How can I simplify a ratio without changing what it compares?

Find a number that divides both terms evenly, then divide each term by that same number. Keep the quantities in the same order and label them so the comparison remains clear. For example, if a bowl has 12 red counters and 8 blue counters, the ratio of red counters to blue counters is 12:8. Both numbers are divisible by 4, so 12:8 becomes 3:2. That means 3 red counters for every 2 blue counters—not 3 counters in total. Dividing both terms by the same common factor preserves the ratio; changing only one term does not. [S1] [S2]

Dividing both terms by 4 scales the two quantities equally: 12 red and 8 blue are represented as 3 red and 2 blue. The equivalent-fraction idea behind this is that dividing the top and bottom by the same nonzero factor keeps the value unchanged. [S1] Subtracting 4 from each gives 8:4, which is 2:1, so it changes the comparison. In the counter example, 12:8 and 3:2 describe the same balance; 8:4 does not. Likewise, reducing only one term would change the relationship. The operation is division by a shared factor, not simply making both numbers smaller.

Sure. Suppose, just for this example, there are 18 green beads and 12 yellow beads. To simplify the ratio of green beads to yellow beads:

1. Write the comparison in order: 18 green : 12 yellow.
2. Find a common factor. Both 18 and 12 divide evenly by 6.
3. Divide both terms by 6: (18 ÷ 6) green : (12 ÷ 6) yellow.
4. Write the result with its labels: 3 green : 2 yellow.

The shortened ratio still compares green beads with yellow beads, in that order. You can check the arithmetic by multiplying both simplified terms by 6: 3 × 6 = 18 and 2 × 6 = 12. This example shows how the numbers reduce; the labels tell you what the quantities represent.

You are finished when the two terms have no common whole-number factor greater than 1. For instance, 3 and 2 have no such factor, so 3:2 cannot be simplified further using whole-number division. If the only common factor is 1, keep the ratio as it is. Preserve the order and labels when you write your result: 3 green beads to 2 yellow beads, for example, is not the same written comparison as 2 yellow beads to 3 green beads. OpenStax’s examples establish the shared-factor method and equivalent-fraction reasoning; they do not tell you which quantities a real-world question intends you to compare. Check the wording of the problem before simplifying.