MathsICSE Class 8 · Direct and Inverse Variation

Inverse Variation: Constant Product, Workers and Days, Speed and Time (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

Two quantities vary inversely if one increases while the other decreases so that their product $xy$ stays constant. For two pairs, $x_1 y_1 = x_2 y_2$. Typical cases are workers and days, speed and time over a fixed distance, and people and the days food lasts.

Notice the rectangles are different shapes but have the same area: when men increase, days decrease so that men × days stays 180.

Illustration drawn for QuestionBuddy.

Understand Inverse Variation Problems

The CISCE curriculum asks for simple, direct word problems on direct and inverse variation.

Inverse variation. $x$ and $y$ vary inversely when $x y = k$ for a constant $k$ , that is $y = \frac{k}{x}$ . If $x$ doubles, $y$ halves; if $x$ is multiplied by $3$ , $y$ is divided by $3$ .

Testing a table. Multiply each pair. If every product is the same, the variation is inverse.

Method. Use $x_{1} y_{1} = x_{2} y_{2}$ , or find the total (the constant) first. For example, 12 men finish a job in 15 days, so the job is $12 \times 15 = 180$ man-days; with 20 men it takes $\frac{180}{20} = 9$ days.

Standard situations.

Number of workers and days to finish the job (same hours).
Speed and time for the same distance.
Number of people and days the same stock of food lasts.
Number of pipes and time to fill the same tank.

Direct, inverse or neither. Ask: "If one doubles, what happens to the other?" Doubles: direct. Halves: inverse. Neither pattern: neither (for instance age and height).

Combined variation (extension). When several factors change, find the total work in "man-days" or "man-hour-days", then divide by the new combined rate.

Provisions after a change. If the group changes midway, compute the part used, then the remaining stock, then divide by the new number of consumers.

Inverse Variation Problems: formulas to remember

Inverse variation: $x y = k (constant)$; Product is constant.
Two situations: $x_{1} y_{1} = x_{2} y_{2}$; Solve for the unknown.
Man-days: $total work = men \times days$; Constant for a fixed job.

How to solve Inverse Variation Problems questions

Find the constant product (total work or total stock), then divide by the new quantity.

1Confirm that more of one means less of the other.
2Compute the constant product from the complete pair.
3Divide the constant by the new value of the known quantity.
4For midway changes, compute the part used and the remainder separately.
5Check that the answer moves in the correct direction.

Inverse Variation Problems in real life

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.

Journey time

Suppose a train covers a fixed route in 4 h 30 min at 60 km/h.

Find the time needed at 90 km/h, and the speed needed to do the route in 2 h 15 min.

Try it first, then show the solutionSolution

Distance $= 60 \times 4.5 = 270$ km.
At 90 km/h: $270 \div 90 = 3$ h. In $2.25$ h: $270 \div 2.25 = 120$ km/h.

Answer: 3 hours; 120 km/h.

Memory hook: Fixed distance: speed times time is constant.

Tricks to remember Inverse Variation Problems

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

Fix the total first
Multiply the given pair to get the total work, stock or distance.
Use it when: All inverse problems.
Opposite directions
If one goes up the other must go down; check your answer obeys this.
Use it when: Spotting errors.

Inverse Variation Problems: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Constant product · 1 mark · MCQBoard level

$x$ and $y$ vary inversely. If $x = 6$ when $y = 8$ , then $y$ when $x = 12$ is:
(a) 4 (b) 16 (c) 2 (d) 12

Try it first, then show the solutionSolution

$x y = 48$ ; $y = \frac{48}{12} = 4$ .

Answer: (a)

Example 2: Workers and days · 3 marks · Short answerBoard level

12 men can finish a job in 15 days. In how many days can 20 men finish it?

Try it first, then show the solutionSolution

Total work $= 12 \times 15 = 180$ man-days.
Days $= \frac{180}{20} = 9$ .

Answer: 9 days

Example 3: People join · 3 marks · Short answerHarder board level

A hostel has food for 45 students for 20 days. If 15 more students join, how long will the food last?

Try it first, then show the solutionSolution

Total $= 45 \times 20 = 900$ student-days.
New number $= 60$ ; days $= \frac{900}{60} = 15$ .

Answer: 15 days

Example 4: Speed and time · 4 marks · Short answerHarder board level

A bus takes 4 h 30 min at 60 km/h. (i) Find the time at 90 km/h. (ii) Find the speed needed to cover the distance in 2 h 15 min.

Try it first, then show the solutionSolution

Distance $= 60 \times 4.5 = 270$ km.
(i) Time $= \frac{270}{90} = 3$ h.
(ii) $2$ h $15$ min $= 2.25$ h; speed $= \frac{270}{2.25} = 120$ km/h.

Answer: (i) 3 h; (ii) 120 km/h

Example 5: Provisions with a change · 4 marks · Long answerHOTS / competency

A fort has provisions for 200 soldiers for 24 days. After 6 days, 40 soldiers leave. For how many more days will the remaining provisions last for the remaining soldiers?

Try it first, then show the solutionSolution

Total $= 200 \times 24 = 4800$ soldier-days. Used in 6 days: $200 \times 6 = 1200$ .
Remaining $= 3600$ soldier-days.
Soldiers left $= 160$ ; days $= \frac{3600}{160} = 22.5$ .

Answer: 22.5 days

Where students lose marks in Inverse Variation Problems

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

Mistake: Treating an inverse problem as direct
More workers means fewer days.
Wrong working
12 men, 15 days; 20 men: $\frac{15 \times 20}{12} = 25$ days.
Right working
$\frac{12 \times 15}{20} = 9$ days.
Do this: Ask: should the answer be bigger or smaller?
Usually costs: Answer mark.
Mistake: Forgetting to subtract the part already used
In midway-change problems the remaining stock is smaller than the total.
Wrong working
Remaining days $= \frac{4800}{160} = 30$ .
Right working
Remaining stock $= 4800 - 1200 = 3600$ ; days $= 22.5$ .
Do this: Write "used" and "remaining" as separate lines.
Usually costs: Answer and method marks.

Exam tips

State "inverse variation, so product is constant".
Convert hours and minutes into one unit before multiplying.
Sanity-check: more people should mean fewer days.

Practice questions on Inverse Variation Problems

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 · MCQBoard level
Which pair of quantities varies inversely?
(a) cost and number of items (b) speed and time for a fixed distance (c) side and area of a square (d) height and age
Hint
Fixed total.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
8 pipes fill a tank in 36 minutes. How long will 12 pipes (of the same kind) take?
Hint
Constant product.
Answer
$8 \times 36 = 288$ ; $\frac{288}{12} = 24$ minutes.
Q3 · 3 marks · Short answerHarder board level
The table gives $x = 2, 4, 5, 10$ and $y = 20, 10, 8, 4$ . Show that $x$ and $y$ vary inversely and find $y$ when $x = 16$ .
Hint
Multiply each pair.
Answer
Products: $40, 40, 40, 40$ . Constant, so inverse. $y = \frac{40}{16} = 2.5$ .
Q4 · 3 marks · Short answerHarder board level
A hostel has food for 120 students for 30 days. For how many students will the same food last 45 days?
Hint
Total student-days.
Answer
$120 \times 30 = 3600$ ; students $= \frac{3600}{45} = 80$ .
Q5 · 4 marks · Long answerHOTS / competency
18 workers can paint a building in 25 days working 8 hours a day. In how many days can 15 workers do it if they work 10 hours a day?
Hint
Total = workers x days x hours.
Answer
Total $= 18 \times 25 \times 8 = 3600$ worker-hours. New rate $= 15 \times 10 = 150$ worker-hours per day. Days $= \frac{3600}{150} = 24$ .

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.What stays constant in inverse variation?Show answer
The product xy.
2.If 6 men do a job in 10 days, how many man-days?Show answer
60.
3.If speed doubles, time for a fixed distance?Show answer
Halves.
4.Are age and height inversely related?Show answer
No; neither direct nor inverse.

Inverse Variation: Constant Product, Workers and Days, Speed and Time (ICSE Class 8)

Understand Inverse Variation Problems

Inverse Variation Problems: formulas to remember

How to solve Inverse Variation Problems questions

Inverse Variation Problems in real life

Tricks to remember Inverse Variation Problems

Inverse Variation Problems: solved examples

Where students lose marks in Inverse Variation Problems

Exam tips

Practice questions on Inverse Variation Problems

Test your recall

Inverse Variation Problems: frequently asked questions

How do I tell direct from inverse?

Can inverse variation have a zero value?