A recap, and ratios on a map

A quick recap first. When two related quantities change by the same factor, the relationship is proportional. Idli batter is one: a common recipe is 2 cups of rice to 1 cup of urad dal, written 2 : 1.

Viswanath mixes 6 cups of rice with 3 of urad dal; Puneet mixes 4 with 2. Will their idlis taste the same?

Test by cross multiplication. For 6 : 3 and 4 : 2, multiply across: 6 × 2 = 12 and 3 × 4 = 12. Equal, so the ratios are proportional, and the idlis should taste the same (if everything else matches too).

In general, a : b and c : d are proportional when a × d = b × c, or equally when a/c = b/d.

Now a place where ratios do real work. Look at the bottom corner of almost any map and you'll find something like 1 : 60,00,000. What's that?

It's the Representative Fraction (RF): the ratio between a distance on the map and the matching real distance on the ground.

So 1 : 60,00,000 means 1 cm on paper stands for 60,00,000 cm on the Earth. Convert: 60,00,000 cm ÷ 100 = 60,000 m, ÷ 1000 = 60 km. One centimetre is sixty kilometres.

One caution the book repeats: this is geographical distance (a straight line, as a bird flies), not road distance, which is always longer because roads bend. Measure between two cities with a ruler, multiply by the RF, and you have the distance. Try the same pair on a map with a different scale: the answers should agree, since the ground hasn't moved.