Simplifying Ratios

A ratio compares two amounts, and the same comparison can be written many ways. If you divide both sides by the same number, the comparison doesn't change — you just describe it with smaller numbers. That gives an equivalent ratio:

6 : 4 \;=\; 3 : 2 \qquad (\text{both divided by } 2)

To reach the simplest form, divide both parts by their highest common factor (HCF) — the biggest number that goes into both. The HCF of 6 and 4 is 2, so 6 : 4 simplifies all the way to 3 : 2, which can't be reduced any further.

A special shape is the "1 : n" form: divide both parts by the first number, so the first part becomes 1. For example 4 : 12 = 1 : 3 (both divided by 4) — handy for seeing "how many times bigger" the second amount is.

To simplify a ratio:

equivalent ratios scale both parts equally — multiply or divide both sides by the same number and the comparison is unchanged;
the simplest form divides both parts by their highest common factor, so they share no common factor left;
the 1 : n form divides both parts by the first part, making the first number 1;
the two parts must be in the same units before you simplify — convert first (e.g. 1\text{ m} : 50\text{ cm} is really 100 : 50 = 2 : 1).

See it shrink

Both bars are cut by the same factor, so they keep the same proportion — only the numbers get smaller.