NCERT Solutions Ganita Prakash (Part 2) Chapter 3 –68Section 3.6 Inverse Proportions — Figure it Out

Book page 67 Updated on2026-09-05

Q1.

Which of the following pairs of quantities are in inverse proportion? (i) The number of taps filling a water tank and the time taken to fill it. (ii) The number of painters hired and the days needed to paint a wall of fixed size. (iii) The distance a car can travel and the amount of petrol in the tank. (iv) The speed of a cyclist and the time taken to cover a fixed route. (v) The length of cloth bought and the price paid at a fixed rate per metre. (vi) The number of pages in a book and the time required to read it at a fixed reading speed.

Answer

In each case, ask what is being held fixed, and whether the second quantity grows or shrinks when the first grows.

	What stays fixed	If the first grows…	Type
(i) taps, time	the volume of the tank	time falls	Inverse
(ii) painters, days	the area of the wall	days fall	Inverse
(iii) distance, petrol	km per litre	petrol needed rises	Direct
(iv) speed, time	the length of the route	time falls	Inverse
(v) cloth, price	the rate per metre	price rises	Direct
(vi) pages, reading time	pages read per hour	time rises	Direct

Inverse: (i), (ii) and (iv). Direct: (iii), (v) and (vi).

Why it happens: Look at what the fixed quantity is in each row. In (i), (ii) and (iv) the fixed thing is the whole job — one tankful, one wall, one route — so it must be shared out, and more sharers means less each: the product stays constant. In (iii), (v) and (vi) the fixed thing is a rate — km per litre, rupees per metre, pages per hour — so each extra unit adds its own fixed amount: the quotient stays constant. That single question, “is a total fixed or is a rate fixed?”, decides every one of these six without any calculation.

Tip: Beware of (vi). Reading looks like work, and work problems are often inverse — but here the reading speed is fixed and it is the book that grows. A longer book takes proportionally longer to read.

Q2.

If 24 pencils cost ₹120, how much will 20 such pencils cost?

Answer

Fewer pencils cost less, so this is a direct proportion.

Cost of 1 pencil = 120 ÷ 24 = ₹5
Cost of 20 pencils = 20 × 5 = ₹100

Or with the rule of three, 24 : 20 :: 120 : x, so x = (20 × 120) ÷ 24 = 2400 ÷ 24 = ₹100.

Why it happens: The rate — ₹5 per pencil — does not change when you buy fewer. So cost ÷ number stays at 5: 120/24 = 5 and 100/20 = 5. Had this been inverse, the answer would have been 120 × 24 ÷ 20 = ₹144, which would mean 20 pencils cost more than 24. Checking whether the answer moves in a sensible direction catches that kind of slip at once.

Q3.

A tank on a building has enough water to supply 20 families living there for 6 days. If 10 more families move in there, how long will the water last? What assumptions do you need to make to work out this problem?

Answer

More families share the same water, so the days must fall — an inverse proportion.

Total families now = 20 + 10 = 30
families × days = constant
20 × 6 = 30 × x
x = 120 ÷ 30 = 4 days

The constant 120 is the store of water measured in family-days — enough for one family for 120 days, or 30 families for 4.

Why it happens: The tank holds a fixed amount of water. Nothing the families do adds to it, so the only question is how fast the fixed store is drawn down. Half as many days for one-and-a-half times as many families: 30 ÷ 20 = 1.5, and 6 ÷ 1.5 = 4.

Assumptions we have made:

Every family uses the same amount of water each day, and the new families use as much as the old ones.
Daily use does not change — nobody starts saving water because the tank is filling up more slowly.
The tank is not refilled during these days, and there is no leakage or overflow.
All the families have the same number of members — otherwise a “family” is not a fair unit to count in.

Real life rarely obeys all four. That is why the answer is a good estimate rather than a guarantee.

Q4.

Fill in the average number of hours each living being sleeps in a day by looking at the charts. Select the appropriate hours from this list : 15, 2.5, 20, 8, 3.5, 13, 10.5, 18.

Answer

Each ring stands for one full day of 24 hours, and the blue arc is the sleeping time. Convert an angle to hours in one step.

Whole circle = 360° = 24 hours
So 1 hour = 360° ÷ 24 = 15°, and hours = angle ÷ 15°

Living being (as printed, left to right)	Blue arc	Fraction of the day	Hours of sleep
Giraffe	a little over one-tenth	2.5/24	2.5
Elephant	a small wedge	3.5/24	3.5
Human child	120°, one-third	8/24 = 1/3	8
Dog	just under half	10.5/24	10.5
Cat	just over half	13/24	13
Squirrel	225°, five-eighths	15/24 = 5/8	15
Snake (python)	270°, three-quarters	18/24 = 3/4	18
Bat	300°, five-sixths	20/24 = 5/6	20

Every number in the given list is used exactly once, which is a useful check.

Why it happens: Read the easy landmarks first and the rest fall into place. The snake’s ring is exactly three-quarters blue, and 3/4 of 24 is 18. The human child’s is one-third, giving 8. The cat’s is a little more than half, so it must be 13 rather than 10.5, while the dog’s is a little less than half, so 10.5. Ordering the eight rings from least blue to most blue matches the ordered list 2.5, 3.5, 8, 10.5, 13, 15, 18, 20 term by term.

Did you know? Large grazing animals such as the giraffe and the elephant must spend most of the day eating and staying alert, so they sleep very little. Bats sleep almost the whole day and hunt at night. To turn any of these back into an angle, multiply by 15°: the bat’s 20 hours is 20 × 15° = 300°.

Q5.

The pie chart on the right shows the result of a survey carried out to find the modes of transport used by children to go to school. Study the pie chart and answer the following questions. (i) What is the most common mode of transport? (ii) What fraction of children travel by car? (iii) If 18 children travel by car, how many children took part in the survey? How many children use taxis to travel to school? (iv) By which two modes of transport are equal numbers of children travelling?

Answer

The chart marks four angles — Bus 120°, Walk 90°, Cycle 60° and Two-wheeler 60°. The Car slice is unmarked, so find it from the total.

Car = 360° − (120° + 90° + 60° + 60°) = 360° − 330° = 30°

The five slices, with the Car slice worked out as 30°.

(i) The bus, with the largest slice of 120° — a third of all the children.

(ii) Fraction travelling by car:

30°/360° = 1/12

(iii) If 1/12 of the children is 18, then

Total children = 18 × 12 = 216
Taxi: the chart has no slice for taxis, so 0 children use a taxi.

The full survey then reads:

Mode	Angle	Fraction	Children
Bus	120°	1/3	72
Walk	90°	1/4	54
Cycle	60°	1/6	36
Two-wheeler	60°	1/6	36
Car	30°	1/12	18
Total	360°	1	216

(iv) Cycle and two-wheeler — both slices are 60°, so 36 children use each.

Why it happens: A pie chart shows only proportions, never counts. The single extra fact “18 children travel by car” is what fixes the size of the survey: it tells us that 30° stands for 18 children, so 1° stands for 0.6 of a child and 360° stands for 216. The taxi question is a fair trap — an absent slice means an absent category, not a small one.

Q6.

Three workers can paint a fence in 4 days. If one more worker joins the team, how many days will it take them to finish the work? What are the assumptions you need to make?

Answer

The fence is fixed, so workers and days are inversely proportional.

workers × days = constant
3 × 4 = 4 × x
x = 12 ÷ 4 = 3 days

The constant 12 is the size of the job in worker-days: painting this fence is 12 days of work for one person.

Why it happens: The number of workers rose by a factor of 4/3, so the time falls by the inverse factor 3/4, and 4 × 3/4 = 3 days. Notice that the days did not fall by one just because one worker was added — inverse proportion works by multiplying factors, not by adding or subtracting.

Assumptions we have made:

All four workers paint at the same rate, and each works the same hours per day.
They can all work at once without getting in each other’s way, and there is enough paint and equipment for four.
The work can be divided freely — no part of the fence has to wait for another part to dry first.

Q7.

It takes 6 hours to fill 2 tanks of the same size with a pump. How long will it take to fill 5 such tanks with the same pump?

Answer

The same pump filling more tanks needs more time — a direct proportion.

Time for 1 tank = 6 ÷ 2 = 3 hours
Time for 5 tanks = 5 × 3 = 15 hours

Why it happens: Here the pump is fixed, so its rate is fixed, and the job is what grows. That makes time ÷ number of tanks constant: 6/2 = 3 and 15/5 = 3. Contrast this with Q6, where the job was fixed and the workers grew — that was inverse. Same story of pumps and tanks, opposite kind of proportion, decided entirely by which quantity is held still.

Tip: Question 10 also has pumps and a tank, but there the tank is fixed and the number of pumps changes. Read what is fixed before you decide.

Q8.

A given set of chairs are arranged in 25 rows, with 12 chairs in each row. If the chairs are rearranged with 20 chairs in each row, how many rows does this new arrangement have?

Answer

The set of chairs does not change, so rows × chairs per row stays constant.

Total chairs = 25 × 12 = 300
25 × 12 = x × 20
x = 300 ÷ 20 = 15 rows

Why it happens: Rows and chairs per row are the two sides of a rectangle of chairs whose area — the total number — is fixed at 300. Making each row longer by a factor of 20/12 = 5/3 makes the number of rows shrink by the inverse factor 3/5: 25 × 3/5 = 15. This is the clearest picture of inverse proportion in the whole exercise, because you can literally see the rectangle change shape while keeping its area.

Check it yourself: 15 × 20 = 300 chairs, the same as before. ✓ Not every rearrangement works, though — with 8 chairs per row you would need 37.5 rows, which is impossible. The number of chairs in a row must divide 300.

Q9.

A school has 8 periods a day, each of 45 minutes duration. How long is each period, if the school has 9 periods a day, assuming that the number of school hours per day stays the same?

Answer

The school day is fixed in length, so periods and their duration are inversely proportional.

Total teaching time = 8 × 45 = 360 minutes (6 hours)
8 × 45 = 9 × x
x = 360 ÷ 9 = 40 minutes

Why it happens: The question itself supplies the constant — “the number of school hours per day stays the same”. Six hours must be cut into 9 pieces instead of 8, so each piece is shorter. The number of periods grew by the factor 9/8, so the length falls by 8/9: 45 × 8/9 = 40 minutes.

Q10.

A small pump can fill a tank in 3 hours, while a large pump can fill the same tank in 2 hours. If both pumps are used together, how long will the tank take to fill?

Answer

Times cannot be added, so work with what each pump does in one hour.

Small pump in 1 hour = 1/3 of the tank
Large pump in 1 hour = 1/2 of the tank
Together in 1 hour = 1/3 + 1/2 = 2/6 + 3/6 = 5/6 of the tank
Time for the whole tank = 1 ÷ 5/6 = 6/5 = 1.2 hours = 1 hour 12 minutes

Why it happens: This is Example 6 in the disguise of pumps. Adding 3 and 2 to get 5 hours would be nonsense — two pumps together must be faster than either alone, so the answer must be less than 2 hours. What can be added is the fraction of the tank each pump fills in the same hour, because those are shares of one and the same job. Once the combined rate 5/6 tank per hour is known, the time is its reciprocal, since rate and time for a fixed job are inversely proportional.

Check it yourself: In 1.2 hours the small pump fills 1.2 × 1/3 = 0.4 of the tank and the large one fills 1.2 × 1/2 = 0.6. Together 0.4 + 0.6 = 1 whole tank. ✓

Q11.

A factory requires 42 machines to produce a given number of toys in 63 days. How many machines are required to produce the same number of toys in 54 days?

Answer

The same number of toys in fewer days needs more machines — inverse proportion.

machines × days = constant
42 × 63 = x × 54
2646 = 54x
x = 2646 ÷ 54 = 49 machines

Why it happens: The job is fixed at 2646 machine-days. The time was cut by the factor 54/63 = 6/7, so the number of machines must rise by the inverse factor 7/6: 42 × 7/6 = 49. Doing it that way is quicker than the long division, and it shows the answer must be more than 42 before you calculate anything.

Tip: Cancel before multiplying. 42 × 63 ÷ 54 = 42 × (63 ÷ 54) = 42 × 7/6 = 7 × 7 = 49.

Q12.

A car takes 2 hours to reach a destination, travelling at a speed of 60 km/h. How long will the car take if it travels at a speed of 80 km/h?

Answer

The destination does not move, so the distance is the constant and speed and time are inversely proportional.

Distance = 60 × 2 = 120 km
60 × 2 = 80 × x
x = 120 ÷ 80 = 1.5 hours = 1 hour 30 minutes

Why it happens: The speed rose by the factor 80/60 = 4/3, so the time falls by the inverse factor 3/4: 2 × 3/4 = 1.5 hours. The car saves half an hour. This is exactly the Lucknow-to-Kanpur situation of Section 3.6, with 120 km in place of 90 km as the constant k in the relation xy = k.

Check it yourself: 80 × 1.5 = 120 km, the same journey. ✓

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