NCERT Solutions Ganita Prakash (Part 2) Chapter 1 –30Section 1.3 Using Percentages — Figure it Out

Book page 28 Updated on2026-09-05

Q1.

The 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

Tip: 250% means “two and a half times”. The increase alone is 150%, i.e. 75 lakhs — the city grew by 75 lakh people in 25 years.

Q2.

The 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%	no exact match — nearest is 2%

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%

Note: The option list has no 4% card, so the USA's share of about 4.2% cannot be matched properly — the closest of the nine options is the second 2%. Take the USA's share as approximately 4%; the remaining options (13%, 8%, 10%, 35%, 0.1%) are distractors.

Why it happens: The hint about standard form matters because the figures are given in two different units. Comparing 83 million with 8.2 billion directly invites a slip of a factor of 1000. Once everything is in billions the arithmetic is easy, and the benchmark “1% of the world is 82 million people” lets you read off every answer by eye.

Q3.

The 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

(v) 8250 × 1.18 = 9735 ✓
(vi) 8250 + 8250 × 0.18 = 8250 + 1485 = 9735 ✓

Option	Value	What is wrong
(i) 8250 + 18	8268	adds ₹18, not 18%
(ii) 8250 + 1800	10,050	18% of 8250 is 1485, not 1800
(iii) 8250 + 18/100	8250.18	adds 0.18 rupees
(iv) 8250 × 18	1,48,500	18 times the price
(vii) 1.8 × 8250	14,850	an 80% tax, not 18%

Why it happens: (v) and (vi) are the same calculation written two ways — 8250 × 1.18 is 8250 × (1 + 0.18), which the distributive law expands into 8250 + 8250 × 0.18. Option (vii) is the sharpest trap: 1.8 and 1.18 look alike but mean 180% and 118% of the price. The decimal for a percentage always needs two places shifted — 18% is 0.18, so the multiplier is 1.18.

Q4.

The 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

Why it happens: +5, –2 and –3 add up to zero, which is why option (iv) looks so plausible. But percentage changes are multiplications, not additions, and each one is measured on a different population. The +5% was applied to p, while the –2% and –3% were applied to larger populations — so the falls remove slightly more mice than the rise put in. The net effect, ×0.99813, leaves the colony just under where it began.

Q5.

A 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.

Why it happens: The 35% and the 30% are measured on different bases. The 35% is added to the cost price; the 30% is taken off the marked price, which is larger. So the discount removes more rupees than the margin added — 30% of ₹1350 is ₹405, while the margin was only ₹350. For the shopkeeper to break even, the discount would have to be 35/135 = 25.9% at most.

Q6.

What percentage of area is occupied by the region marked ‘E’ in the figure? [A rectangle on a dot grid, divided into 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.

Why it happens: The diagonal cuts the lower-left 4 × 4 square into two congruent right triangles, so D and E are equal — each is half of 16, that is 8. You do not need a formula for the triangles beyond “half the square”. Check the whole thing: 16 + 24 + 8 + 8 + 8 = 64, and 25 + 37.5 + 12.5 + 12.5 + 12.5 = 100%.

Q7.

What 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.

Why it happens: Each expression is the same three numbers multiplied and divided — x, y and 100 — and multiplication does not care about order. The percentage sign hides this, but once both are written as xy/100 the identity is obvious. It is genuinely useful: 4% of 75 looks awkward, but 75% of 4 is three quarters of 4, which is 3. Swapping is often the fastest route to a mental answer.

Try This: Use the swap on 8% of 50 (= 50% of 8 = 4), 2% of 350 (= 350% of 2 = 7) and 16% of 25 (= 25% of 16 = 4).

Q8.

A 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

Working the same thing step by step: 40% of 160 = 64 Grade 8 students, and 60% of 64 = 38.4.

Careful: 38.4 is not a whole number, so the figures in the question cannot all be exact — you cannot have four-tenths of a student. Taken at face value the answer is about 38 Grade 8 girls. For the numbers to come out whole, the total would need to be a multiple of 25 (100% ÷ 4), for example 150 students → 36 girls, or 200 students → 48 girls.

Why it happens: The 60% is a percentage of the Grade 8 group, not of the whole party — which is why it must be multiplied by 40%, not added to it. Adding would suggest the girls are 60% of everyone, more than twice the true figure. Whenever a percentage is taken of a group that is itself a percentage, the two multiply.

Q9.

A 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%

With numbers: if a pencil costs ₹3, then 3 pencils sell for the cost of 5, i.e. ₹15, so one sells for ₹5. Profit per pencil ₹2 on a cost of ₹3 — that is 66.67%.

Why it happens: It is a profit because he needs to sell only 3 pencils to recover what 5 cost him — he is charging more per pencil than he paid. A useful way to see the size of it: the money from 3 pencils covers the cost of 5, so 2 pencils out of every 3 sold are pure profit in cost terms, giving 2/3 = 66.67%. Notice the letter c cancels out entirely, so the answer does not depend on what a pencil actually costs.

Q10.

The 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.

Why it happens: The extra 0.12% is 4% of last year's ₹3 rise. This year's increase is charged on the fare as it stood after last year's rise, not on the original fare — so the two percentages compound instead of adding. The effect is small over two years and small percentages, but it is the same mechanism as compound interest, and over many years it becomes large.

Q11.

If 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

Decrease = b – 10b/11 = b/11
Percentage decrease = (b/11)/b × 100 = 100/11
= 9 1/11 % ≈ 9.09%

Why it happens: To cancel a multiplication by 1.1 you must divide by 1.1, and 1/1.1 is 0.909…, not 0.9. Cutting the breadth by a full 10% would give 1.1 × 0.9 = 0.99, an area 1% smaller than before. This is the same asymmetry seen with Ariba's marbles: an increase of r is undone by a decrease of r/(1 + r), which is always a little less than r.

Check it yourself: Take a 10 cm × 11 cm rectangle, area 110 cm². Increase the length to 11 cm and the breadth must become 10 cm — a drop of 1 cm from 11, which is 1/11 = 9.09%.

Q12.

The 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. [Nutritional information: Potato 70%, Vegetable oil 24%, Salt 3%, Spices 3%; Net qty 65 g.]

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

Did you know? Almost a quarter of the packet, 15.6 g, is oil. The percentages add to 100 and so do the weights — a check worth running on any label, exactly as with the badam drink mixes earlier in the chapter.

Q13.

Three 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

Why it happens: “Buy 1 get 1 free” is often read as a 100% discount, because the free item costs nothing. But the discount must be measured against the full price of everything you carry out — ₹200 worth of goods for ₹100 is half price, 50%. The hint points to exactly this: compare the free items to the total items received, not to the ones you paid for. Shops B and C tie at ₹300 for 4 items even though B's percentage discount is higher, because B's offer only fits 3 items at a time and the fourth is bought at full price.

Q14.

In 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

Why it happens: The answer feels far too large — surely dropping 1 percentage point should cost one or two people? But 99% and 98% are not one step apart in this sense; they are “1 in 100” and “1 in 50”. The room has to shrink to half its size for that lone right-hander to double his share. Tracking the small group is much easier than tracking the large one, because his number never changes — only the denominator does. Near 100%, tiny changes in the percentage mean huge changes in the total.

Check it yourself: 49 out of 50 is 98%, and 1 out of 50 is 2%. ✓

Q15.

Look at the following graph. [Ability to use computer by age and gender (2023). Female then Male, per age group — Children 4% / about 5%; Teenage 24% / 29%; Twenties 26% / 37%; Thirties 14% / 25%; Forties 7% / 14%; Fifties 4% / 9%; Seniors 2% / 4%. Source: NSS Round 79, Comprehensive Annual Modular Survey, National Statistics Office.] 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%

Why it happens: (iii) is the important one to reject. Each bar answers “what share of this age group can use a computer?”, so every group's bars are measured against a different base. A graph of percentages cannot tell you the size of the groups themselves — for that you would need the actual population counts. (iv) is a reminder to read “more than” strictly: 25% is a quarter, so it is not more than a quarter, and it applies to men only.

Did you know? The widest gender gap in the graph is in the thirties — 25% of men against 14% of women, nearly double. The gap is narrowest among children and teenagers, which suggests it may close for later generations.

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