Finding a percentage of a quantity

Madhu's biscuits have 25% sugar; Madhav's have 35%. Who ate more sugar?

Careful. A percentage is a share of something, so comparing bare percentages only works if the wholes are the same. If both ate 100 g, Madhav ate 35 g against Madhu's 25 g: easy. But suppose Madhu ate 120 g and Madhav 95 g. Now it has to be worked out.

Madhu: 25% of 120. As a proportion, 25 : 100 :: ? : 120. Or in steps: 25 g of sugar per 100 g of biscuit means 5 g per 20 g, so 30 g per 120 g. Or solve 25/100 = s/120, giving s = (25/100) × 120 = 30 g.

Madhav: 35% of 95. The numbers are less friendly, so go through one gram. 100 g holds 35 g of sugar, so 1 g holds 35/100 g, and 95 g holds (35/100) × 95 = 33.25 g.

So Madhav ate more sugar (33.25 g against 30 g), even though he ate less biscuit. In general, y% of a value is (y/100) × that value. (And x% of y always equals y% of x: 80% of 75 = 75% of 80 = 60. Use whichever is easier.)

A lot of this can be done in your head if you know a few anchors:

Two relations make the rest easy. First, 20% of a value is double 10% of it, since 20 parts out of 100 is twice 10 parts out of 100. Second, percentages add: (20% of y) + (5% of y) = 25% of y, because (20/100)y + (5/100)y = (25/100)y.

Put those together:

• 15% = 10% + 5%
• 40% = 4 × 10%
• 75% = 50% + 25% (or divide by 4 and multiply by 3)
• 90% = 100% − 10% (90% of 70 is 70 − 7 = 63)