Proportional Reasoning – 1

Overview

This chapter introduces ratio and proportion as ways of comparing quantities. We learn to write and simplify ratios, find equivalent ratios, and solve problems using the unitary method and proportions.

Key concepts

• A ratio a : b compares two quantities of the same kind.
• Two ratios that are equal form a proportion: a : b = c : d.
• In a proportion, the product of the extremes equals the product of the means.
• The unitary method first finds the value of one unit.

Important formulae

• a : b = c : d means a × d = b × c.
• To divide N in ratio a : b, parts are (a ÷ (a+b)) × N and (b ÷ (a+b)) × N.
• Percentage = (part ÷ whole) × 100.

Solved example

1. If 5 pens cost ₹60, find the cost of 8 pens.
2. Cost of 1 pen = 60 ÷ 5 = ₹12.
3. Cost of 8 pens = 12 × 8 = ₹96.

Important questions

1. Simplify the ratio 45 : 60 to its lowest terms.
2. Divide ₹1200 between two people in the ratio 3 : 5.
3. If a : b = 2 : 3 and b : c = 4 : 5, find a : b : c.
4. Check whether 6, 9, 10 and 15 are in proportion.

Quick revision

Simplify ratios like fractions and check proportions by cross multiplication. The unitary method — find one, then scale — solves most word problems. Sharing in a ratio means splitting into a + b equal parts.