AFCAT Numerical Ability · Ratio and Proportion

Ratios in Algebra: Substitution and Componendo-Dividendo

Work out an expression from a given ratio, find x : y from an equation, and use componendo-dividendo and the k method on equal ratios.

All Ratio and Proportion notes

Why this matters

AFCAT's 2019 paper asked the first kind: given x : y, evaluate an expression in x and y. The quick way is to put the ratio's numbers straight in, which works whenever every term has the same degree. The rest of the page is the reverse problem and the two standard properties of equal ratios.

Concept 1 of 4: Evaluating an expression from a ratio

If every term of an expression has the same degree, the

k

x = pk

y = qk

cancels. So put the ratio's numbers in directly: for

x : y = 3 : 4

, use

x = 3

and

y = 4

Definition

If $x : y = p : q$ , write $x = pk$ and $y = qk$ . In an expression where every term has the same power (all first powers, or all squares), $k$ cancels.
So substitute $x = p$ , $y = q$ and work out the numbers.
Keep the sign: with $x : y = 3 : 4$ , $(x + y) : (x - y)$ is $7 : -1$ .
Three letters work the same way: $a : b : c = 1 : 2 : 3$ gives $(a + b) : c = 3 : 3 = 1 : 1$ .

Substituting the ratio

x : y = p : q \;\Rightarrow\; \dfrac{ax + by}{cx + dy} = \dfrac{ap + bq}{cp + dq}

Worked example

x : y = 2 : 5

, find

(3x + 4y) : (5x - y)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (II) 2019 — Numerical Ability · 04-301Easy

Example 1 · Ratio and Proportion · Algebraic Ratio Expressions

x : y = 5 : 6

, then

(3x^2 - 2y^2) : (y^2 - x^2)

Keep the minus sign

With

x : y = 3 : 7

(x + y) : (x - y) = 10 : -4 = -5 : 2

. Dropping the sign gives

5 : 2

, which is often one of the options.

Only when every term has the same degree

x : y = 2 : 3

does not fix

(x + 1) : y

, because the 1 carries no

k

. Put

x = 2k

y = 3k

and the

k

stays.

Drill 13 more on evaluating an expression from a ratio

Concept 2 of 4: Finding x : y from an equation

When an equation links

x

and

y

, collect the

x

terms on one side and the

y

terms on the other. What is left is

\dfrac xy

as a number.

Definition

From a ratio of two expressions, cross-multiply first, then collect $x$ and $y$ .
Percentages and decimals: '40% of $x$ equals 25% of $y$ ' means $0.4x = 0.25y$ , so $x : y = 0.25 : 0.4 = 5 : 8$ .
'The ratio $x : y$ is 9 times $y : x$ ' means $\dfrac{x^2}{y^2} = 9$ , so $x : y = 3 : 1$ for positive numbers.

Collecting x and y

ax + by = cx + dy \;\Rightarrow\; \dfrac xy = \dfrac{d - b}{a - c}

Worked example

(2x + 3y) : (x + 5y) = 5 : 7

, find

x : y

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Ratio and Proportion · Algebraic Ratio Expressions

If (3x + 2y):(2x + 3y) = 5:4, then x:y is:

Collect before you read off the ratio

From

9x = 4y

x : y = 4 : 9

. Reading

9 : 4

off the coefficients turns the ratio round.

Drill 5 more on finding x : y from an equation

Concept 3 of 4: Componendo and dividendo

\dfrac ab = \dfrac cd

, adding and subtracting the terms on each side gives

\dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}

. It turns a sum over a difference into a plain ratio in one step.

Definition

Componendo: $\dfrac ab = \dfrac cd \Rightarrow \dfrac{a + b}{b} = \dfrac{c + d}{d}$ .
Dividendo: $\dfrac ab = \dfrac cd \Rightarrow \dfrac{a - b}{b} = \dfrac{c - d}{d}$ .
Both together: $\dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}$ . It works in reverse: from $\dfrac{x + y}{x - y} = \dfrac pq$ , $\dfrac xy = \dfrac{p + q}{p - q}$ .
Look for it whenever a fraction is a sum over a difference, including roots: $\dfrac{\sqrt a + \sqrt b}{\sqrt a - \sqrt b}$ .
$x^3 + 3x$ and $3x^2 + 1$ are the two halves of $(x + 1)^3 = x^3 + 3x^2 + 3x + 1$ , so componendo-dividendo turns their ratio into $\left(\dfrac{x + 1}{x - 1}\right)^3$ .

Componendo and dividendo

\dfrac ab = \dfrac cd \;\Rightarrow\; \dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}

Worked example

\dfrac{x + y}{x - y} = \dfrac 73

, find

x : y

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Ratio and Proportion · Algebraic Ratio Expressions

\frac{x^3 + 3x}{3x^2 + 1} = \frac{341}{91}

, then the value of

x

Do the same on both sides

From

\dfrac{x + y}{x - y} = \dfrac 73

, add and subtract on the left and on the right:

\dfrac{2x}{2y} = \dfrac{10}{4}

. Doing it on one side only changes the equation.

Drill 4 more on componendo and dividendo

Concept 4 of 4: Equal ratios: sums and the k method

When several ratios are equal,

\dfrac ab = \dfrac cd = \dfrac ef = k

, every numerator is

k

times its denominator. So a sum of numerators over the matching sum of denominators is still

k

Definition

If $\dfrac ab = \dfrac cd = \dfrac ef = k$ , then $a = bk$ , $c = dk$ and $e = fk$ .
So $\dfrac{a + c + e}{b + d + f} = k$ . With the same multipliers on top and bottom, $\dfrac{2a + 3c}{2b + 3d} = k$ too.
Squares give $k^2$ : $\dfrac{a^2 + c^2}{b^2 + d^2} = k^2$ .

Sum of equal ratios

\dfrac ab = \dfrac cd = \dfrac ef = k \;\Rightarrow\; \dfrac{a + c + e}{b + d + f} = k

Worked example

\dfrac ab = \dfrac cd = \dfrac 35

, find

\dfrac{2a + 7c}{2b + 7d}

and

\dfrac{a^2 + c^2}{b^2 + d^2}

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Ratio and Proportion · Algebraic Ratio Expressions

a : b = c : d = e : f = 1 : 2

, then I.

\frac{3a + 5c + 7e}{3b + 5d + 7f} = \frac{1}{3}

II.

\frac{a^2 + c^2 + e^2}{b^2 + d^2 + f^2} = \frac{1}{2}

Which of the following options is correct?

The multipliers must match

\dfrac{2a + c}{b + 2d}

is not

k

in general, because

a

and

b

carry different multipliers. The rule needs the same multiplier on each top-and-bottom pair.

Drill 1 more on equal ratios: sums and the k method

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)

Evaluating an expression from a ratio
Substituting the ratio
$x : y = p : q \;\Rightarrow\; \dfrac{ax + by}{cx + dy} = \dfrac{ap + bq}{cp + dq}$
Finding x : y from an equation
Collecting x and y
$ax + by = cx + dy \;\Rightarrow\; \dfrac xy = \dfrac{d - b}{a - c}$
Componendo and dividendo
Componendo and dividendo
$\dfrac ab = \dfrac cd \;\Rightarrow\; \dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}$
Equal ratios: sums and the k method
Sum of equal ratios
$\dfrac ab = \dfrac cd = \dfrac ef = k \;\Rightarrow\; \dfrac{a + c + e}{b + d + f} = k$

Watch out for (5)

Keep the minus sign
→ Evaluating an expression from a ratio
Only when every term has the same degree
→ Evaluating an expression from a ratio
Collect before you read off the ratio
→ Finding x : y from an equation
Do the same on both sides
→ Componendo and dividendo
The multipliers must match
→ Equal ratios: sums and the k method

Drill the questions on this topic

27 questions from the bank, with cart and Word export.

Drill all 27 questions