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NCERT Solutions Ganita Prakash (Part 1) Chapter 7 –1777.6 Unit Conversions (chapter-end set) — Figure it Out

Book page 176 Updated on2026-09-05

Q1.
Anagh mixes 600 mL of orange juice with 900 mL of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form.
Answer

2 : 3.

orange : apple = 600 : 900
HCF of 600 and 900 = 300
600⁄300 : 900⁄300 = 2 : 3
Tip: Both quantities were already in millilitres, so no conversion was needed. Had the apple juice been given as 0.9 litre, the first job would have been to write it as 900 mL.
Q2.
Last year, we hired 3 buses for the school trip. We had a total of 162 students and teachers who went on that trip and all the buses were full. This year we have 204 students. How many buses will we need? Will all the buses be full?
Answer

4 buses are needed, and they will not all be full — 12 seats will be empty.

Seats in one bus = 162 ÷ 3 = 54
Buses needed = 204 ÷ 54 = 3.78, so 4 buses
Seats in 4 buses = 4 × 54 = 216
Empty seats = 216 − 204 = 12
Why the answer is not 3.78: The proportion 3 : 162 :: x : 204 gives x = 3.78, and up to that point the arithmetic is fine. But buses come whole. Rounding 3.78 down to 3 would leave 204 − 162 = 42 people standing on the road, so the answer must be rounded up. Whenever a proportion produces a fraction of an object that cannot be cut, ask which way the rounding has to go.
Q3.
The area of Delhi is 1,484 sq. km and the area of Mumbai is 550 sq. km. The population of Delhi is approximately 30 million and that of Mumbai is 20 million people. Which city is more crowded? Why do you say so?
Answer

Mumbai is more crowded — about 36,364 people in every square kilometre, against about 20,216 in Delhi.

Delhi: 30 million ÷ 1484 = 3,00,00,000 ÷ 1484 ≈ 20,216 people per sq km
Mumbai: 20 million ÷ 550 = 2,00,00,000 ÷ 550 ≈ 36,364 people per sq km
Why the bigger population is not the more crowded city: Delhi has 10 million more people, but it also has almost three times the land to spread them over. Crowding is people per unit of area — a ratio, not a count. Once both cities are reduced to “people in one square kilometre”, they can be compared fairly, and Mumbai packs nearly twice as many into the same space.
Did you know? This ratio is called population density. It is why a small island city can feel far more crowded than a large state with many more people in it.
Q4.
A crane of height 155 cm has its neck and the rest of its body in the ratio 4 : 6. For your height, if your neck and the rest of the body also had this ratio, how tall would your neck be?
Answer

Your neck would be 4⁄10 = 2⁄5 of your height. Measure yourself and multiply.

Total groups = 4 + 6 = 10
Neck = 4⁄10 × your height = 0.4 × your height

For the crane: 4⁄10 × 155 = 62 cm of neck
If you are 150 cm tall: 4⁄10 × 150 = 60 cm
If you are 160 cm tall: 4⁄10 × 160 = 64 cm
Why the answer is worth pausing over: A 60 cm neck on a 150 cm person is longer than the whole of your head and chest together. The ratio 4 : 6 is perfectly ordinary for a crane, whose long neck is built for fishing in shallow water; imposed on a human body it is absurd. Ratios always belong to something — copying one from one creature to another has to be done knowing what it will look like.
Q5.
Let us try an ancient problem from Lilavati. At that time weights were measured in a unit named palas and niskas was a unit of money. “If 2½ palas of saffron costs 3⁄7 niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?”
Answer

52.5 palas of saffron.

2½ palas = 5⁄2 palas, cost = 3⁄7 niskas
3⁄7 : 5⁄2 :: 9 : x
Cross multiplying: 3⁄7 × x = 5⁄2 × 9
3⁄7 × x = 45⁄2
x = 45⁄2 × 7⁄3 = 315⁄6 = 52.5 palas
Why the answer is so much bigger: Compare the money first — 9 ÷ 3⁄7 = 9 × 7⁄3 = 21, so the buyer is spending twenty-one times as much as before. The saffron must therefore be twenty-one times as much: 5⁄2 × 21 = 105⁄2 = 52.5 palas. Fractions in the data change nothing about the method; they only mean the factor of change is itself a fraction.
Did you know? Bhāskarāchārya wrote the Līlāvatī in 1150 CE, and problems in it are addressed to the reader as though in conversation — “O expert businessman!”. Its trairāśika problems are exactly the Rule of Three of this chapter.
Q6.
Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain’s age when the ratio of her age to her brother’s age is 1 : 2?
Answer

Harmain will be 4 years old (and her brother 8), that is, after 3 years.

Let the number of years from now be x.
Harmain: 1 + x, brother: 5 + x
(1 + x) : (5 + x) = 1 : 2
Cross multiplying: 2(1 + x) = 1(5 + x)
2 + 2x = 5 + x
x = 3
Harmain's age = 1 + 3 = 4 years, brother = 5 + 3 = 8 ✓ and 4 : 8 = 1 : 2 ✓
Why the ratio of ages keeps changing: The age gap of 4 years never changes, but each year adds 1 to both terms — and adding the same number to both terms of a ratio does change it. Today it is 1 : 5, in three years 1 : 2, and later 10 : 14 = 5 : 7, always drifting towards 1 : 1. The moment when it is exactly 1 : 2 is when the brother's age is twice Harmain's, and since the gap is 4, that must be 4 and 8.
Q7.
The mass of equal volumes of gold and water are in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of 1 litre of gold?
Answer

18.5 kg.

gold : water = 37 : 2 for equal volumes
37 : 2 :: x : 1
2 × x = 37 × 1
x = 37⁄2 = 18.5 kg
Why “equal volumes” is essential: The ratio 37 : 2 compares masses only when the two samples occupy the same space. It says nothing about a kilogram of gold against a kilogram of water — those have equal mass by definition. What it tells you is that gold is 18.5 times as dense as water, so one litre of gold is as heavy as eighteen and a half litres of water.
Did you know? A one-litre bottle full of gold would weigh 18.5 kg — heavier than a full school bag. This is the property Archimedes is said to have used to test whether a crown was pure gold.
Q8.
It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy? (Please refer to the section on Unit Conversions earlier in this chapter).
Answer

About 22.96 tonnes, that is roughly 22,957 kg of cow manure.

Area of plot = 200 × 500 = 1,00,000 sq ft
1 acre = 43,560 sq ft
Plot in acres = 100000 ÷ 43560 = 2.2957 acres
Manure = 10 × 2.2957 = 22.957 tonnes ≈ 22.96 tonnes
In kilograms: 22.957 × 1000 ≈ 22,957 kg
Why the conversion cannot be skipped: The recommended dose is quoted per acre, but the plot is measured in feet. Pairing 10 tonnes with 1,00,000 square feet directly would compare tonnes-per-acre with tonnes-per-square-foot — two different rates. Converting the plot to acres puts both quantities on the same scale before the Rule of Three is applied.
Tip: Manure is sold by the tractor-load, so a farmer would round up and buy 23 tonnes. As with the buses, the mathematics gives the exact figure and the situation decides how to round it.
Q9.
A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 mL. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity?
Answer

300 seconds, that is 5 minutes.

10 litres = 10 × 1000 = 10,000 mL
Number of mugfuls = 10000 ÷ 500 = 20
Time = 20 × 15 = 300 seconds = 5 minutes
By the Rule of Three:
500 : 15 :: 10000 : t
500 × t = 10000 × 15
t = 150000 ÷ 500 = 300 seconds
Why volume and time are proportional here: The tap runs at a steady rate — 500 ÷ 15 = 33⅓ mL per second. Nothing about the tap changes when a bigger vessel is put under it, so twenty times the water needs exactly twenty times the time.
Q10.
One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land?
Answer

About ₹82,645.

1 acre = 43,560 sq ft costs ₹15,00,000
43560 : 1500000 :: 2400 : x
43560 × x = 2400 × 15,00,000
x = 36,00,00,00,000 ÷ 43560
x = ₹82,644.63 ≈ ₹82,645
Why the answer is so small a share: 2,400 square feet is only 2400⁄43560 ≈ 5.5% of an acre, so it should cost about 5.5% of ₹15,00,000 — near ₹82,500, which matches. Doing that rough check first tells you at once whether the long division has gone astray.
Tip: The rate per square foot is 1500000 ÷ 43560 ≈ ₹34.44. Multiplying 2400 × 34.44 gives ₹82,656 — the small difference is only the rounding of the rate, which is why it is safer to divide at the very end.
Q11.
A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take him if he decides to use a tractor instead?
Answer

Oxen: 120 hours. Tractor: 30 hours.

Oxen: 6 hours for 1 acre
1 : 6 :: 20 : t ⇒ t = 20 × 6 = 120 hours

The tractor is 4 times faster, so it needs one-fourth of the time:
Per acre: 6 ÷ 4 = 1.5 hours
For 20 acres: 20 × 1.5 = 30 hours, or simply 120 ÷ 4 = 30 hours
Two different relationships in one question: Area and time rise together — twice the field, twice the hours — so that part is a straight Rule of Three. Speed and time do the opposite: four times the speed means one-quarter of the time. Handling both correctly in the same problem is the point of the question, and confusing them would give the tractor 480 hours instead of 30.
Did you know? 120 hours is fifteen working days of eight hours; 30 hours is under four. This is the arithmetic behind mechanisation changing the rhythm of the farming year.
Q12.
The ₹10 coin is an alloy of copper and nickel called ‘cupro-nickel’. Copper and nickel are mixed in a 3 : 1 ratio to get this alloy. The mass of the coin is 7.74 grams. If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin?
Answer

Copper ≈ ₹5.26, nickel ≈ ₹2.59, together about ₹7.85.

Total groups = 3 + 1 = 4, each group = 7.74 ÷ 4 = 1.935 g
Copper = 3 × 1.935 = 5.805 g
Nickel = 1 × 1.935 = 1.935 g

Copper: ₹906 per 1000 g → 5.805 × 906 ÷ 1000 = 5259.33 ÷ 1000 = ₹5.26
Nickel: ₹1341 per 1000 g → 1.935 × 1341 ÷ 1000 = 2594.835 ÷ 1000 = ₹2.59
Total metal cost ≈ ₹7.85
Why a ₹10 coin is not made of ₹10 of metal: The metal in the coin is worth about ₹7.85, comfortably under its face value. That gap matters: if the metal were worth more than ₹10, it would pay to melt coins down rather than spend them, and the coin would vanish from circulation. Mints choose the alloy and the mass with exactly this ratio in mind.
Check it yourself: Copper is 3⁄4 of the mass but only 5.26⁄7.85 ≈ 67% of the cost, because nickel is the dearer metal — ₹1,341 against ₹906 per kilogram. Mass ratio and cost ratio are two different ratios, and they need not agree.
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