MathsICSE Class 8 · Chapter 9

Direct and Inverse Variation ICSE Class 8: Proportion Word Problems Solved

Updated 15 October 2026 · Fact-checked

Two quantities vary directly when they increase or decrease together in the same ratio, so x/y stays constant. They vary inversely when one increases as the other decreases in the same ratio, so x × y stays constant. ICSE Class 8 uses these to solve simple word problems on cost, speed, workers and food.

What Direct and Inverse Variation is about

Many everyday quantities are linked. Buy more pens and the cost goes up; send more workers and a job takes fewer days. Variation describes how two quantities change together.

Direct variation: $x$ and $y$ increase (or decrease) together in the same ratio. Then $\frac{x}{y} = k$ , a constant. We write $x \propto y$ .
Inverse variation: as $x$ increases, $y$ decreases in the same ratio. Then $x y = k$ , a constant. We write $x \propto \frac{1}{y}$ .

The CISCE Class VIII curriculum asks for simple and direct word problems on both kinds. You must first decide which kind of variation a situation shows, then solve by the unitary method (find the value for one unit) or by proportion.

A warning: not every pair of increasing quantities varies directly. Your age and your height both increase, but not in a fixed ratio, so they do not vary directly.

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Direct and Inverse Variation sits in the Ratio and Proportion theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Direct variation

$x$ varies directly as $y$ if $\frac{x}{y}$ is the same for every pair of values. So if $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are two pairs:
$\frac{x _{1}}{y _{1}} = \frac{x _{2}}{y _{2}}$ .

Examples: cost and number of items (at a fixed price), distance and time (at a fixed speed), fuel used and distance (at a fixed mileage), wages and days worked.

Doubling one quantity doubles the other; halving one halves the other.

Worked example

5 pens cost ₹60. Find the cost of 8 pens.

Try it first, then show the solutionSolution

Cost varies directly as the number of pens.
Unitary method: 1 pen costs $60 \div 5 = ₹12$ .
8 pens cost $8 \times 12 = ₹96$ .

Answer: ₹96

Testing a table for direct variation

Divide each $y$ by its $x$ (or each $x$ by its $y$ ). If every quotient is the same, the quantities vary directly and that quotient is the constant $k$ .

Then any missing value comes from $y = k x$ .

If even one quotient is different, the table does not show direct variation.

Worked example

Do $x$ and $y$ vary directly? $x$ : 2, 4, 6, 9; $y$ : 10, 20, 30, 45. Find $y$ when $x = 11$ .

Try it first, then show the solutionSolution

$\frac{10}{2} = \frac{20}{4} = \frac{30}{6} = \frac{45}{9} = 5$ .
All equal, so they vary directly with $y = 5 x$ .
When $x = 11$ , $y = 55$ .

Answer: Yes; $y = 55$ .

Solving direct variation by proportion

Write the two pairs as equal ratios and cross-multiply:
$\frac{x _{1}}{y _{1}} = \frac{x _{2}}{y _{2}} \Rightarrow x_{1} y_{2} = x_{2} y_{1}$ .

Put the unknown on the left once you have cross-multiplied, then divide. Keep the same units in both pairs (both in km, both in hours).

Worked example

A car travels 150 km on 2.5 litres of petrol. How much petrol is needed for 720 km?

Try it first, then show the solutionSolution

Petrol varies directly as distance: $\frac{2.5}{150} = \frac{x}{720}$ .
$150 x = 2.5 \times 720 = 1800$ .
$x = 12$ litres.

Answer: 12 litres

Inverse variation

$x$ varies inversely as $y$ if the product $x y$ is the same for every pair: $x_{1} y_{1} = x_{2} y_{2}$ .

Examples: number of workers and days to finish a job; speed and time for a fixed distance; number of people and days a food stock lasts; number of taps and time to fill a tank.

Doubling one quantity halves the other.

The product often has a meaning: workers × days = worker-days of work, speed × time = distance.

Worked example

12 workers can build a wall in 10 days. How long would 15 workers take?

Try it first, then show the solutionSolution

More workers, fewer days: inverse variation.
Work $= 12 \times 10 = 120$ worker-days.
Days for 15 workers $= 120 \div 15 = 8$ .

Answer: 8 days

Direct, inverse or neither?

Ask two questions:
1. When one quantity goes up, does the other go up or down?
2. Does it change in the same ratio (doubling one doubles or halves the other)?

Situation	Type
Number of tickets and total cost	Direct
Speed and time for a fixed journey	Inverse
Distance and time at a fixed speed	Direct
Workers and days for a job	Inverse
Age and height of a child	Neither

Always write the type before solving; it decides whether you multiply or divide.

Worked example

State the type of variation: (i) the number of students sharing a fixed cost and the share of each (ii) the length of cloth and its cost.

Try it first, then show the solutionSolution

(i) More students, smaller share, and doubling the students halves the share: inverse.
(ii) More cloth, more cost in the same ratio: direct.

Answer: (i) Inverse (ii) Direct

Testing a table for inverse variation

Multiply each $x$ by its $y$ . If every product is the same, the quantities vary inversely and the product is $k$ .

Then a missing value is $y = \frac{k}{x}$ .

Worked example

$x$ : 2, 3, 5, 6; $y$ : 30, 20, 12, ?. Check for inverse variation and fill the gap.

Try it first, then show the solutionSolution

$2 \times 30 = 3 \times 20 = 5 \times 12 = 60$ .
So $x y = 60$ , inverse variation.
Missing $y = 60 \div 6 = 10$ .

Answer: Inverse; missing value 10.

Maths in real life: Direct and Inverse Variation

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Direct variation

Suppose 6 bananas cost ₹42 at a fruit stall.

How much do 15 bananas cost?

Try it first, then show the solutionSolution

One banana costs $42 \div 6 = ₹7$ .
15 bananas cost $15 \times 7 = ₹105$ .

Answer: ₹105

Memory hook: More items, more cost, in the same ratio.

Inverse variation

Imagine 4 friends can paint a fence in 6 hours, all working at the same speed.

How long would 8 friends take?

Try it first, then show the solutionSolution

Work $= 4 \times 6 = 24$ person-hours.
8 friends take $24 \div 8 = 3$ hours.

Answer: 3 hours

Memory hook: Twice the people, half the time.

Direct variation with time

Suppose your phone charges steadily, gaining 20% of its charge in 15 minutes.

How long does it take to gain 80%?

Try it first, then show the solutionSolution

Charge gained varies directly as time.
$80%$ is 4 times $20%$ , so time $= 4 \times 15 = 60$ minutes.

Answer: 60 minutes

Memory hook: Four times the charge needs four times the time.

Topics in this chapter

Direct and Inverse Variation formula sheet

Direct variation: $\frac{x}{y} = k$; x ∝ y.
Direct proportion: $\frac{x _{1}}{y _{1}} = \frac{x _{2}}{y _{2}}$; Cross-multiply to solve.
Inverse variation: $x y = k$; x ∝ 1/y.
Inverse proportion: $x_{1} y_{1} = x_{2} y_{2}$; Products are equal.
Speed, distance, time: $d = s \times t$; Fixed d: s and t inverse; fixed s: d and t direct.

How the school exam asks Direct and Inverse Variation

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Identify the variation	State whether two quantities vary directly, inversely or neither.	Up–up in the same ratio is direct; up–down in the same ratio is inverse.
Complete a table	Check whether x and y vary directly or inversely and fill in the missing values.	Test quotients for direct, products for inverse.
Direct word problems	Cost of items, map scales, fuel and distance, shadows at the same time of day.	Unitary method or equal ratios.
Inverse word problems	Workers and days, speed and time, food stock and people, taps and filling time.	Find the constant product, then divide.
Changes part-way	Workers leave or people join after some days.	Find the work or stock left, then share it among the new number.

Tricks to remember Direct and Inverse Variation

Short memory aids. Say each one aloud once, then use it on a question the same day.

Same way divide, opposite way multiply
Direct: the quotient x/y stays fixed. Inverse: the product xy stays fixed.
Use it when: Choosing the equation.
More men, less days
If more of one means less of the other, it is inverse.
Use it when: Spotting inverse variation in words.
Name the product
In inverse problems, give the constant a name: worker-days, student-days, tap-minutes.
Use it when: Problems where people join or leave.
Sense check
Before writing the answer, ask: should it be bigger or smaller than the given value?
Use it when: Catching a wrong variation type.

Direct and Inverse Variation: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Spot the inverse pair · 1 mark · MCQExam level

Which pair of quantities varies inversely?
(a) Number of pens and their total cost
(b) Distance travelled and petrol used
(c) Number of workers and the days they take to finish a job
(d) Length of cloth and its cost

Try it first, then show the solutionSolution

(a), (b) and (d) increase together: direct.
More workers finish in fewer days: inverse.

Answer: (c)

Example 2: Cost of cloth · 3 marks · Short answerExam level

12 m of cloth cost ₹1,080. Find the cost of 17 m of the same cloth.

Try it first, then show the solutionSolution

Cost varies directly as length.
1 m costs $1080 \div 12 = ₹90$ .
17 m cost $17 \times 90 = ₹1530$ .

Answer: ₹1,530

Example 3: Is the table direct? · 3 marks · Short answerExam level

$x$ : 3, 5, 8, 12; $y$ : 15, 25, 40, 60. Show that $x$ and $y$ vary directly and find $x$ when $y = 75$ .

Try it first, then show the solutionSolution

$\frac{15}{3} = \frac{25}{5} = \frac{40}{8} = \frac{60}{12} = 5$ .
So $y = 5 x$ : direct variation.
$75 = 5 x$ , so $x = 15$ .

Answer: $x = 15$

Example 4: Speed and time · 3 marks · Short answerHarder exam level

A car takes 5 hours for a journey at 60 km/h. How long will it take at 75 km/h?

Try it first, then show the solutionSolution

Fixed distance: speed and time vary inversely.
$60 \times 5 = 75 \times t$ .
$t = \frac{300}{75} = 4$ hours.

Answer: 4 hours

Example 5: Map scale · 4 marks · Long answerHarder exam level

On a map, 3 cm represents 75 km. (i) What actual distance does 7.2 cm represent? (ii) What map distance shows 130 km?

Try it first, then show the solutionSolution

Map distance varies directly as actual distance; 1 cm represents $75 \div 3 = 25$ km.
(i) $7.2 \times 25 = 180$ km.
(ii) Map distance $= 130 \div 25$ .
$= 5.2$ cm.

Answer: (i) 180 km (ii) 5.2 cm

Example 6: Food stock · 4 marks · Long answerHarder exam level

A hostel has enough food for 120 students for 35 days. If 20 more students join, how long will the food last?

Try it first, then show the solutionSolution

More students, fewer days: inverse variation.
Food $= 120 \times 35 = 4200$ student-days.
New number of students $= 140$ .
Days $= 4200 \div 140 = 30$ .

Answer: 30 days

Example 7: Taps added part-way · 5 marks · Long answerHOTS / competency

6 identical taps fill a tank in 40 minutes. The tank is empty and the 6 taps are opened; after 10 minutes, 2 more identical taps are opened. How long does it take to fill the tank in all? How long would 8 taps take from the start?

Try it first, then show the solutionSolution

Taps and time vary inversely. Whole tank $= 6 \times 40 = 240$ tap-minutes.
In the first 10 minutes: $6 \times 10 = 60$ tap-minutes done.
Left: $240 - 60 = 180$ tap-minutes, shared by 8 taps: $180 \div 8 = 22.5$ minutes.
Total time $= 10 + 22.5 = 32.5$ minutes.
With 8 taps from the start: $240 \div 8 = 30$ minutes.

Answer: 32.5 minutes; 30 minutes with 8 taps from the start

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Planning a school trip

Suppose a school bus is booked for a 240 km trip. It runs 6 km on one litre of diesel, and the diesel costs ₹90 per litre (prices are for this question only). The diesel cost is shared equally by the students who go.

(i) · 3 marks
How many litres of diesel are needed, and what is its total cost?
Answer
Answer: 40 litres; ₹3,600
Diesel varies directly as distance: $240 \div 6 = 40$ L. Cost $= 40 \times 90 = ₹3600$ .
(ii) · 3 marks
The trip takes 5 hours at 48 km/h. How long would it take at 60 km/h?
Answer
Answer: 4 hours
Fixed distance, so speed and time vary inversely: $48 \times 5 = 60 \times t$ , $t = 4$ hours.
(iii) · 4 marks
If 40 students share the cost, each pays ₹90. How much does each pay if 45 students go? Which kind of variation is this?
Answer
Answer: ₹80; inverse variation
Number of students × share $= ₹3600$ is constant, so inverse variation. Share $= 3600 \div 45 = ₹80$ .

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

9 notebooks cost ₹315. Find the cost of 14 such notebooks. (3 marks)

Step	Marks
Direct variation stated	1
1 notebook costs $315 \div 9 = ₹35$	1
14 notebooks cost $14 \times 35 = ₹490$	1

15 pumps can empty a tank in 12 hours. How many pumps are needed to empty it in 9 hours? (4 marks)

Step	Marks
Fewer hours need more pumps: inverse variation	1
Pump-hours $= 15 \times 12 = 180$	1
Pumps $= 180 \div 9$	1
$= 20$ pumps	1

A hostel has food for 400 students for 60 days. After 15 days, 50 more students join. How long will the remaining food last? (5 marks)

Step	Marks
Inverse variation between students and days	1
After 15 days, food left for 400 students for $60 - 15 = 45$ days	1
Food left $= 400 \times 45 = 18000$ student-days	1
Students now $= 450$	1
Days $= 18000 \div 450 = 40$	1

Where students lose marks in Direct and Inverse Variation

The slips examiners see most often, and what to write instead.

Mistake: Treating an inverse situation as direct
More workers do not take more days.
Wrong working
12 workers: 10 days, so 15 workers: $\frac{15}{12} \times 10 = 12.5$ days
Right working
$\frac{12 \times 10}{15} = 8$ days
Do this: State the type of variation first and sense-check the answer.
Usually costs: The whole question.
Mistake: Forgetting the days already used
When people join part-way, only the food (or work) that is left gets shared.
Wrong working
$\frac{400 \times 60}{450}$
Right working
$\frac{400 \times 45}{450} = 40$
Do this: Subtract the days already passed before sharing.
Usually costs: 2 to 3 marks.
Mistake: Mixing units
Minutes with hours, or metres with km, give wrong ratios.
Do this: Change both pairs to the same units before setting up the proportion.
Usually costs: The answer.
Mistake: Calling any 'both increase' pair direct
Direct variation needs the same ratio; age and height both increase but not in proportion.
Do this: Check the quotient is constant.
Usually costs: 1 mark in identification questions.
Mistake: No statement of the method
In word problems, examiners give a mark for stating direct or inverse variation.
Do this: Write a line such as 'more men, fewer days: inverse variation'.
Usually costs: 1 mark.

Practice questions on Direct and Inverse Variation

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQExam level
If $x$ varies directly as $y$ , and $x = 6$ when $y = 9$ , then $x$ when $y = 15$ is:
(a) 8
(b) 10
(c) 12
(d) 22.5
Hint
$\frac{x}{y} = \frac{2}{3}$ .
Answer
(b)
Q2 · 2 marks · Short answerExam level
25 kg of rice cost ₹1,250. Find the cost of 18 kg.
Hint
Cost of 1 kg.
Answer
₹50 per kg; 18 kg cost ₹900.
Q3 · 2 marks · Short answerExam level
$x$ varies inversely as $y$ , and $x = 12$ when $y = 5$ . Find $y$ when $x = 15$ .
Hint
$x y = 60$ .
Answer
$y = 60 \div 15 = 4$ .
Q4 · 3 marks · Short answerHarder exam level
A train covers 210 km in 3 hours at a steady speed. How far does it go in 5 hours, and how long does it take for 595 km?
Hint
Speed is 70 km/h.
Answer
350 km; $595 \div 70 = 8.5$ hours.
Q5 · 1 mark · Assertion-reasonHarder exam level
Assertion (A): For a fixed distance, speed and time vary inversely.
Reason (R): Speed × time = distance, which is constant.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Constant product means inverse variation.
Answer
(c)
Q6 · 3 marks · Short answerHarder exam level
36 men can dig a trench in 25 days. How many men are needed to dig it in 15 days?
Hint
Man-days are fixed.
Answer
$36 \times 25 = 900$ man-days; $900 \div 15 = 60$ men.
Q7 · 3 marks · Short answerHarder exam level
$x$ : 4, 6, 8, 12; $y$ : 18, 12, 9, ?. Show that $x$ and $y$ vary inversely, fill the gap, and find $x$ when $y = 24$ .
Hint
Products.
Answer
$4 \times 18 = 6 \times 12 = 8 \times 9 = 72$ . Missing $y = 6$ ; when $y = 24$ , $x = 3$ .
Q8 · 2 marks · Short answerHarder exam level
A photocopier makes 80 copies in 4 minutes. How many copies does it make in $1 \frac{1}{2}$ hours?
Hint
Change hours to minutes.
Answer
20 copies per minute × 90 minutes $= 1800$ copies.
Q9 · 3 marks · Short answerHarder exam level
A school day has 8 periods of 45 minutes each. If the day stays the same length but has 9 periods, how long is each period?
Hint
Periods × length is constant.
Answer
$8 \times 45 = 360$ minutes; $360 \div 9 = 40$ minutes.
Q10 · 3 marks · Short answerHarder exam level
At a certain time of day, a 15 m pole casts a 10 m shadow. At the same time, a tree casts an 18 m shadow. How tall is the tree?
Hint
Height varies directly as shadow at the same time.
Answer
$\frac{15}{10} = \frac{h}{18}$ , so $h = 27$ m.
Q11 · 4 marks · Long answerHOTS / competency
24 workers can finish a job in 30 days. After 10 days, 4 workers leave. How many more days will the rest take to finish? What is the total time?
Hint
Work left is 24 workers for 20 days.
Answer
Work left $= 24 \times 20 = 480$ worker-days; 20 workers take $480 \div 20 = 24$ more days. Total $= 10 + 24 = 34$ days.
Q12 · 4 marks · Long answerHOTS / competency
A cyclist covers a distance in 4 hours at 15 km/h. (i) At what speed must he ride to cover it in 3 hours? (ii) How much farther would he go in 4 hours at this new speed?
Hint
Distance is 60 km.
Answer
(i) $15 \times 4 = v \times 3$ , $v = 20$ km/h. (ii) In 4 h he covers 80 km, which is 20 km farther.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.If x varies directly as y, what is constant?Show answer
x/y.
2.If x varies inversely as y, what is constant?Show answer
xy.
3.Speed and time for a fixed distance vary how?Show answer
Inversely.
4.If 3 kg of apples cost ₹240, what do 5 kg cost?Show answer
₹400.
5.If 6 men do a job in 8 days, how long do 4 men take?Show answer
12 days.
6.Do age and weight of a person vary directly?Show answer
No.
7.x ∝ y and x = 4 when y = 10. Find x when y = 25.Show answer
10.
8.xy = 36; find y when x = 9.Show answer
4.

Best order to study it

Learn what direct variation means and practise the unitary method.
Test tables for direct variation and fill missing values.
Learn inverse variation and test tables using products.
Practise deciding direct, inverse or neither from words.
Finish with word problems, including changes part-way through.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Did you state whether the variation is direct or inverse?
For direct, did you keep the quotient constant?
For inverse, did you keep the product constant?
Did you use the same units in both pairs?
In part-way problems, did you remove the work already done?
Does your answer make sense: bigger or smaller as expected?

Direct and Inverse Variation: frequently asked questions

What does x ∝ y mean?

It means x varies directly as y: x = ky for some constant k.

Is every 'one up, other down' pair an inverse variation?

No. It is inverse only if the product stays the same. For example, as a candle burns, time goes up and length goes down, but length × time is not constant.

Which method is better, unitary or proportion?

Both give the same answer. The unitary method is easy to explain; proportion is quicker once you are confident. Show clear working either way.

Why do speed and time vary inversely?

For a fixed distance, speed × time = distance is constant. So doubling the speed halves the time.

Are time and work problems related to this chapter?

Yes. Workers and days vary inversely, and the next chapter, Time and Work, builds on this idea using each person's work per day.

Based on: CISCE Curriculum for Upper Primary Classes (VI-VIII), Mathematics (Class VIII); CISCE RDCD: Curriculum Pre-School to Class VIII