AFCAT Numerical Ability · Ratio and Proportion

Direct, Inverse and Joint Variation

Write the variation as an equation with one constant, find the constant from the values given, and use it for the new values; then spot the same pattern in work, speed and wage problems.

All Ratio and Proportion notes

Why this matters

Every computational question here is the same three steps: write the equation with a constant, find the constant, substitute. The HARD ones add a power, a second variable or one more condition, and the word problems (men and days, speed and time) are inverse variation in disguise.

Concept 1 of 4: Direct variation

x

varies directly as

y

when

x = ky

: double

y

and

x

doubles. Every direct-variation question is the same three steps. Write

x = ky

, find

k

from the pair given, and use it for the new value.

Definition

$x \propto y$ means $x = ky$ for a constant $k$ , or equally $\dfrac{x_1}{y_1} = \dfrac{x_2}{y_2}$ .
Powers and roots: $x \propto y^2$ means $x = ky^2$ , and $x \propto \sqrt y$ means $x = k\sqrt y$ .
Chains: if $x \propto y^m$ and $y \propto z^n$ , then $x \propto z^{mn}$ .
A value that varies as the square of the weight: break the object, and each piece is valued by its own square, so the total value falls.

Direct variation

x \propto y \;\text{means}\; x = ky, \text{ so } \dfrac{x_1}{y_1} = \dfrac{x_2}{y_2}

Worked example

x

varies as

y^2

, and

x = 27

when

y = 3

. Find

x

when

y = 5

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 1 · Ratio and Proportion · Variation and Work

The price of a diamond is directly proportional to the square of its weight. If a diamond worth

₹8100

is broken into two pieces whose weights are in the ratio

2:1

, find the loss in value.

A square doubles twice

x \propto y^2

and

y

doubles,

x

becomes four times as large, not twice.

Drill 8 more on direct variation

Concept 2 of 4: Inverse variation

x

varies inversely as

y

when their product stays fixed: double

y

and

x

halves.

Definition

$x \propto \dfrac 1y$ means $xy = k$ , or equally $x_1 y_1 = x_2 y_2$ .
With a power: $x \propto \dfrac{1}{y^2}$ means $xy^2 = k$ .

Inverse variation

x \propto \dfrac 1y \;\text{means}\; xy = k, \text{ so } x_1 y_1 = x_2 y_2

Worked example

a

varies inversely as

b^2

, and

a = 18

when

b = 2

. Find

a

when

b = 3

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Ratio and Proportion · Variation and Work

x

varies inversely as the square of

y

in such a way that, if

x = 1

, then

y = 6

. I. If

y = 3

, then

x = 4

II. If

y = 6

, then

x = 1

Which of the following options is correct?

Inverse means dividing, not subtracting

y

rises from 4 to 8, an inversely varying

x

halves. It does not fall by 4.

Drill 5 more on inverse variation

Concept 3 of 4: Joint and combined variation

Several variations at once multiply into one formula:

x

varies directly as

y

and inversely as

z^2

means

x = \dfrac{ky}{z^2}

. There is still only one constant to find.

Definition

Directly as $y$ and inversely as $z$ : $x = \dfrac{ky}{z}$ .
Jointly as $b$ and $c$ : $a = kbc$ .
' $y$ is the sum of two parts, one varying as $x$ and one inversely as $x$ ': $y = ax + \dfrac bx$ . That has two constants, so it needs two pairs of values.

Combined variation

x \propto \dfrac{y}{z^2} \;\text{means}\; x = \dfrac{ky}{z^2}

Worked example

p

varies directly as

q

and inversely as

r

p = 6

when

q = 4

and

r = 2

. Find

p

when

q = 9

and

r = 3

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Ratio and Proportion · Variation and Work

Let

y

be equal to the sum of two quantities of which one varies directly as

x

and the other inversely as

x

. If

y = 6

when

x = 4

and

y = \frac{10}{3}

when

x = 3

, then what is the relation between

x

and

y

One relation, one constant

x

varies as

y

and inversely as

z

' is one relation,

x = \dfrac{ky}{z}

, with one

k

. Giving

y

and

z

a constant each leaves an equation you cannot solve.

Drill 4 more on joint and combined variation

Concept 4 of 4: Variation in word problems: work, speed and wages

Many word problems are a variation in disguise. For a fixed job, workers and days vary inversely. Over a fixed distance, speed and time vary inversely. Wages go with the work done.

Definition

A fixed job: workers × days stays the same (man-days).
A fixed distance: speed × time stays the same, so the times are in the inverse ratio of the speeds.
Wages are shared in the ratio of the work done. For people working together, that is the ratio of their daily rates $\dfrac 1a : \dfrac 1b$ .
A wage bill is workers × wage, so it changes by the compound ratio of the two changes.

Man-days

M_1 D_1 = M_2 D_2

Worked example

18 workers finish a wall in 10 days. How many days do 12 workers take?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Ratio and Proportion · Variation and Work

The wages of labourers in a factory has increased in the ratio 22 : 25 and their number decreased in the ratio 3 : 2. What was the original wage bill of the factory, if the present bill is ₹5000?

More workers, fewer days

12 workers take 10 days, so 20 workers take 6 days. Setting it up as a direct proportion,

12 : 20 = 10 : x

, gives 16.7 days: longer with more people.

Drill 6 more on variation in word problems: work, speed and wages

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 (4)

Direct variation
Direct variation
$x \propto y \;\text{means}\; x = ky, \text{ so } \dfrac{x_1}{y_1} = \dfrac{x_2}{y_2}$
Inverse variation
Inverse variation
$x \propto \dfrac 1y \;\text{means}\; xy = k, \text{ so } x_1 y_1 = x_2 y_2$
Joint and combined variation
Combined variation
$x \propto \dfrac{y}{z^2} \;\text{means}\; x = \dfrac{ky}{z^2}$
Variation in word problems: work, speed and wages
Man-days
$M_1 D_1 = M_2 D_2$

Watch out for (4)

A square doubles twice
→ Direct variation
Inverse means dividing, not subtracting
→ Inverse variation
One relation, one constant
→ Joint and combined variation
More workers, fewer days
→ Variation in word problems: work, speed and wages

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