In algebra, a ratio is the relationship between two quantities. It is expressed as the quotient of one magnitude divided by another, or as a relation between several variables.
- If a school has a twenty-to-one student-teacher ratio, that means that there are twenty times as many students as teachers.
- In a benzene ring, atoms of carbon and hydrogen exist in a one-to-one ratio with each other; there are the same number of each. The carbon:hydrogen ratio in naphthalene is 5:4. The ratio of oxygen atoms to hydrogen atoms in water is 1:2.
Jupiter is 318 times the size of Earth.
- The musical interval of a perfect fifth, the pitch ratio 3:2, consists of two pitches, one approximately 1.5 times the frequency of another.
Note the use of words such as "times", "parts", "number", etc. This occurs because ratios are unitless; the units cancel out of the ratio. e.g. 3 kg /5 kg = 3 000 g. /5 000 g = 3/5.
Ratios are not exactly the same thing as fractions. For example, if I have three pennies and five nickels, then the ratio of pennies to nickels is 3:5 or 3/5. This indicates that there are three fifths as many pennies as nickels. However the fraction of coins which are pennies is 3/(3+5) = 3/8. Thus the chances of a randomly selected coin being a penny are three in eight.
The most common thing to do with ratios is multiply them. For example, if there are two snorts for every giggle, and three giggles for every guffaw, then (since 2sn/1gig × 3gig/guf = 6sn/guf) there are six snorts for every guffaw. Note that the intermediate unit "giggle" canceled out of the expression. Note also that each fraction in the expression was equal to one, which is how we know that their product (6sn/guf) is also equal to one. Similarly, if I know there are two guffaws, then I can multiply by 6 snorts per guffaw (6sn/guf = one, the multiplicative identity) to learn that there are 12 snorts.