You may encounter two ratios containing a common element. To combine the ratios, you can use a process similar to creating a common denominator for fractions.
Because ratios act like fractions, you can multiply both sides of a ratio (or all sides, if there are more than two) by the same number, just as you can multiply the numerator and denominator of a fraction by the same number. You can change fractions to have common denominators. Likewise, you can change ratios to have common terms corresponding to the same quantity. Consider the following problem:
In a box containing action figures of the three Fates from Greek mythology, there are three figures of Clotho for every two figures of Atropos, and five figures of Clotho for every four figures of Lachesis.
- What is the least number of action figures that could be in the box?
- What is the ratio of Lachesis figures to Atropos figures?
(a) In symbols, this problem tells you that C : A = 3 : 2 and C : L = 5 : 4. You cannot instantly combine these ratios into a single ratio of all three quantities, because the terms for C are different. However, you can fix that problem by multiplying each ratio by the right number, making both C’s into the least common multiple of the current values:
| C : A : L | C : A : L | |||
| 3 : 2 | → | Multiply by 5 | → | 15 : 10 |
| 5 : : 4 | → | Multiply by 3 | → | 15 : : 12 |
| This is the combined ratio: 15 : 10 : 12 | ||||
The actual numbers of action figures are these three numbers times an Unknown Multiplier, which must be a positive integer. Using the smallest possible multiplier, 1, there are 15 + 12 + 10 = 37 action figures.
(b) Once you have combined the ratios, you can extract the numbers corresponding to the quantities in question and disregard the others: L:A = 12:10, which reduces to 6:5.
A school has 3 freshmen for every 4 sophomores and 5 sophomores for every 4 juniors. If there are 240 juniors in the school, how many freshmen are there?