SSC CGL Maths · Ratio, Proportion and Variation

Direct and Inverse Variation

Two quantities vary directly when their ratio stays fixed and inversely when their product stays fixed, so one known pair of values gives the constant and the constant gives every other value.

All Ratio, Proportion and Variation notes

Why this matters

Thirteen questions in the bank, nine EASY and four MODERATE: seven are direct proportion (shadows, recipes, cloth by the metre, a machine's output) and six are inverse or joint variation. All four MODERATE ones sit in the second group, where several quantities or a square act at once; multiplying by each factor's ratio clears most of them in one line.

Concept 1 of 2: Direct proportion and the unitary method

If one pen costs ₹12, two cost ₹24 and ten cost ₹120: the cost grows in step with the number. That is direct variation. The ratio cost : number never changes, so the cost of one (the unit) unlocks every other case.

Definition

y varies directly as x (written $y \propto x$ ) when $y = kx$ for a fixed k. Doubling x doubles y.
Unitary method: find the value for one unit, then multiply. 6 litres for 84 km is 14 km a litre.
Ratio method: $y_2 = y_1 \times \frac{x_2}{x_1}$ . Often quicker, since $\frac{x_2}{x_1}$ may be a neat number.
Shadows at the same time of day vary directly as the heights of the objects that cast them.
y varies as (x + c): write $y = k(x + c)$ and find k from the given pair.
Sense check: the answer must move the same way as the cause. More people need more flour.

Direct variation

y = kx \qquad \frac{y_2}{y_1} = \frac{x_2}{x_1}

kthe constant of variation, $\frac{y}{x}$

Worked example

A 6 m tree casts a 4.5 m shadow. At the same time, a tower casts a 30 m shadow. How tall is the tower?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 22 Sep 2025, Shift 3 · Q56Easy

Example 1 · Ratio, Proportion and Variation · Direct and Inverse Variation

A recipe requires

0.625

liters of milk for

5

servings. How many servings can be made with

3.75

liters of milk?

Turning a direct proportion upside down

If 4 kg of rice feeds 10 people, 25 people need

4 \times \frac{25}{10} = 10

kg, not

4 \times \frac{10}{25} = 1.6

kg. Ask first whether the answer should go up or down.

Drill 6 more on direct proportion and the unitary method

Concept 2 of 2: Inverse and joint variation

Filling a tank with smaller buckets takes more of them: the number of buckets times the size of a bucket is always the tank. When the product stays fixed, the variation is inverse. When a quantity depends on several others at once, each one acts by its own power, and the changes simply multiply.

Definition

y varies inversely as x when $xy = k$ . Doubling x halves y.
Joint variation: $y = \frac{k x^a z^b}{w^c}$ . Find k from the one full set of values given, then put in the new values.
Fast route: skip k. Multiply the old y by each factor's ratio raised to its power, flipping the ratio for a quantity in the denominator: $y_2 = y_1 \left(\frac{x_2}{x_1}\right)^a \left(\frac{w_1}{w_2}\right)^c$ .
Value as the square of weight (gems, diamonds): pieces in the ratio of their weights are worth the squares of those terms, which add to less than the square of the total. That gap is the loss.
Read the stem for the power: "as the square of", "as the cube of".

Inverse and joint variation

xy = k \qquad y = \frac{k x^a}{w^c} \qquad y_2 = y_1 \left(\frac{x_2}{x_1}\right)^a \left(\frac{w_1}{w_2}\right)^c

a, cthe powers stated in the stem
wa quantity that y varies inversely with

Worked example

z varies directly as the square of x and inversely as y. z = 6 when x = 3 and y = 12. Find z when x = 5 and y = 10.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 25 Sep 2024, Shift 3 · Q58Moderate

Example 2 · Ratio, Proportion and Variation · Direct and Inverse Variation

‘x’ varies directly with the cube of ‘y’, and inversely with the square of ‘z’. If

x = \frac{1}{36}

when

y = 2

and

z = 3

, then what is the value of

800x

when

y = 3

and

z = 5

Treating an inverse variation as direct

If each bucket holds one third as much, the tank needs 3 times as many buckets, not one third as many. Smaller buckets mean more trips.

Forgetting the power

If y varies as the square of x and x becomes 3 times as large, y becomes 9 times as large, not 3 times. Apply the stated power to the ratio of the x values.

Drill 5 more on inverse and joint variation

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Direct proportion and the unitary method
Direct variation
$y = kx \qquad \frac{y_2}{y_1} = \frac{x_2}{x_1}$
Inverse and joint variation
Inverse and joint variation
$xy = k \qquad y = \frac{k x^a}{w^c} \qquad y_2 = y_1 \left(\frac{x_2}{x_1}\right)^a \left(\frac{w_1}{w_2}\right)^c$

Watch out for (3)

Turning a direct proportion upside down
→ Direct proportion and the unitary method
Treating an inverse variation as direct
→ Inverse and joint variation
Forgetting the power
→ Inverse and joint variation

Test yourself on Ratio, Proportion and Variation

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Ratio, Proportion and Variation chapter test Drill all 13 questions