Ratio and Proportion

Q: Find the third proportional to 16 and 24.

c = (242)/(16) = (576)/(16) = 36.

ICSE Class 10 · Mathematics · Chapter 7

What this chapter covers

Ratio: the language of comparison
Proportion
Continued proportion
Mean proportion
Invertendo and alternendo
Componendo and dividendo
Using the properties together
Direct simple applications on proportions
Common mistakes and how to avoid them

Key definitions

Continued proportion is the relation between three or more quantities in which the ratio of each quantity to the next one is the same.

Invertendo is the property that if a : b = c : d, then b : a = d : c.

Alternendo is the property that if a : b = c : d, then a : c = b : d.

Componendo is the property that if a : b = c : d, then (a + b) : b = (c + d) : d.

Dividendo is the property that if a : b = c : d, then (a - b) : b = (c - d) : d.

Componendo and dividendo is the property that if a : b = c : d with a ≠ b, then (a + b) : (a - b) = (c + d) : (c - d).

Key points

A ratio a : b compares two quantities of the same kind by division; b ≠ 0 and both quantities must be in the same unit.
Four non-zero quantities are in proportion, a : b :: c : d, when a : b = c : d; the extremes are a, d and the means are b, c.
In a proportion, product of extremes = product of means: ad = bc, and the converse also holds.
Fourth proportional to a, b, c: d = (bc)/(a).
a, b, c are in continued proportion when a : b = b : c, so b² = ac; third proportional to a and b is (b²)/(a).
Mean proportional between a and c: b = √(ac) (not the average).
Invertendo: a : b = c : d ⇒ b : a = d : c.
Alternendo: a : b = c : d ⇒ a : c = b : d (all four of the same kind).
Componendo: (a + b) : b = (c + d) : d; dividendo: (a - b) : b = (c - d) : d.
Componendo and dividendo: (a + b) : (a - b) = (c + d) : (c - d) for a ≠ b, and its converse is used to solve equations.
The k-method (a = bk, c = dk) proves any identity about a proportion when no single property is obvious.
Applications: form the proportion in words, use ad = bc, solve, and check the answer in the original condition.

Questions answered

Define a proportion, and name the extremes and the means in p : q :: r : s.

A proportion is a statement that two ratios are equal. In p : q :: r : s the extremes are p and s and the means are q and r.

Are 0.4, 0.6, 1.2 and 1.8 in proportion? Give a reason.

Yes. Product of extremes = 0.4 × 1.8 = 0.72 and product of means = 0.6 × 1.2 = 0.72, and the two are equal.

Find the fourth proportional to 6, 9 and 14.

6 : 9 = 14 : d, so 6d = 126 and d = 21.

Find the third proportional to 16 and 24.

c = (24²)/(16) = (576)/(16) = 36.

Find the mean proportional between 9 and 25, and explain why it is not 17.

√(9 × 25) = 15. The number 17 is the average of 9 and 25; the mean proportional is the square root of the product, not half the sum.

State alternendo and use it to rewrite 3 : 5 = 12 : 20.

Alternendo: if a : b = c : d, then a : c = b : d. Here 3 : 12 = 5 : 20, and both ratios equal 1 : 4.

Prove componendo starting from (a)/(b) = (c)/(d).

Adding 1 to both sides gives (a)/(b) + 1 = (c)/(d) + 1, that is (a + b)/(b) = (c + d)/(d).

If (7x + 4y)/(7x - 4y) = (9)/(5), find x : y.

Componendo and dividendo gives (14x)/(8y) = (14)/(4), so (x)/(y) = (14 × 8)/(4 × 14) = 2, and x : y = 2 : 1.

Past board exam questions

ICSE 2027 sample paper · 1 mark

12, 18 and x are in continued proportion. The value of x is:

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ICSE 2027 sample paper · 4 marks

Given, (a + 2b + x)/(a - 2b + x) = ((a + b)²)/((a - b)²), a ≠ 0 Using properties of proportion, find the value of x.

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Applying componendo and dividendo: ((a + 2b + x) + (a - 2b + x))/((a + 2b + x) - (a - 2b + x)) = ((a + b)² + (a - b)²)/((a + b)² - (a - b)²) (2(a + x))/(4b) = (2(a² + b²))/(4ab) a + x = (a² + b²)/(a) ⇒ x = (b²)/(a)

ICSE 2026 board paper · 1 mark

(x + 3), 1, (3x - 7) and -5 are in proportion. The value of x is:

-1
1
-5
5

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-1

ICSE 2026 sample paper · 3 marks

If x, y and z are in continued proportion, prove that: (x)/(y².z²) + (y)/(z².x²) + (z)/(x².y²) = (1)/(x³) + (1)/(y³) + (1)/(z³)

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x, y, z in continued proportion ⟹ (x)/(y) = (y)/(z) ⇒ y² = xz. LHS = (x³ + y³ + z³)/(x²y²z²). Since y² = xz, x²y²z² = x²z² · xz = x³z³. LHS = (x³)/(x³z³) + (y³)/(x³z³) + (z³)/(x³z³) = (1)/(z³) + (y³)/((y²)³) + (1)/(x³) (as x³z³ = (xz)³ = y⁶) = (1)/(z³) + (1)/(y³) + (1)/(x³) = RHS.

ICSE 2026 board paper · 3 marks

If x = (5ab)/(a - b), a ≠ b, (a) Find: (x)/(a) (b) Using properties of proportion, find: (x + a)/(x - a)

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(a) (x)/(a) = (5b)/(a - b). (b) By componendo and dividendo, (x + a)/(x - a) = (5b + (a - b))/(5b - (a - b)) = (a + 4b)/(6b - a).

ICSE 2025 sample paper · 1 mark

A mixture of paint is prepared by mixing 2 parts of red pigments with 5 parts of the base. Using the given information in the following table, find the values of a, b & c to get the required mixture of paint. | Parts of red pigment | 2 | 4 | b | 6 | |---|---|---|---|---| | Parts of base | 5 | a | 12.5 | c |

a = 10, b = 10, c = 10
a = 5, b = 2, c = 5
a = 10, b = 5, c = 10
a = 10, b = 5, c = 15

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a = 10, b = 5, c = 15

ICSE 2025 board paper · 3 marks

Using properties of proportion, find the value of 'x': (6x² + 3x - 5)/(3x - 5) = (9x² + 2x + 5)/(2x + 5); x ≠ 0

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By dividendo, (6x² + 3x - 5 - (3x - 5))/(3x - 5) = (9x² + 2x + 5 - (2x + 5))/(2x + 5) (6x²)/(3x - 5) = (9x²)/(2x + 5). Since x ≠ 0, divide by 3x²: (2)/(3x - 5) = (3)/(2x + 5) 4x + 10 = 9x - 15 ⇒ 5x = 25 ⇒ x = 5.

ICSE 2025 compartment paper · 4 marks

If a, b, c and d are in continued proportion, prove that ad(c² + d²) = c³(b + d)

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Let (a)/(b) = (b)/(c) = (c)/(d) = k. Then c = dk, b = ck = dk², a = bk = dk³. LHS = ad(c² + d²) = dk³ · d · (d²k² + d²) = d⁴k³(k² + 1). RHS = c³(b + d) = d³k³(dk² + d) = d⁴k³(k² + 1). LHS = RHS. Hence proved.