Comparing, increasing and decreasing

Eesha scored 42 out of 50 in English and 70 out of 80 in Science. She thinks English went better, because she lost only 8 marks there against 10 in Science. Reema says Science, because the score is higher. Vishu says you can't compare at all, since the maximums differ.

Vishu has spotted the real problem, and percentages solve it. English: (42/50) × 100 = 84%. Science: (70/80) × 100 = 87.5%. Science was the better performance, and neither of the first two arguments was sound.

The same trick reads a food label. Buying a badam drink mix, how do you tell which brand has more badam and fewer chemicals? Turn every ingredient's weight into a percentage of the total. For brand DEF, sugar is (99/150) × 100 = 66%: two-thirds of the packet is sugar! Do it for each ingredient, and check that the percentages add up to 100. The book calls this KYC: Know Your Contents.

Percentages also describe change:

Percentage increase = (amount of increase ÷ original amount) × 100.

Tomatoes cost ₹30 three years ago and ₹42 now. The increase is ₹12, so the percentage increase is (12/30) × 100 = 40%. Prices rose by 40%.

Percentage decrease works the same way. A theatre's footfall fell from 160 to 100, a drop of 60, so (60/160) × 100 = 37.5%. Footfall fell by 37.5%.

Note the word original. The change is always measured against where you started, not where you ended: the original always goes underneath. Get that base wrong and every answer is wrong.

Finally, two sentences that sound different and aren't. 'The population in 1991 is 165% of that in 1961' means q = (165/100)p = 1.65p. 'The population increased by 65% from 1961 to 1991' means q = p + 0.65p = 1.65p. Identical.