Ratio, Proportion, Partnership & Ages

high importance~2 Q in Tier 123 formulas⚡ 15 shortcuts5 subtopics

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Ratio basics & dividing amounts

⏱ 4 min read🧩 5 question types🎯 12 practice Q

The idea in one minute

A ratio $a : b$ compares two quantities of the same kind. Scaling both terms by the same number leaves it unchanged: $2:3 = 8:12$ .

Write the quantities as $ax$ and $bx$ . The multiplier $x$ turns every extra fact — sum, difference, product — into a small equation.

One idea: the multiplier x

"Numbers are in ratio $a : b$ " means the numbers are $ax$ and $bx$ . The $x$ is one shared unit.

Sum given: $(a+b)x = S$ .
Difference given: $(b-a)x = D$ .
Product given: $abx^2 = P$ .

Two numbers in $5 : 7$ differ by 12: $2x = 12$ , so $x = 6$ and the sum is $12x = 72$ .

Rule: Never argue with the ratio itself. Convert to $ax$ , $bx$ , and solve for x.

Dividing an amount

Split N in $a : b : c$ : one part $= \dfrac{N}{a+b+c}$ , then multiply by each term.

Rs 780 in $3 : 4 : 6$ : one part $= \dfrac{780}{13} = 60$ , so the shares are 180, 240 and 360.

Questions often hide the total. "A gets Rs 300 more than B" gives $(b-a)x = 300$ — recover x, then the total is $(a+b+c)x$ .

A difference works the same way. Rs 1,870 in $4 : 7$ : eleven parts of 170, so the shares are 680 and 1,190 and Y gets $\dfrac{1870}{11} \times 3 = 510$ more than X. One division answers every share, every difference and the total.

Tip: Compute the value of one part first; every share follows by multiplication.

The LCM bridge

Given $A : B = 2 : 3$ and $B : C = 4 : 5$ , make the two B-terms equal. The LCM of 3 and 4 is 12, so scale the first ratio by 4 and the second by 3: $A : B = 8 : 12$ and $B : C = 12 : 15$ .

Read them together: $A : B : C = 8 : 12 : 15$ .

In symbols, $A:B = m:n$ and $B:C = p:q$ give $A:B:C = mp : np : nq$ .

The equality form

" $2A = 3B = 4C$ " — take the LCM of 2, 3, 4, which is 12. Each letter equals $\dfrac{12}{\text{its multiplier}}$ , so $A : B : C = 6 : 4 : 3$ .

Watch: Each term is LCM ÷ its own multiplier. A bigger multiplier means a smaller term.

When the ratio changes

"Add k and the ratio becomes $p : q$ ": write $\dfrac{ax + k}{bx + k} = \dfrac{p}{q}$ and cross-multiply once.

Boys : girls $= 4 : 5$ . After 100 boys join it is $6 : 5$ . Now $\dfrac{4x + 100}{5x} = \dfrac{6}{5}$ gives $20x + 500 = 30x$ , so $x = 50$ and the school had $9 \times 50 = 450$ students.

Both ratio statements describe the same x. That is the whole trick.

Which ratio is larger

To compare $a : b$ with $c : d$ , cross-multiply: $a : b$ is larger exactly when $ad > bc$ .

$3 : 4$ or $5 : 7$ ? Compare $3 \times 7 = 21$ with $4 \times 5 = 20$ . So $3 : 4$ is the larger ratio.

Squares and roots of ratios

The duplicate ratio of $a : b$ is $a^2 : b^2$ ; the sub-duplicate is $\sqrt{a} : \sqrt{b}$ . Squares with sides in $2 : 3$ have areas in $4 : 9$ — one squaring step.

Note: Ratio terms must share one unit. 3 kg to 500 g is $3000 : 500 = 6 : 1$ , never $3 : 500$ .

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Divide an amount in a given ratio

How to spot it:

A sum of money, a prize or sweets is split in m : n : p; a share, a difference or the total is asked.

\text{share} = \frac{\text{term}}{\text{sum of terms}} \times N, \quad \text{1 part} = \frac{N}{m + n + p}

Method

Add the ratio terms for the total parts.
One part = amount ÷ total parts.
Multiply by the asked term; for a difference use the gap between terms.

Why it works:

The ratio fixes each share as a set fraction of the whole.

Try this

Rs 780 is divided among A, B and C in the ratio 3 : 4 : 6. What is A's share?

Show solution

Total parts $= 13$ ; one part $= \dfrac{780}{13} = 60$ .
A $= 3 \times 60 = 180$ .

Answer

Rs 180

Type 2very common3 practice Q

Combine ratios into A : B : C

How to spot it:

'A : B = 2 : 3 and B : C = 4 : 5, find A : B : C' — or the compact form '2A = 3B = 4C'.

A:B = m:n,\ B:C = p:q \Rightarrow A:B:C = mp : np : nq

Method

Two-ratio form: scale so the shared letter matches (LCM bridge).
One-equality form: each term is LCM ÷ its multiplier.
Long chains: multiply the fractions A/B × B/C and reduce.

Why it works:

The shared term must stand for the same number in both ratios before merging.

Try this

If 2A = 3B = 4C, then A : B : C is:

Show solution

LCM(2, 3, 4) $= 12$ .
$A = \dfrac{12}{2},\ B = \dfrac{12}{3},\ C = \dfrac{12}{4}$ .
$A : B : C = 6 : 4 : 3$ .

Answer

6 : 4 : 3

Type 3very common2 practice Q

Multiplier with sum, difference or product

How to spot it:

'Two numbers are in ratio a : b and their sum, difference or product is ...' — one extra fact, one equation.

ax \pm bx = (a \pm b)x, \qquad abx^2 = P \Rightarrow x = \sqrt{P/(ab)}

Method

Write the numbers as ax and bx.
Sum → (a+b)x; difference → (b−a)x; product → abx².
Solve for x, then build whatever is asked.

Why it works:

Every statistic of the pair is a multiple of x, so one fact fixes x.

Try this

The ratio of two numbers is 7 : 11 and their product is 693. Find the sum of the numbers.

Show solution

$77x^2 = 693$ , so $x^2 = 9$ and $x = 3$ .
Numbers are 21 and 33.
Sum $= 54$ .

Answer

Type 4common2 practice Q

Ratio changes when a number joins or leaves

How to spot it:

'Boys to girls is 4 : 5; after 100 boys join it becomes 6 : 5' — a fixed number moves.

\frac{ax \pm k}{bx \pm k} = \frac{p}{q}

Method

Write the original counts as ax, bx.
Add or subtract k only where the question says.
Equate to the new ratio and cross-multiply.

Why it works:

A fixed number breaks the proportionality, so the new ratio pins x exactly.

Try this

The ratio of boys to girls in a school is 4 : 5. If 100 more boys join, the ratio becomes 6 : 5. How many students were there originally?

Show solution

$\dfrac{4x + 100}{5x} = \dfrac{6}{5}$ .
$20x + 500 = 30x$ , so $x = 50$ .
Students $= 9 \times 50 = 450$ .

Answer

450

Type 5common

Comparing two ratios

How to spot it:

Two ratios are given; which is larger, or are they equal, is asked.

a:b > c:d \iff ad > bc

Method

Cross-multiply: first term × other denominator.
Compare the two products.
Equal products mean equal ratios.

Why it works:

Cross-multiplication puts both ratios over a common base in one step.

Try this

Which of the ratios 3 : 4 and 5 : 7 is greater?

Show solution

$3 \times 7 = 21$ .
$4 \times 5 = 20$ .
$21 > 20$ , so $3 : 4$ is greater.

Answer

3 : 4

Formula sheet

Share of the total

\text{share} = \frac{\text{ratio term}}{\text{sum of terms}} \times N

Combining ratios

A:B = m:n,\ B:C = p:q \Rightarrow A:B:C = mp : np : nq

Cross-multiplication test

a:b > c:d \iff ad > bc

Multiplier method

\text{shares } ax, bx,\ (b-a)x = \text{given difference}

Duplicate / sub-duplicate

a:b \Rightarrow a^2:b^2 \text{ (duplicate)},\ \sqrt{a}:\sqrt{b} \text{ (sub-duplicate)}

Shortcuts that save time

⚡ x solves everything

Translate 'ratio a : b' into ax and bx. Every extra fact becomes one small equation in x.

Example

Two numbers are in the ratio 5 : 7 and their difference is 12. Find their sum.

Show solution

Numbers are $5x$ and $7x$ ; $2x = 12$ .
$x = 6$ ; sum $= 12x = 72$ .

Answer

⚡ LCM bridging for three terms

Scale A : B so its B-term matches the B-term of B : C, then read off A : B : C.

Example

If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.

Show solution

Make B = 12 (LCM of 3 and 4).
$A : B = 8 : 12$ and $B : C = 12 : 15$ .
$A : B : C = 8 : 12 : 15$ .

Answer

8 : 12 : 15

⚡ One share known, get the rest

Share = fraction × total. Recover the total from one share, then build any other share.

Example

Rs 1,200 is divided between A and B in the ratio 2 : 3. Find B's share.

Show solution

B's fraction $= \dfrac{3}{5}$ .
$\dfrac{3}{5} \times 1200 = 720$ .

Answer

Rs 720

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Adding ratios term-wise: A:B = 2:3 and B:C = 4:5 read as 2:3:5.

Bridge the shared B: A:B:C = 8:12:15.

Mistake 02

Shifting a ratio by n years or n rupees.

Shift the actual quantities (ax ± n), never the ratio terms.

Mistake 03

Comparing ratios by their differences (b − a).

Cross-multiply: a:b beats c:d exactly when ad > bc.

Mistake 04

Mixing units inside one ratio.

Convert first: 3 kg to 500 g is 3000 : 500 = 6 : 1.

Mistake 05

Reading 'the ratio becomes p : q' as more information about the same x pair.

It is the same x pair — write both statements with one x and equate.

Quick revision

Read this the night before the exam.

Quantities in a ratio are $ax$ , $bx$ — one multiplier for all.
One part $=$ total $\div$ sum of terms.
Bridge two ratios through the shared term (LCM).
$kA = lB = mC$ : each term is LCM $\div$ multiplier.
Changed ratio: same x, one cross-multiplication.
Compare ratios by cross-multiplying, never by gaps.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.

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