Ratio, Proportion and Variation
Ratio, Proportion and Variation
This domain tests comparing quantities by ratio, solving proportions, and recognising direct and inverse variation in words and tables. A ratio must be in simplest form, and both quantities must be in the same unit.
Core rules
- Ratio a:b = a/b (b ≠ 0). Simplify by dividing both terms by their HCF: 18:24 = 3:4.
- A proportion a:b = c:d holds iff ad = bc. In continued proportion a:b = b:c, b² = ac.
- Direct variation: y ∝ x means y = kx, so y/x is constant. Inverse variation: y ∝ 1/x means xy = k.
- For three quantities, make the common term equal: if a:b = 2:3 and b:c = 4:5, write a:b = 8:12 and b:c = 12:15, so a:c = 8:15.
Worked example
If a:b = 3:4 and b:c = 6:7, find a:c. The common term b is 4 in the first ratio and 6 in the second. LCM(4,6) = 12. Multiply a:b by 3: a:b = 9:12. Multiply b:c by 2: b:c = 12:14. Since b is 12 in both, a:c = 9:14.
Traps
- Never add ratios term by term: 2:3 and 4:5 do not give 6:8.
- Convert to a common unit first; 50 paise : ₹2 is 50:200 = 1:4, not 50:2.
- In inverse variation, the product is constant; do not set y/x constant.
- Order matters: a:b is not b:a unless a = b.