Ratio, Proportion and Variation formulas
At a glance
a : b → a + b parts; one part = total ÷ (a + b) a:b and b:c → match b with the LCM; A/C = (A/B)(B/C) kA = mB = nC → A : B : C = 1/k : 1/m : 1/n a : b = c : d ⇔ ad = bc; mean prop. √(ab) a/b = p/q ⇔ (a + b)/(a − b) = (p + q)/(p − q) y ∝ x: y/x fixed; y ∝ 1/x: xy fixed; part fixed: y = a + bx a/b = c/d = (a + c)/(b + d) duplicate a² : b² sub-duplicate √a : √b partnership: share ∝ capital × months
A ratio is a count of parts
a : b → a + b equal parts one part = total ÷ (a + b) A's share = a/(a + b) of the total
Changing a ratio: adding to both terms
Shortcut: If the ratio is 3 : 5, write the numbers as 3k and 5k. One equation in k then settles every 'add x to both' or 'after t years' question.
Combining ratios: a : b and b : c
A : B = a : b, B : C = c : d A : B : C = ac : bc : bd A : C = (a/b) × (c/d) = ac : bd
Shortcut: Need only A : C? Multiply the fractions: A/C = (A/B) × (B/C). For 2 : 3 and 4 : 5 that is 2/3 × 4/5 = 8/15.
Dividing a sum in a given ratio
Shortcut: Clear fractions with the LCM of the denominators: 1/2 : 1/3 : 1/4, times 12, is 6 : 4 : 3.
Proportion and componendo-dividendo
a : b = c : d ⇔ ad = bc mean proportional of a, b = √(ab) e.g. 9, 16 → 12 third proportional to a, b = b²/a e.g. 4, 6 → 9 fourth proportional to a, b, c = bc/a e.g. 2, 3, 8 → 12 a/b = p/q ⇔ (a + b)/(a − b) = (p + q)/(p − q)
| Rule | Result | Example from 2/3 = 4/6 |
|---|---|---|
| Invertendo | b/a = d/c | 3/2 = 6/4 |
| Alternendo | a/c = b/d | 2/4 = 3/6 |
| Componendo | (a + b)/b = (c + d)/d | 5/3 = 10/6 |
| Dividendo | (a − b)/b = (c − d)/d | −1/3 = −2/6 |
| Componendo-dividendo | (a + b)/(a − b) = (c + d)/(c − d) | 5/(−1) = 10/(−2) |
| Addendo | a/b = c/d = (a + c)/(b + d) = (pa + qc)/(pb + qd) | 2/3 = 6/9 = (2 + 4)/(3 + 6) |
Addendo works for any number of equal ratios: if a/b = c/d = e/f = k, then (a + c + e)/(b + d + f) = k too.
| Name | Ratio | From 4 : 9 |
|---|---|---|
| Inverse (reciprocal) | b : a (= 1/a : 1/b) | 9 : 4 |
| Duplicate | a² : b² | 16 : 81 |
| Sub-duplicate | √a : √b | 2 : 3 |
| Triplicate | a³ : b³ | 64 : 729 |
| Sub-triplicate | ∛a : ∛b | ∛4 : ∛9 |
| Compounded with c : d | ac : bd | with 3 : 2 → 12 : 18 = 2 : 3 |
Shortcut: Whenever you see (something + something)/(something − something) = a number, apply componendo-dividendo at once. It strips the expression down to a simple ratio.
Direct and inverse variation
y ∝ x → y = kx → y₁/x₁ = y₂/x₂ y ∝ 1/x → y = k/x → x₁y₁ = x₂y₂ y ∝ x² → y = kx² → x × 3 gives y × 9
Joint and part-fixed variation
y ∝ xz → y = kxz y ∝ x/z → y = kx/z part fixed → y = a + bx (two data points give a and b)
Partnership: capital × time
share ∝ capital × months same time for all → share ∝ capital same capital → share ∝ months
Shortcut: When a partner changes capital midway, add the pieces: ₹20,000 for 6 months then ₹10,000 for 6 months = 1,20,000 + 60,000 = 1,80,000.
Formulas stick when you use them
The Ratio, Proportion and Variation lesson has worked problems and quick checks for every formula here. Free to start.
Start free