Ratio, Proportion and Partnership: Basic Ratio

Concept Explanation

A ratio compares two quantities of the same kind by division. Written as a : b, it tells us how many times one quantity contains the other — not their absolute values.

Key Properties

• Order matters: a : b is generally not equal to b : a.
• Same units: Convert both quantities to the same unit before forming a ratio.
• Simplest form: Divide both terms by their HCF. Example: 24 : 36 = 2 : 3.
• No unit in the final ratio: Units cancel when both terms share the same unit.

Part-to-Part vs Part-to-Whole

If boys : girls = 4 : 5, the whole class has 4 + 5 = 9 equal parts. Boys are 4/9 of the total and girls are 5/9 of the total.

Converting Ratio to Actual Numbers

If a : b = 3 : 5, write the quantities as 3k and 5k for some constant k. Use the given total, difference, or one actual value to find k.

Example: Ratio 3 : 5 and total 40 → 3k + 5k = 40 → k = 5 → numbers are 15 and 25.

Three-Term Ratios

For a : b : c, use ak, bk, and ck. Sum of parts = (a + b + c)k.

Placement Shortcut

To divide amount M in ratio a : b, first share = M × a/(a + b) and second share = M × b/(a + b).