MathsICSE Class 8 · Direct and Inverse Variation

Direct Variation: Tables, Unitary Method and Word Problems (ICSE Class 8 Maths)

Updated 17 October 2026 · Fact-checked

Two quantities vary directly if they increase or decrease together in the same ratio, so $\frac{x}{y}$ is constant. If $x_1, y_1$ and $x_2, y_2$ are two pairs, then $\frac{x_1}{y_1} = \frac{x_2}{y_2}$. Solve with this proportion or by finding the value of one unit.

Understand Direct Variation Problems

The CISCE curriculum covers direct and inverse variation through simple, direct word problems.

Direct variation. $x$ and $y$ vary directly when $y = k x$ for a constant $k$ (the constant of variation), equivalently $\frac{y}{x} = k$ . Double $x$ and $y$ doubles; halve $x$ and $y$ halves.

Testing a table. Compute $\frac{y}{x}$ for each pair. If all ratios are equal, the variation is direct; one unequal ratio rules it out.

Two methods.

Proportion: $\frac{x _{1}}{y _{1}} = \frac{x _{2}}{y _{2}}$ , then cross-multiply.
Unitary method: find the value for one unit, then multiply: if 8 notebooks cost Rs 96, one costs Rs 12, so 15 cost Rs 180.

Standard situations. Cost and quantity, distance and time at a constant speed, wages and days worked, quantity and ingredients, map distance and real distance (scale), height and shadow length at the same time of day.

Careful. Direct variation does not mean "addition": it means equal ratios. Also check the units match on both sides, such as hours with hours, cm with cm.

Linking to the graph (idea). Plotting $y$ against $x$ for direct variation gives points on a straight line through the origin (not required to draw, but it explains why $y = 0$ when $x = 0$ ).

Direct Variation Problems: formulas to remember

Direct variation: $\frac{y}{x} = k (constant)$; Same ratio for every pair.
Proportion: $\frac{x _{1}}{y _{1}} = \frac{x _{2}}{y _{2}}$; Cross-multiply to solve.
Unitary method: $value of 1 unit \times number of units$; Divide first, then multiply.

How to solve Direct Variation Problems questions

Check the quantities increase together, then use the constant ratio.

1Decide whether the two quantities increase or decrease together.
2Write the ratio $\frac{y}{x}$ from the complete pair.
3Set it equal to the unknown ratio and solve.
4Or find the value of one unit, then scale.
5Check the units and the reasonableness of the answer.

Direct 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.

Shadow and height

Suppose at a certain time a 1.5 m pole casts a 2 m shadow, and a tower casts a 36 m shadow at the same time.

Find the height of the tower.

Try it first, then show the solutionSolution

$\frac{h}{36} = \frac{1.5}{2}$ .
$h = 36 \times 0.75 = 27$ m.

Answer: 27 m.

Memory hook: Same time of day, same ratio of height to shadow.

Tricks to remember Direct Variation Problems

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

Together up, together down
If more of one means more of the other, it is direct.
Use it when: Deciding the type.
Unit rate first
Divide to get 1, then multiply to get the asked number.
Use it when: Quick MCQs.

Direct Variation Problems: solved examples

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

Example 1: Cost of notebooks · 1 mark · MCQBoard level

If 8 notebooks cost Rs 96, the cost of 15 notebooks is:
(a) Rs 160 (b) Rs 170 (c) Rs 180 (d) Rs 190

Try it first, then show the solutionSolution

One notebook: $96 \div 8 = 12$ ; fifteen: $12 \times 15 = 180$ .

Answer: (c)

Example 2: Distance and time · 3 marks · Short answerBoard level

A car covers 135 km in 3 hours at a steady speed. How far will it go in 5 hours?

Try it first, then show the solutionSolution

Speed $= \frac{135}{3} = 45$ km/h.
Distance $= 45 \times 5 = 225$ km.
Check by proportion: $\frac{135}{3} = \frac{x}{5}$ gives $x = 225$ .

Answer: 225 km

Example 3: Testing two tables · 3 marks · Short answerHarder board level

Table A: $x = 2, 6, 10$ with $y = 7, 21, 35$ . Table B: $x = 4, 6, 9$ with $y = 10, 16, 22$ . Which table shows direct variation?

Try it first, then show the solutionSolution

Table A: $\frac{7}{2} = 3.5$ , $\frac{21}{6} = 3.5$ , $\frac{35}{10} = 3.5$ . Constant.
Table B: $\frac{10}{4} = 2.5$ , $\frac{16}{6} \approx 2.67$ . Not equal.

Answer: Table A is direct; B is not.

Example 4: Map scale · 4 marks · Short answerHarder board level

On a map, 3 cm represents 12 km. (i) A road measures 8.5 cm on the map. Find its actual length. (ii) A town is 50 km away. Find its distance on the map.

Try it first, then show the solutionSolution

Scale: $1$ cm $= 4$ km.
(i) $8.5 \times 4 = 34$ km.
(ii) $50 \div 4 = 12.5$ cm.

Answer: (i) 34 km; (ii) 12.5 cm

Example 5: Shadow problem · 4 marks · Case-basedHOTS / competency

A 1.5 m pole casts a 2 m shadow. At the same time a tower casts a 36 m shadow and a boy casts a 3.2 m shadow. Find the height of the tower and of the boy.

Try it first, then show the solutionSolution

Ratio height : shadow $= \frac{1.5}{2} = 0.75$ .
Tower: $36 \times 0.75 = 27$ m.
Boy: $3.2 \times 0.75 = 2.4$ m.
Both obey the same ratio, so the variation is direct.

Answer: Tower 27 m; boy 2.4 m

Where students lose marks in Direct Variation Problems

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

Mistake: Adding instead of scaling
Direct variation keeps ratios equal, not differences.
Wrong working
12 kg costs 540, so 20 kg costs $540 + 8$ .
Right working
20 kg costs $\frac{540}{12} \times 20 = 900$ .
Do this: Always compute a unit rate or a ratio.
Usually costs: Method and answer marks.
Mistake: Mixing units in the proportion
Hours and minutes (or cm and m) cannot sit in the same ratio unchanged.
Wrong working
$\frac{135}{3 h} = \frac{x}{90 min}$ with $x = 4050$ .
Right working
Convert $90$ min to $1.5$ h: $x = 45 \times 1.5 = 67.5$ km.
Do this: Convert units before forming the ratio.
Usually costs: Answer mark.

Exam tips

Name the two quantities and their direction at the start.
Check by substituting back into the ratio.
Where a decimal answer is natural (like 8.75), give it as it is.

Practice questions on Direct 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
If $x$ and $y$ vary directly and $y = 10$ when $x = 4$ , then $y$ when $x = 10$ is:
(a) 4 (b) 25 (c) 40 (d) 100
Hint
$\frac{y}{x} = \frac{10}{4} = 2.5$ .
Answer
(b)
Q2 · 2 marks · Short answerBoard level
12 kg of sugar costs Rs 540. Find the cost of 20 kg.
Hint
Find the price of 1 kg.
Answer
1 kg $= 45$ ; 20 kg $= 900$ . Rs 900.
Q3 · 3 marks · Short answerHarder board level
A worker earns Rs 1,950 for 6 days. For how many days must he work to earn Rs 5,525?
Hint
Daily wage first.
Answer
Daily wage $= \frac{1950}{6} = 325$ . Days $= \frac{5525}{325} = 17$ .
Q4 · 3 marks · Short answerHarder board level
250 g of flour makes 15 biscuits. How much flour is needed for 54 biscuits?
Hint
Flour per biscuit is not a whole number; use the proportion.
Answer
$\frac{250}{15} = \frac{x}{54}$ , so $x = \frac{250 \times 54}{15} = 900$ g.
Q5 · 4 marks · Case-basedHOTS / competency
A bus travels 180 km on 15 litres of diesel. Find (i) the diesel needed for 468 km, (ii) the distance on 22.5 litres, (iii) the fuel cost per km if diesel costs Rs 105 per litre.
Hint
Distance per litre first.
Answer
180 km per 15 L is 12 km per L. (i) $468 \div 12 = 39$ L. (ii) $22.5 \times 12 = 270$ km. (iii) $105 \div 12 =$ Rs 8.75 per km.

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 direct variation?Show answer
The ratio y/x.
2.If 5 pens cost Rs 40, cost of 9 pens?Show answer
Rs 72.
3.Does a larger distance take a longer or shorter time at constant speed?Show answer
Longer: direct variation.
4.What is the graph of direct variation like?Show answer
A straight line through the origin.

Direct Variation: Tables, Unitary Method and Word Problems (ICSE Class 8 Maths)

Understand Direct Variation Problems

Direct Variation Problems: formulas to remember

How to solve Direct Variation Problems questions

Direct Variation Problems in real life

Tricks to remember Direct Variation Problems

Direct Variation Problems: solved examples

Where students lose marks in Direct Variation Problems

Exam tips

Practice questions on Direct Variation Problems

Test your recall

Direct Variation Problems: frequently asked questions

Is every increasing relationship direct?

Can I use unitary method for all direct problems?