CDS Mathematics · Ratio, Proportion and Variation

Direct and Inverse Variation

'y varies as x' means y = kx; 'inversely' means y = k/x; find k from one pair of values and use it for another.

All Ratio, Proportion and Variation notes CDS Mathematics strategy

Why this matters

Eighteen PYQs, the largest page in the chapter. Every computational one is the same three steps: write the equation with k, find k from the given pair, substitute the new values. The statement questions ask which expressions still vary together — substitute x = ky and see whether the ratio is constant.

Concept 1 of 2: Finding the constant of variation

'Varies as' is shorthand for 'is a constant multiple of'. Once that constant is known from one situation, every other situation follows. Inverse variation keeps the PRODUCT constant.

Definition

$y \propto x$ : $y = kx$ . $y \propto x^n$ : $y = kx^n$ . $y \propto \dfrac1x$ : $xy = k$ .
Joint variation: $p \propto \dfrac{q}{r^2}$ means $p = \dfrac{kq}{r^2}$ ; percentage changes multiply through.
Fixed stock shared among men for days: men $\times$ days is constant.
A quantity 'reduced by an amount varying as $\sqrt n$ ': $v = v_0 - c\sqrt n$ .

Direct and inverse

y = kx^n, \qquad x^n y = k

Worked example

y

varies inversely as

x^2

, and

y = 12

when

x = 2

. Find

y

when

x = 4

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q75Moderate

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

Given

y

is inversely proportional to

\sqrt{x}

, and

x = 36

when

y = 36

. What is the value of

x

when

y = 54

Inverse means the product is constant

y

varies inversely as

\sqrt x

, then

y\sqrt x

is constant, so a bigger

y

needs a SMALLER

x

. An answer larger than the starting

x

signals a direct-variation slip.

Drill 10 more on finding the constant of variation

Concept 2 of 2: Which quantities vary together

To test whether

A

varies as

B

, substitute the given relation (

x = ky

) and look at

\dfrac AB

. If it is a constant — no

x

y

left — they vary together.

Definition

If $x = ky$ , then any two expressions of the same degree in $x, y$ vary together: $x^2 + y^2 = (k^2 + 1)y^2$ .
$\dfrac{x^3}{y^4} = \dfrac{k^3}{y}$ varies INVERSELY as $y$ .
If $x \propto yz$ , then $y \propto \dfrac xz$ , i.e. $y$ varies inversely as $\dfrac zx$ .
If $P^2 \propto R$ and $Q^2 \propto R$ , then $P^2 \pm Q^2$ and $PQ$ also vary as $R$ .

Test for variation

A \propto B \iff \dfrac AB = \text{constant}

Worked example

x

varies as

y

, does

x^2 - xy

vary as

y^2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (I) 2026 — Elementary Mathematics · Q13Moderate

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

If x varies directly as y, then which of the following is/are correct ? I.

(x^2 + y^2)

varies directly as

y^2

. II.

\left(\frac{x^3}{y^4}\right)

varies directly as y. Select the answer using the code given below :

Count the degree

After

x = ky

, an expression of degree

n

becomes a constant times

y^n

. So

\dfrac{x^3}{y^4}

(degree

-1

) varies inversely as

y

, not directly.

Drill 6 more on which quantities vary together

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)

Finding the constant of variation
Direct and inverse
$y = kx^n, \qquad x^n y = k$
Which quantities vary together
Test for variation
$A \propto B \iff \dfrac AB = \text{constant}$

Watch out for (2)

Inverse means the product is constant
→ Finding the constant of variation
Count the degree
→ Which quantities vary together

Test yourself on Ratio, Proportion and Variation

20 past CDS questions from this chapter, timed at 24 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 18 questions