JEE Mathematics Ratio Notes Set 01

Meaning of Ratio

A ratio compares two quantities of the same kind by division. If student A has 6 note books and student B has 10, the ratio of their note books is 6 : 10, which simplifies to 3 : 5. The ratio of a to b is written a : b or \( \frac{a}{b} \); a is the antecedent (first term) and b is the consequent (second term). Quantities must be in the same units: 2 metres to 300 cm is 200 : 300 = 2 : 3. Multiplying or dividing both terms by the same non-zero number does not change a ratio, so numbers in the ratio 2 : 5 can be taken as 2x and 5x.

Kinds of Ratio

1. If a > b the ratio a : b is a ratio of greater inequality; if a < b it is a ratio of lesser inequality; if a = b it is a ratio of equality.
2. Compounded ratio of a : b and c : d is ac : bd.
3. The duplicate ratio of a : b is a² : b² and the triplicate ratio is a³ : b³.
4. To compare two ratios, convert them to fractions with a common denominator.

Effect of Adding the Same Quantity to Both Terms

For positive c, adding c to both terms brings a ratio closer to 1. A ratio of greater inequality decreases: if a > b then \( \frac{a + c}{b + c} < \frac{a}{b} \). A ratio of lesser inequality increases: if a < b then \( \frac{a + c}{b + c} > \frac{a}{b} \). For example 4 : 3 becomes 5 : 4, which is smaller, and 2 : 3 becomes 3 : 4, which is larger. The proof uses \( \frac{a}{b} - \frac{a + c}{b + c} = \frac{c(a - b)}{b(b + c)} \).

Ratios as Percentages

The ratio \( \frac{a}{b} \) is the percentage \( \frac{a}{b} \times 100 \)%. A profit of Rs. 6,000 on a capital of Rs. 1,00,000 is 6%. The ratio 3 : 40 is 7.5% and \( 16\frac{2}{3} \)% equals \( \frac{50}{3 \times 100} = \frac{1}{6} \).

Ratio from Equations

1. Homogeneous equations: for 2x² − 3xy + y² = 0 divide by y² and put w = x/y to get 2w² − 3w + 1 = 0, so (2w − 1)(w − 1) = 0 and x : y = 1 : 2 or 1 : 1.
2. Two equations in three unknowns give only ratios. From x − 2y + 3z = 0 and 2x + 3y − z = 0, eliminating one variable at a time gives x : y : z = −1 : 1 : 1.
3. If x : y : z = 2 : 3 : 4, write x = 2k, y = 3k, z = 4k and substitute.

Worked Examples

Example 1. Two brothers' ages are in the ratio 3 : 4. After 12 years the ratio will be 5 : 6. Find their present ages.
Let the ages be 3x and 4x. Then 6(3x + 12) = 5(4x + 12), so 18x + 72 = 20x + 60, giving x = 6. The ages are 18 and 24.

Example 2. The angles of a triangle are in the ratio 2 : 5 : 11. Find them.
2x + 5x + 11x = 180 gives x = 10, so the angles are 20°, 50° and 110°. For a quadrilateral with angles in the ratio 1 : 2 : 3 : 4 the sum is 360°, so x = 36 and the angles are 36°, 72°, 108°, 144°.

Example 3. Find the number to be subtracted from each term of 38 : 31 so that the ratio becomes 4 : 3.
3(38 − x) = 4(31 − x), so 114 − 3x = 124 − 4x and x = 10. Check: 28 : 21 = 4 : 3.

Example 4. What common quantity added to both terms of a² : b² makes the ratio a : b?
b(a² + x) = a(b² + x) gives x(b − a) = ab² − a²b = −ab(a − b), so x = ab (for a ≠ b).

Example 5. A man earns Rs. 7650 and spends Rs. 7038. What percentage of his income does he save?
Saving = Rs. 612, and \( \frac{612}{7650} \times 100 = 8 \)%.