NCERT Solutions Ganita Prakash (Part 2) Chapter 1 –14Section 1.2 Percentage of Some Quantity — Figure it Out

Book page 12 Updated on2026-09-05

Q1.

Estimate 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. [(i) a bar of 5 equal parts, one part marked 20%; the second bar of 5 parts totals 75 with 4 parts marked 60. (ii) bars of 10 equal parts, one part marked ?; the second bar totals 90 with 6 parts marked ?. (iii) bars of 4 equal parts, one part marked ?; the second bar totals 140 with 3 parts marked ?.]

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

Why it happens: The bar is one whole, so cutting it into n equal parts makes each part 100/n per cent — no matter what number the bar stands for. That is why the same picture works for 75 in (i) and for 140 in (iii): the percentages depend only on how the bar is cut, and the values depend on what the whole bar is worth.

Check it yourself: In (iii), 105 out of 140 should come back to 75%. 105/140 = 3/4 = 75%. It does.

Q2.

Find 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

Two bar models: a percentage below 100 shades part of the bar; a percentage above 100 runs past the end of it.

For the rest, draw a bar for the whole and shade the part: 16% of 250 shades roughly a sixth of the bar; 62% of 360 shades a little under two-thirds; 1% of an hour is a sliver; 7% of 10 kg is a thin strip near the start.

Why it happens: (i) and (ii) both come to 40 even though the wholes differ — 40 is a quarter of 160 but only about a sixth of 250. This is the chapter's central warning: a percentage is meaningless until you say of what. And (iv) shows why bar models must be allowed to extend past 100% — the answer, 56, is larger than the whole, 40.

Q3.

Surya 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.

Check it yourself: 45 ml out of 60 ml is 45/60 = 3/4 = 75%, and 15/60 = 25%. The two shares add to 60 ml and to 100%.

Q4.

Pairs 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

Why it happens: (i) and (ii) need no arithmetic at all, because only one of the two factors changes. (iii) needs none either — 29% of 43 is under half of 43, so under 22, while 92% of 110 is nearly all of 110. (v) and (vi) are worth a rough check: in (v) 45% of 200 is nearly half of 200, while 10% of 490 is only 49; in (vi) both the percentage and the base are larger on the left, so the left must win.

Tip: A comparison only becomes a calculation when the percentage rises while the base falls (or the reverse). Otherwise the answer can be seen.

Q5.

Fill 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

None of these needs you to find k, m or n first — the percentages themselves are in simple ratios.

(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) 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)

Why it happens: Every one of these values is the same unknown multiplied by a hundredth, so the values are proportional to the percentages. Double the percentage and the value doubles; take a tenth of the percentage and the value becomes a tenth. Solving for k (which is 700/3, not a whole number) would only make the arithmetic worse.

Q6.

Fill 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

Why it happens: Each blank sits in a different place of the same relationship, part = (per cent ÷ 100) × whole. In (i) the part and the whole are known; in (ii) the per cent and the whole; in (iii) the part and the per cent. Knowing any two always gives the third.

Careful: In (iii) do not take 80% of 40. That would give 32 — a number smaller than 40, when the whole must be larger than 40, since 40 is only 80% of it.

Q7.

Is 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

Why it happens: A week is only 7 times a day, but 10% is 10 times 1%. Ten beats seven, so the smaller base with the larger percentage wins. Had the comparison been 10% of a day against 1% of a fortnight (14 days), the fortnight would have won — 1% of 14 days is 201.6 minutes.

Try This: Is 5% of a kilometre longer than 50% of a metre? Is 2% of a kilogram heavier than 25% of 100 g? Is 1% of a year longer than 20% of a fortnight?

Q8.

Mariam’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%.

Why it happens: The bull always leaves exactly 1 unit uneaten, every single day. What changes is not the waste but the size of the meal it is compared with. The percentage left over is 1/(n + 1) × 100, and as the meal grows, that one wasted unit becomes a smaller and smaller share of it. Since 1/(n + 1) is never 0, the percentage eaten is never 100% — it only creeps closer. This is exactly why a fixed amount looks serious in a small budget and trivial in a large one.

Q9.

Workers 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

Or as a proportion:

20/100 = 18/d → d = 18 × 100 / 20 = 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.

Why it happens: Proportional reasoning is a claim about the world, not just about the numbers. If the first 20% happened to be the easiest, flattest part of the plantation, the remaining 80% would take more than 72 days and the answer would be an underestimate. Stating the assumption is part of the answer.

Q10.

The 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

Check: 9 + 72 + 9 = 90 minutes, and 10 + 80 + 10 = 100%.

Tip: Once you have 10% of 90, which is 9, everything else follows — 80% is eight times that, 8 × 9 = 72. Finding 10% first is almost always the cheapest first move.

Q11.

An 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.

Tip: Use whatever current figure your class agrees on; the method does not change. With 8 billion the answer is 7.2 billion, with 8.2 billion it is 7.38 billion. Since the 90% itself is only an estimate, quoting the answer as “about 7.4 billion” is honest — writing 7,380,000,000 would suggest a precision the data do not have.

Q12.

A 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.

Suppose 4 people need 100 g in all: rava 40 g, sugar 40 g, ghee 20 g
For 8 people: rava 80 g, sugar 80 g, ghee 40 g, total 200 g
Rava's share = 80/200 = 40% — unchanged

(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 ✓

Why it happens: A recipe is a ratio, 2 : 2 : 1. Scaling every part of a ratio by the same factor leaves the ratio — and therefore the percentages — untouched. This is exactly why recipes are written in proportions in the first place: one recipe then serves any number of people.

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