JEE Mathematics Proportion Notes Set 01

Proportion

Four quantities a, b, c, d are in proportion when the ratio a : b equals the ratio c : d, written a : b = c : d or \( \frac{a}{b} = \frac{c}{d} \). It is read as "a is to b as c is to d". Here a and b must be of the same kind and c and d must be of the same kind, but a and c may differ in kind: 4 metres : 5 metres = 8 hours : 10 hours. The first and last terms (a and d) are the extremes, and the middle terms (b and c) are the means.

Product of Extremes and Means

If a : b = c : d, then ad = bc, that is, the product of the extremes equals the product of the means. Conversely, if ad = bc the four quantities are in proportion.

Mean, Third and Fourth Proportional

1. If a : b = b : c then b² = ac. b is the mean proportional (geometric mean) between a and c, \( b = \sqrt{ac} \), and c is the third proportional to a and b.
2. If a : b = c : d, then d is the fourth proportional to a, b, c, and \( d = \frac{bc}{a} \).
3. If \( \frac{a}{b} = \frac{b}{c} = \frac{c}{d} = … \) the quantities are in continued proportion. Writing the common ratio as k, such a sequence is geometric.

Properties of Proportion

If a : b = c : d then each of the following also holds.
1. Invertendo: b : a = d : c.
2. Alternando: a : c = b : d.
3. Componendo: (a + b) : b = (c + d) : d.
4. Dividendo: (a − b) : b = (c − d) : d.
5. Componendo and dividendo: (a + b) : (a − b) = (c + d) : (c − d).
Each follows by adding or subtracting 1 on both sides of a/b = c/d. For the converse of the last one, cross-multiplying (a + b)(c − d) = (c + d)(a − b) gives bc = ad.

The Constant-Ratio Method

When a/b = c/d = k, write a = bk and c = dk and substitute. If a/b = c/d = e/f = …, then each ratio also equals \( \frac{a + c + e + …}{b + d + f + …} \) and, more generally, \( \frac{ma + nc + pe + …}{mb + nd + pf + …} \) for any multipliers m, n, p. For example, if a : b = c : d, then \( \frac{a + c}{b + d} = \frac{a - c}{b - d} = \frac{a}{b} \).

Worked Examples

Example 1. Find the fourth proportional to 7, 8 and 28.
7 : 8 = 28 : x gives 7x = 8 × 28, so x = 32.

Example 2. What number added to each of 3, 4, 9, 11 makes them proportional?
(3 + x)(11 + x) = (4 + x)(9 + x) gives 33 + 14x = 36 + 13x, so x = 3. The numbers 6, 7, 12, 14 satisfy 6 : 7 = 12 : 14.

Example 3. The sum of the means of a proportion is 7, the sum of the extremes is 8 and the sum of the squares of all four terms is 65. Find the proportion.
Let b + c = 7, a + d = 8 and bc = ad. Squaring and adding, (b + c)² + (a + d)² = 113, so 65 + 2bc + 2ad = 113, giving bc = ad = 12. Then b, c are the roots of t² − 7t + 12 = 0, namely 3 and 4, and a, d are the roots of t² − 8t + 12 = 0, namely 2 and 6. The proportion is 2 : 3 = 4 : 6 (or 6 : 4 = 3 : 2).

Example 4. Find the least integer which, added to both terms of 5 : 9, makes the ratio greater than 7 : 10.
(5 + x)/(9 + x) > 7/10 gives 50 + 10x > 63 + 7x, so x > 13/3. The least integer is 5.

Example 5. Solve \( \frac{x^3 + 27x}{9x^2 + 27} = \frac{7}{9} \).
By componendo and dividendo, \( \frac{(x + 3)^3}{(x - 3)^3} = \frac{7 + 9}{7 - 9} = -8 \). Taking cube roots, \( \frac{x + 3}{x - 3} = -2 \), so x + 3 = −2x + 6 and x = 1.