18,650 × 1.025 = 5,31,616.25
In-text Questions — Page 24Section 1.3 — Growth and Compounding
Math Talk
Q1Suppose we want to know the expression/formula to find the total interest amount gained at the end of the maturity period. What would be the formula for each of the two options?
Answer
Interest is whatever the deposit has gained above the principal, so in both cases subtract p from the maturity amount.
Amount at maturity
Total interest
Without compounding
p(1 + rt)
p(1 + rt) – p = prt
With compounding
p(1 + r)t
p(1 + r)t – p = p[(1 + r)t – 1]
Check with the chapter’s numbers — p = 6000, r = 0.1, t = 3
Without compounding: prt = 6000 × 0.1 × 3 = ₹1800 ✓ (7800 – 6000)
With compounding: 6000[(1.1)³ – 1] = 6000 × 0.331 = ₹1986 ✓ (7986 – 6000)
In-text Questions — Page 25Section 1.3 — Tricky Percentages: Would You Rather?
Math Talk
Q1You have won a contest. The organisers offer you two options to choose from: Option A: You deposit ₹100 and you get back ₹300. Option B: You deposit ₹1000 and you get back ₹1500. What is the percentage gain each option gives? You can choose any option only once. Which option would you choose? Why?
Answer
Option A: gain = 300 – 100 = ₹200 on a deposit of ₹100
Percentage gain = 200/100 × 100 = 200%
Option B: gain = 1500 – 1000 = ₹500 on a deposit of ₹1000
Percentage gain = 500/1000 × 100 = 50%
Which to choose? Option A wins on percentage, but Option B puts more money in your pocket — ₹500 against ₹200. Since you may use an option only once, Option B is the better choice, provided you have ₹1000 to deposit. If you only have ₹100, Option B is not available at all and A is the answer.
Q1A provision store is offering a stock clearance sale. Customers can choose one of the two options — 20% discount or ₹50 discount — for any purchase above ₹150. Which option would you choose if you want to: (i) buy items worth ₹180 (ii) buy items worth ₹225 (iii) buy items worth ₹300
Answer
Purchase
20% discount
₹50 discount
Better choice
You pay
(i) ₹180
₹36
₹50
₹50 off
₹130
(ii) ₹225
₹45
₹50
₹50 off
₹175
(iii) ₹300
₹60
₹50
20% off
₹240
The two offers are worth the same at one particular bill:
20% of x = 50
0.2x = 50 → x = ₹250
Q2A bakery called Cakely is offering a 30% + 20% discount on all cakes. Another bakery called Cakify is offering a 50% discount on all cakes. Would you rather choose Cakely or Cakify if you want the cheaper cost?
Answer
Cakify — its single 50% discount is genuinely bigger than Cakely’s “30% + 20%”.
For a cake worth ₹200: Cakely: 30% off ₹200 → 200 – 60 = ₹140
then 20% off ₹140 → 140 – 28 = ₹112
Cakify: 50% off ₹200 → ₹100
Cakely’s offer is really a single discount of 44%, not 50%:
Price paid = 0.70 × 0.80 × MP = 0.56 × MP
So the customer pays 56% and the discount is 44%
In-text Questions — Page 27Section 1.3 — A Mishap
Try This
Q1Ariba and Arun have some marbles. Ariba says, “The number of marbles with me is 120% of the marbles Arun has”. What would be an appropriate statement Arun could make comparing the number of marbles he has with Ariba’s?
Answer
Arun could say: “The number of marbles with me is 83⅓% of the marbles Ariba has” — or, equivalently, “I have 16⅔% fewer marbles than Ariba.”
Let Arun have n marbles. Then Ariba has 1.2n.
Arun as a fraction of Ariba = n / 1.2n = 1/1.2 = 10/12 = 5/6
5/6 × 100 = 500/6 = 83.33%
Shortfall = 100% – 83.33% = 16.67% (exactly 16⅔%)
Figure it Out — Pages 28–30Section 1.3 Using Percentages — chapter-end set
Try This
Q1The population of Bengaluru in 2025 is about 250% of its population in 2000. If the population in 2000 was 50 lakhs, what is the population in 2025?
Answer
Population in 2025 = 250% of 50 lakhs
= 2.5 × 50 lakhs
= 125 lakhs = 1.25 crore
Q2The population of the world in 2025 is about 8.2 billion. The populations of some countries in 2025 are given. Match them with their approximate percentage share of the worldwide population. [Hint: Writing these numbers in the standard form and estimating can help]. [Germany 83 million, India 1.46 billion, Bangladesh 175 million, USA 347 million; options 13%, 8%, 18%, 10%, 1%, 35%, 2%, 2%, 0.1%.]
Answer
Write every population in billions first, then divide by 8.2.
Country
Population
In billions
Share of 8.2 billion
Option
Germany
83 million
0.083
0.083/8.2 = 1.01%
1%
India
1.46 billion
1.46
1.46/8.2 = 17.8%
18%
Bangladesh
175 million
0.175
0.175/8.2 = 2.13%
2%
USA
347 million
0.347
0.347/8.2 = 4.23%
about 4% — no matching option
Estimating instead of dividing:
1% of 8.2 billion = 82 million → Germany, at 83 million, is almost exactly 1%
10% of 8.2 billion = 820 million; India’s 1460 million is nearly twice that → about 18%
Bangladesh’s 175 million is a little over 2 × 82 million → about 2%
USA’s 347 million is a little over 4 × 82 million → about 4%
Q3The price of a mobile phone is ₹8,250. A GST of 18% is added to the price. Which of the following gives the final price of the phone including the GST? (i) 8250 + 18 (ii) 8250 + 1800 (iii) 8250 + 18/100 (iv) 8250 × 18 (v) 8250 × 1.18 (vi) 8250 + 8250 × 0.18 (vii) 1.8 × 8250
Answer
(v) and (vi) — both give the correct final price.
GST = 18% of 8250 = 0.18 × 8250 = ₹1485
Final price = 8250 + 1485 = ₹9735
Q4The monthly percentage change in population (compared to the previous month) of mice in a lab is given: Month 1 change was +5%, Month 2 change was –2%, and Month 3 change was –3%. Which of the following statement(s) are true? The initial population is p. (i) The population after three months was p × 0.05 × 0.02 × 0.03. (ii) The population after three months was p × 1.05 × 0.98 × 0.97. (iii) The population after three months was p + 0.05 – 0.02 – 0.03. (iv) The population after three months was p. (v) The population after three months was more than p. (vi) The population after three months was less than p.
Answer
(ii) and (vi) are true.
+5% → multiply by 1.05
–2% → multiply by 0.98
–3% → multiply by 0.97
Population after 3 months = p × 1.05 × 0.98 × 0.97
1.05 × 0.98 = 1.029
1.029 × 0.97 = 0.99813
= 0.99813p, which is less than p
Statement
Verdict
Reason
(i)
False
0.05 × 0.02 × 0.03 uses the changes, not the multipliers
(ii)
True
each change multiplies the running population
(iii)
False
adds bare decimals to a population count
(iv)
False
0.99813p ≠ p
(v)
False
0.99813p < p
(vi)
True
the population ends 0.187% below where it started
Q5A shopkeeper initially set the price of a product with a 35% profit margin. Due to poor sales, he decided to offer a 30% discount on the selling price. Will he make a profit or a loss? Give reasons for your answer.
Answer
He makes a loss of 5.5%.
Let the cost price be x.
Marked price with a 35% margin = 1.35x
After a 30% discount, he receives 70% of that:
0.70 × 1.35x = 0.945x
0.945x < x, so it is a loss
Loss = x – 0.945x = 0.055x
Loss % = 0.055x / x × 100 = 5.5%
With numbers, if the cost price were ₹1000: marked price ₹1350, discount ₹405, sale price ₹945 — a loss of ₹55, which is 5.5% of ₹1000.
Q6What percentage of area is occupied by the region marked ‘E’ in the figure?
Page 29 — the rectangle drawn on a dot grid and cut into the regions A, B, C, D and E.
Answer
12.5%.
Counting on the dot grid, the whole rectangle is 8 units wide and 8 units tall, so its area is 64 square units.
Region
Shape and size
Area (sq units)
Percentage
A
rectangle 4 × 4
16
25%
B
rectangle 4 × 6
24
37.5%
C
rectangle 4 × 2
8
12.5%
D
half of the lower-left 4 × 4 square
8
12.5%
E
the other half of that square
8
12.5%
Total
8 × 8
64
100%
E = 8/64 × 100 = 1/8 × 100 = 12.5%
The 8 × 8 rectangle on the dot grid. Region E, shaded, is half of the lower-left 4 × 4 square — 8 of the 64 square units.
Q7What is 5% of 40? What is 40% of 5? What is 25% of 12? What is 12% of 25? What is 15% of 60? What is 60% of 15? What do you notice? Can you make a general statement and justify it using algebra, comparing x% of y and y% of x?
Answer
5% of 40 = 0.05 × 40 = 2 40% of 5 = 0.40 × 5 = 2
25% of 12 = 0.25 × 12 = 3 12% of 25 = 0.12 × 25 = 3
15% of 60 = 0.15 × 60 = 9 60% of 15 = 0.60 × 15 = 9
What we notice: in every pair the two answers are equal.
General statement: for any numbers x and y, x% of y = y% of x.
Justification:
x% of y = (x/100) × y = xy/100
y% of x = (y/100) × x = xy/100
Both equal xy/100, so they are equal.
Q8A school is organising an excursion for its students. 40% of them are Grade 8 students and the rest are Grade 9 students. Among these Grade 8 students, 60% are girls. [Hint: Drawing a rough diagram can help]. (i) What percentage of the students going to the excursion are Grade 8 girls? (ii) If the total number of students going to the excursion is 160, how many of them are Grade 8 girls?
Answer
(i) 24%.
Grade 8 students = 40% of all students
Grade 8 girls = 60% of those = 60% of 40%
= 0.60 × 0.40
= 0.24 = 24% of all students
Picture it as a bar: cut the whole school party into 40% and 60%; then cut the 40% strip again, 60% of it being girls. The girls’ piece is 24 out of every 100 students.
(ii)
24% of 160 = 0.24 × 160 = 38.4
So there are about 38 Grade 8 girls. (38.4 is not a whole number, so the percentages in the question must be approximate — there cannot be 0.4 of a student.)
Q9A shopkeeper sells pencils at a price such that the selling price of 3 pencils is equal to the cost of 5 pencils. Does he make a profit or a loss? What is his profit or loss percentage?
Answer
He makes a profit of 66.67% (exactly 66⅔%).
Let the cost price of one pencil be c.
3 × (selling price) = 5c
SP of one pencil = 5c/3
Profit per pencil = 5c/3 – c = 2c/3
Profit % = (2c/3) / c × 100 = 2/3 × 100 = 66.67%
Q10The bus fares were increased by 3% last year and by 4% this year. What is the overall percentage price increase in the last 2 years?
Answer
7.12% — not 7%.
Let the original fare be f.
After last year: f × 1.03
After this year: f × 1.03 × 1.04
1.03 × 1.04 = 1.0712
Overall increase = 1.0712 – 1 = 0.0712 = 7.12%
On a fare of ₹100: ₹100 → ₹103 → ₹107.12.
Q11If the length of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) does the breadth decrease by?
Answer
The breadth decreases by 100/11 % = 9 1/11 % ≈ 9.09% — not 10%.
Let the original length be l and breadth b, so area = lb.
New length = 1.1l. Let the new breadth be b′.
For the area to be unchanged:
1.1l × b′ = lb
b′ = b/1.1 = 10b/11 = 0.9090…b
Q12The percentage of ingredients in a 65 g chips packet is shown in the picture. Find out the weight each ingredient makes up in this packet.
Page 29 — the nutritional information printed on the 65 g chips packet, redrawn.
Answer
Ingredient
Percentage
Working
Weight
Potato
70%
0.70 × 65
45.5 g
Vegetable oil
24%
0.24 × 65
15.6 g
Salt
3%
0.03 × 65
1.95 g
Spices
3%
0.03 × 65
1.95 g
Total
100%
65 g
Quick route: 1% of 65 g = 0.65 g
70% → 70 × 0.65 = 45.5 g | 24% → 24 × 0.65 = 15.6 g | 3% → 1.95 g
Q13Three shops sell the same items at the same price. The shops offer deals as follows: Shop A: “Buy 1 and get 1 free”; Shop B: “Buy 2 and get 1 free”; Shop C: “Buy 3 and get 1 free”. Answer the following: (i) If the price of one item is ₹100, what is the effective price per item in each shop? Arrange the shops from cheapest to costliest. (ii) For each shop, calculate the percentage discount on the items. [Hint: Compare the free items to the total items you receive.] (iii) Suppose you need 4 items. Which shop would you choose? Why?
Answer
(i) Divide what you pay by how many items you take home.
Shop
Items paid for
Items received
You pay
Effective price per item
A
1
2
₹100
₹50
B
2
3
₹200
₹66.67
C
3
4
₹300
₹75
Cheapest to costliest: A, then B, then C.
(ii) The discount is the free item as a share of everything you receive.
Shop A: 1 free out of 2 received = 1/2 = 50%
Shop B: 1 free out of 3 received = 1/3 = 33.33%
Shop C: 1 free out of 4 received = 1/4 = 25%
(iii) For exactly 4 items, choose Shop A.
Shop
How you get 4 items
Total paid
A
buy 2, get 2 free
₹200
B
buy 2 get 1 (3 items), then buy 1 more
₹300
C
buy 3, get 1 free
₹300
Q14In a room of 100 people, 99% are left-handed. How many left-handed people have to leave the room to bring that percentage down to 98%?
Answer
50 left-handed people must leave.
The trick is to follow the person who does not leave.
At the start: 99 left-handed, 1 right-handed, 100 in all
Only left-handed people leave, so that 1 right-handed person stays.
For left-handers to be 98%, right-handers must be 2%.
1 person = 2% of the room
→ the room now holds 1 ÷ 0.02 = 50 people
→ left-handed people now = 50 – 1 = 49
Left-handed people who left = 99 – 49 = 50
Q15Look at the following graph.
Page 30 — “Ability to use computer by age and gender (2023)”, redrawn from the printed bar graph. The subtitle on the graph reads: “The ability to use computers is highest among those in their twenties and teenagers.” Source: NSS Round 79, Comprehensive Annual Modular Survey, National Statistics Office.
Age group
Female
Male
Children
4%
about 5% (not labelled)
Teenage
24%
29%
Twenties
26%
37%
Thirties
14%
25%
Forties
7%
14%
Fifties
4%
9%
Seniors
2%
4%
Based on the graph, which of the following statement(s) are valid? (i) People in their twenties are the most computer-literate among all age groups. (ii) Women lag behind in the ability to use computers across age groups. (iii) There are more people in their twenties than teenagers. (iv) More than a quarter of people in their thirties can use computers. (v) Less than 1 in 10 aged 60 and above can use computers. (vi) Half of the people in their twenties can use computers.
Answer
Valid statements: (i), (ii) and (v).
Statement
Verdict
Reason from the graph
(i)
Valid
Twenties lead in both bars — 26% female and 37% male — above teenagers’ 24% and 29%
(ii)
Valid
the female bar is shorter than the male bar in every one of the seven age groups
(iii)
Not valid
the graph shows percentages within each age group, and says nothing about how many people are in each group
(iv)
Not valid
thirties are 14% (female) and 25% (male); 25% is a quarter exactly, not more, and the female figure is well below
(v)
Valid
seniors are 2% and 4%, both under 10%, i.e. under 1 in 10
(vi)
Not valid
26% and 37% — neither is anywhere near 50%
Puzzle Time — Page 32Peaceful Knights
Q1Place 8 knights on the chess board so that no knight attacks another. A knight moves in an ‘L-shape’. It can move either (a) two steps vertically and one step horizontally, or (b) two steps horizontally and one step vertically. Possible moves of a knight are shown below.
Page 32 — the board printed in the book (10 squares by 10), and a knight with the 8 squares it can move to.
Answer
Put all 8 knights in one row — for example, on the first 8 squares of the top row.
One answer: 8 knights side by side in the top row.
Every knight move goes 1 or 2 rows up or down, so a knight can never land on a square in its own row. So no knight attacks another.