Ratio, Proportion, Partnership & Ages

The multiplier x solves everything

"Numbers are in the ratio $a : b$" means the numbers are $ax$ and $bx$. Every extra fact is a one-line equation in x:

• Sum given: $(a + b)x = S$.
• Difference given: $(b - a)x = D$.
• Product given: $abx^2 = P$ → take a square root. Once x is known, both numbers, their sum, difference and product are immediate. Sum of parts $= (a+b)x$, so 1 part $= \frac{S}{a + b}$.

Dividing an amount in a ratio

For $N$ split in $a : b : c$, the shares are $\frac{a}{a+b+c}N$, $\frac{b}{a+b+c}N$, $\frac{c}{a+b+c}N$. Fastest form: value of one part $= \frac{N}{a+b+c}$, then multiply. Questions hide the total: "A gets ₹300 more than B" → $(b - a)x = 300$ → recover x → total $= (a + b + c)x$.

Combining two ratios (the LCM bridge)

Given $A : B = m : n$ and $B : C = p : q$, make the B-terms equal: scale the first by p, the second by n (or both to the LCM of the two B values). Then $A : B : C = mp : np : nq$. Read the answer from the middle out: the A-term comes from the first ratio, the C-term from the second, and B is the common bridge value. The one-equality version: $2A = 3B = 4C$ → take the LCM of 2, 3, 4 (here 12): $A : B : C = 6 : 4 : 3$. Each term = LCM ÷ its multiplier.

Long chains

For $A : B$, $B : C$, $C : D$ given and $A : D$ asked, multiply the fractions: $\frac{A}{D} = \frac{A}{B} \times \frac{B}{C} \times \frac{C}{D}$, then reduce.

When a ratio changes

"Add k to each term and the ratio becomes p : q": $\frac{ax + k}{bx + k} = \frac{p}{q}$ — one cross-multiplication gives x. Subtractions work the same with −k. The two ratio statements share the SAME x; that is the whole trick.

Comparing two ratios

To compare $a : b$ with $c : d$, cross-multiply: $a : b$ is greater exactly when $ad > bc$. Example: which is larger, $3 : 4$ or $5 : 7$? Compare $3 \times 7 = 21$ with $4 \times 5 = 20$ → $3 : 4$ wins. Duplicate and sub-duplicate ratios come up in harder questions: $a : b$ squared is $a^2 : b^2$ (duplicate), $\sqrt{a} : \sqrt{b}$ is the sub-duplicate. "The ratio of the areas of two squares whose sides are in ratio 2 : 3" is simply $4 : 9$.

Same kind, same unit

Ratio terms must measure the same kind of quantity in the same unit — ₹ with ₹, kg with kg. A question giving "3 kg to 500 g" expects $3000 : 500 = 6 : 1$, not $3 : 500$. Converting units FIRST is half the answer in mixed-unit questions (speeds km/h vs m/s, hours vs minutes, ₹ vs paise).

Quick revision

• Ratio terms are ax, bx — x is the universal unknown.
• 1 part = total ÷ (sum of ratio terms).
• Bridge: equalise the shared letter; kA = kB = kC → LCM route.
• Changed ratio → equate the two expressions of the same quantity.