Figure it Out — Direct and Inverse Proportion Problems — Proportional Reasoning-2

1. 13 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 67

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

   Hint. For each one ask: if I double the first quantity, does the second double or halve?

   The test to apply to each pair. Double the first quantity. If the second doubles, they are directly proportional; if it halves, they are inversely proportional.

   (i) Taps and filling time — INVERSE. Twice as many taps deliver water twice as fast, so the tank fills in half the time. Their product (taps × time) is the fixed volume of the tank.

   (ii) Painters and days — INVERSE. Twice as many painters finish the same wall in half the time. The product (painters × days) is the fixed total amount of work.

   (iii) Distance and petrol — DIRECT. Twice as much petrol carries the car twice as far. Here the quotient is fixed — it is the car's mileage in km per litre.

   (iv) Speed and time — INVERSE. For a fixed route, cycling twice as fast halves the time. The product (speed × time) is the fixed length of the route.

   (v) Cloth and price — DIRECT. Twice as much cloth costs twice as much. The fixed quotient is the rate per metre.

   (vi) Pages and reading time — DIRECT. Twice as many pages take twice as long to read. The fixed quotient is the reading speed in pages per hour.

   The pattern to carry away. Inverse proportion appears when something is fixed and shared out — one tank, one wall, one route — so having more of the first quantity means each unit does less. Direct proportion appears when there is a fixed rate — mileage, price per metre, pages per hour — and more input simply buys more output.

   ✦ Inversely proportional: (i), (ii) and (iv). Directly proportional: (iii), (v) and (vi).

2. 22 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 67

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

   Hint. Fewer pencils should cost less — decide which kind of proportion that is before calculating.

   Which proportion? Buying more pencils costs more and buying fewer costs less, so cost and quantity move together — a direct proportion at a fixed rate per pencil.

   Find the rate. ₹120 ÷ 24 pencils = ₹5 per pencil

   Scale up to 20 pencils. 20 × ₹5 = ₹100

   By the rule of three instead. Writing 24 : 20 :: 120 : x, x = (20 × 120) ÷ 24 = 2400 ÷ 24 = ₹100 ✓

   Check the direction. 20 pencils is fewer than 24, and ₹100 is less than ₹120 — the answer moved the way it should. Had the answer come out above ₹120, the ratio would have been used upside down.

   ✦ 20 pencils cost ₹100, at ₹5 per pencil.

3. 33 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 67

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

   Hint. The tank holds a fixed amount, and it is being shared among more people.

   Which proportion? The tank holds a fixed quantity of water. More families share the same water, so it runs out sooner — families and days are inversely proportional.

   Count the families after the move. 20 + 10 = 30 families. (Answering with 10 instead of 30 is the usual slip — the ten families join, they do not replace anyone.)

   Apply the constant product. families₁ × days₁ = families₂ × days₂ 20 × 6 = 30 × x 120 = 30x x = 4 days

   Sense check. 30 families is 1.5 times 20 families, so the water should last 6 ÷ 1.5 = 4 days ✓ — and 4 days is less than 6, as more people must mean fewer days.

   The assumptions this rests on.

   1. Every family uses the same amount of water per day. A family of two and a family of eight plainly do not, so the answer is an average.
   2. Daily use does not change. Nobody rations water once they notice the tank emptying, and nobody uses extra.
   3. The tank is not refilled during those days, and there is no leak or other loss.
   4. All 30 families are present for the whole period.

   Naming these matters. The arithmetic is only as reliable as the assumptions, and in a real building the first one is almost never exactly true.

   ✦ The water will last 4 days, assuming every family uses the same fixed amount of water each day and the tank is neither refilled nor leaking.

4. 44 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 67

   Fill in the average number of hours each living being sleeps in a day by reading the eight ring charts (giraffe, elephant, human, dog, cat, squirrel, python, bat). Choose from this list: 15, 2.5, 20, 8, 3.5, 13, 10.5, 18.

   Hint. In each ring the coloured arc is a fraction of the full day. Compare the arcs and put the eight beings in order first.

   How to read the charts. Each ring stands for one whole day of 24 hours, and the shaded arc is the sleeping part. So

   sleeping hours = (fraction of the ring shaded) × 24

   You do not need a protractor. The list gives eight values for eight rings, so each value is used exactly once — put the rings in order from the smallest shaded arc to the largest, put the list in increasing order, and match them off.

   The list in increasing order: 2.5, 3.5, 8, 10.5, 13, 15, 18, 20.

   Matching, smallest arc to largest:

   Living being	Shaded arc	Sleep per day
   Giraffe	about 1/10 of the ring	2.5 hours
   Elephant	a little more	3.5 hours
   Human	about 1/3	8 hours
   Dog	a little under half	10.5 hours
   Cat	just over half	13 hours
   Squirrel	about 5/8	15 hours
   Python	three-quarters	18 hours
   Bat	about 5/6	20 hours

   Check a few against the arithmetic. Human: 8 ÷ 24 = 1/3 of the ring ✓ Cat: 13 ÷ 24 ≈ 0.54, just over half ✓ Bat: 20 ÷ 24 ≈ 0.83, five-sixths ✓

   What the chart is teaching. Large grazing animals such as the giraffe and elephant sleep very little — they must keep eating and stay alert to predators. Small animals and hunters that eat concentrated food sleep the most. The pie-chart form makes the comparison instant in a way a column of numbers does not, which is the real point of the exercise.

   ✦ Giraffe 2.5 h, elephant 3.5 h, human 8 h, dog 10.5 h, cat 13 h, squirrel 15 h, python 18 h, bat 20 h.

5. 55 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 68

   The pie chart shows the modes of transport children use to go to school: Bus 120°, Walk 90°, Cycle 60°, Two-wheeler 60°, and Car the remaining slice. (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?

   Hint. Find the missing angle first — the five slices must add to 360°.

   Find the missing slice. The five slices fill the circle: Car = 360° − 120° − 90° − 60° − 60° = 30°

   Mode	Angle
   Bus	120°
   Walk	90°
   Cycle	60°
   Two-wheeler	60°
   Car	30°
   Total	360° ✓

   (i) Most common mode. The largest slice is the most popular, and Bus has the largest angle at 120°. So the most common mode is the bus — used by a third of the children, since 120 ÷ 360 = 1/3.

   (ii) Fraction travelling by car. 30 ÷ 360 = 1/12

   (iii) Total number of children. One-twelfth of the children travel by car, and that is 18 children, so total ÷ 12 = 18 total = 18 × 12 = 216 children

   Filling in the whole survey as a check:

   Mode	Angle	Children
   Bus	120°	72
   Walk	90°	54
   Cycle	60°	36
   Two-wheeler	60°	36
   Car	30°	18
   Total	360°	216 ✓

   How many use taxis? None — zero children. Taxi is not one of the five modes in the chart, and the five slices shown already account for the whole 360°, so every child surveyed is in one of those five groups. There is no room left for a sixth mode. (The word "taxis" looks like a slip for "two-wheeler" in the printed question. If that is what was meant, the answer is 36 children. The answer as the question actually reads is zero, and it is worth saying why rather than quietly answering a different question.)

   (iv) Equal numbers. Equal slices mean equal numbers, and Cycle and Two-wheeler are both 60°. So cycle and two-wheeler carry equal numbers — 36 children each.

   ✦ (i) Bus, at 120°. (ii) 1/12. (iii) 216 children took part; no children use taxis, since taxi is not one of the five modes shown and the five slices already total 360°. (iv) Cycle and two-wheeler, 36 children each.

6. 63 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 68

   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?

   Hint. The fence is a fixed job being shared among more workers.

   Which proportion? The fence is a fixed amount of work. More workers finish it sooner, so workers and days are inversely proportional.

   Count the workers after the change. 3 + 1 = 4 workers.

   Apply the constant product. workers₁ × days₁ = workers₂ × days₂ 3 × 4 = 4 × x 12 = 4x x = 3 days

   The constant 12 has a meaning: it is 12 worker-days, the total amount of labour the fence needs. Four workers supply that in three days, three workers in four days, and six workers would do it in two.

   Sense check. More workers, fewer days — 3 days is less than 4 ✓

   The assumptions this rests on.

   1. All four workers paint at the same rate. A faster painter would change the answer.
   2. They work the same number of hours each day as before.
   3. They do not obstruct one another. Only so many people can reach a fence at once; past a point, extra workers add nothing.
   4. The work divides freely — no part of the job has to wait for another part to finish first.

   Assumption 3 is where this model breaks in real life. It predicts that 12 workers finish in 1 day and 24 workers in half a day, which cannot be true of a single fence. Inverse proportion describes the arithmetic, not the crowd.

   ✦ Four workers will finish in 3 days, assuming all workers paint at the same steady rate, work the same hours, and do not get in each other's way.

7. 73 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 68

   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?

   Hint. Careful — this one is not inverse. Ask whether more tanks means more time or less.

   Stop and check the direction. This question sits among inverse-proportion problems, but it is not one of them. There is a single pump working at a fixed rate, and more tanks to fill means more time, not less. Tanks and time are directly proportional.

   Treating it as inverse would give 2 × 6 = 5 × x and x = 2.4 hours — filling five tanks faster than two, which is impossible. That absurd answer is the signal that the wrong relationship was assumed.

   Find the rate. 6 hours ÷ 2 tanks = 3 hours per tank

   Scale up. 5 tanks × 3 hours = 15 hours

   By the rule of three. 2 : 5 :: 6 : x, so x = (5 × 6) ÷ 2 = 15 hours ✓

   How to tell the two kinds apart here. The number of pumps against time is inverse, because the job is fixed and gets shared. The number of tanks against time is direct, because the job itself is growing while the pumping rate stays the same. Ask which quantity is being held constant before you write anything.

   ✦ It will take 15 hours, since one tank takes 3 hours and tanks and time are directly proportional for a single pump.

8. 82 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 68

   A given set of chairs is 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?

   Hint. The number of chairs does not change — that fixed total is the constant.

   Which proportion? The same chairs are being rearranged, so the total is fixed. Putting more chairs in each row means fewer rows are needed, so rows and chairs-per-row are inversely proportional.

   Find the constant. It is simply the number of chairs: 25 × 12 = 300 chairs

   Apply it to the new arrangement. rows × 20 = 300 rows = 300 ÷ 20 = 15 rows

   Check. 15 rows × 20 chairs = 300 chairs ✓ — the same 300 chairs, just laid out differently.

   Why this one is easy to trust. Unlike the workers-and-days problems, the constant here is a real countable thing rather than an abstract "amount of work". You can see immediately that 300 is right, and that makes the inverse relationship obvious rather than something to be taken on faith.

   ✦ The new arrangement has 15 rows.

9. 92 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 68

   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 the number of school hours per day stays the same?

   Hint. The total teaching time each day is fixed — that is the constant.

   Which proportion? The school day is fixed in length. Cutting it into more periods makes each period shorter, so the number of periods and the length of a period are inversely proportional.

   Find the constant — the length of the school day. 8 × 45 = 360 minutes (which is 6 hours)

   Apply it to nine periods. 9 × length = 360 length = 360 ÷ 9 = 40 minutes

   Check. 9 × 40 = 360 minutes ✓ — the same six-hour day.

   Reading the result. Adding a ninth period costs every existing period 5 minutes. That is the trade-off the arithmetic makes visible, and it is exactly the kind of decision a timetable committee faces: an extra subject slot, or longer lessons.

   ✦ Each period would be 40 minutes long.

10. 104 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 68

    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?

    Hint. You cannot add or average the times. Work out how much of the tank each pump fills in one hour.

    Why the times cannot simply be combined. Adding them gives 5 hours and averaging gives 2.5 hours, and both are absurd — two pumps together must be faster than either alone, so the answer has to be less than 2 hours. Times do not add; rates do.

    Step 1 — find each pump's rate. Take the full tank as 1 unit of work. Small pump: fills the tank in 3 hours, so in 1 hour it fills 1/3 of it. Large pump: fills it in 2 hours, so in 1 hour it fills 1/2 of it.

    Step 2 — add the rates. 1/3 + 1/2 = 2/6 + 3/6 = 5/6 of the tank per hour

    Step 3 — turn the rate back into a time. If 5/6 of a tank takes 1 hour, then one whole tank takes 1 ÷ (5/6) = 6/5 = 1.2 hours

    Step 4 — convert. 0.2 hour = 0.2 × 60 = 12 minutes, so the answer is 1 hour 12 minutes.

    Check it lands in the right window. The combined time must be less than the faster pump's 2 hours and more than half of it (since the slower pump is not as good as a second large pump), so it must lie between 1 hour and 2 hours. 1 hour 12 minutes ✓

    The idea to remember. Whenever two agents work together — pumps, taps, painters, people cutting vegetables — convert each to work done per unit time, add those, and invert at the end. This is the same method the chapter used for Ram and Shyam.

    ✦ Together the pumps fill 5/6 of the tank per hour, so the tank fills in 6/5 = 1.2 hours = 1 hour 12 minutes.

11. 112 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 68

    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?

    Hint. The number of toys is fixed. Fewer days available means more machines are needed.

    Which proportion? The order of toys is fixed. Finishing it in fewer days needs more machines, so machines and days are inversely proportional.

    Predict the direction first. 54 days is less than 63 days, so the answer must be more than 42 machines. Knowing this before calculating catches an upside-down ratio instantly.

    Find the constant. 42 × 63 = 2646 machine-days — the total machine-time the order needs.

    Apply it. machines × 54 = 2646 machines = 2646 ÷ 54 = 49 machines

    Check. 49 × 54 = 2646 ✓ and 49 > 42, as predicted.

    A neater route. The time shrinks in the ratio 63 : 54, which simplifies (divide by 9) to 7 : 6. Machines must therefore grow in the reversed ratio 6 : 7, giving 42 × 7/6 = 49 ✓ Reversing the ratio is what inverse proportion means in practice.

    ✦ 49 machines are required.

12. 122 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 68

    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?

    Hint. The journey is fixed in length, so its distance is the constant.

    Which proportion? The destination does not move, so the distance is fixed. Going faster takes less time — speed and time are inversely proportional.

    Find the constant — the distance. 60 km/h × 2 h = 120 km

    Apply it at the new speed. 80 × time = 120 time = 120 ÷ 80 = 1.5 hours

    Convert. 0.5 hour = 30 minutes, so the journey takes 1 hour 30 minutes.

    Check. 80 km/h × 1.5 h = 120 km ✓, the same journey — and 1.5 hours is less than 2 hours, as a higher speed demands.

    By reversing the ratio instead. The speed rises in the ratio 60 : 80 = 3 : 4, so the time must fall in the reversed ratio 4 : 3: 2 × 3/4 = 1.5 hours ✓

    ✦ The car takes 1.5 hours, that is 1 hour 30 minutes.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 2, Reprint 2026-27 (hegp203.pdf), where this is Chapter 3 (pages 55-69) — the tenth chapter of the Class 8 course. Like the rest of Part 2, this PDF carries NO printed answer key, so every answer was derived from first principles and independently recomputed in Python before being written. TWO FIGURE-ONLY ITEMS WERE RESOLVED BY MEASURING THE RENDERED PAGE rather than guessing: (1) the eight sleep ring charts on page 67 were rendered at 300 dpi and the shaded arc of each ring measured by sampling 3600 points around its circumference — the eight measured fractions map one-to-one onto the eight values the question offers (giraffe 2.96 -> 2.5, elephant 3.84 -> 3.5, human 7.78 -> 8, dog 10.07 -> 10.5, cat 12.99 -> 13, squirrel 15.08 -> 15, python 18.09 -> 18, bat 20.16 -> 20), which is what fixes the assignment; (2) the transport pie chart on page 68 prints only four of its five angles, so the page was rendered and the labels read off before deducing Car = 30 degrees. ONE ITEM IS FLAGGED AS UNANSWERABLE FROM THE PDF: the map distances in section 3.2, because the chapter PDF stamps its map 'Map not to scale' - the method is given and no distance is invented. ONE APPARENT SLIP IN THE BOOK IS FLAGGED: page 68 question 5(iii) asks how many children use 'taxis', but taxi is not one of the five modes in the pie chart and the five slices already total 360 degrees, so the answer as printed is zero; the solution says so and notes that 'two-wheeler' (36 children) was probably intended.. Questions are referenced from the NCERT textbook for identification.