The same mix in smaller numbers
Equivalent ratios describe the same mix, so one mix has many names: 2 : 3, 4 : 6, 6 : 9, and so on. The name with the smallest whole numbers is called the ratio in its simplest form. A ratio is in its simplest form when its numbers have no common factor except 1.
To simplify 4 : 6, find a number that divides both numbers. Both 4 and 6 divide by 2: 4 ÷ 2 = 2 and 6 ÷ 2 = 3. So 4 : 6 = 2 : 3. The only common factor of 2 and 3 is 1, so 2 : 3 is in its simplest form.
The top bar is cut into 10 small parts: 4 colored and 6 plain. Put the small parts together in pairs and it is the bar below, 2 colored parts and 3 plain ones. The colored length has not changed.
Divide by the highest common factor
Simplify 12 : 18. Both numbers are even, so divide both by 2: 12 : 18 = 6 : 9. But 6 and 9 still have the common factor 3, so divide again: 6 : 9 = 2 : 3. Now the only common factor is 1, and the ratio is in its simplest form.
You can do it in one step by dividing both numbers by their Highest Common Factor (HCF). The HCF of 12 and 18 is 6, and 12 ÷ 6 = 2 and 18 ÷ 6 = 3. Dividing by 2 and then by 3 is the same as dividing by 2 × 3 = 6.
This is the same move as simplifying a fraction. simplifies to by dividing the numerator and the denominator by 6, and 12 : 18 simplifies to 2 : 3 by dividing both numbers by 6.
The same bar cut into 30 small parts, then 15, then 5. The colored length is the same in all three, so 12 : 18, 6 : 9 and 2 : 3 are the same mix.
The same unit first
Both amounts in a ratio must be measured in the same unit before you simplify. A ribbon 1.2 m long is cut into a piece 45 cm long and a piece 75 cm long. The ratio of the short piece to the whole ribbon is not 45 : 1.2, because one number is in centimeters and the other is in meters. Write 1.2 m as 120 cm. Then the ratio is 45 : 120, and the HCF of 45 and 120 is 15, so 45 : 120 = 3 : 8.
Both lengths in centimeters, measured in pieces of 15 cm: the short piece is 3 pieces and the whole ribbon is 8, so 45 : 120 = 3 : 8.
The usual mistakes
Dividing only one number. Both numbers must be divided by the same number, or the mix changes. 4 : 6 is not 2 : 6: the ratio 2 : 6 has 1 colored part for every 3 plain parts, but 4 : 6 has 2 colored parts for every 3 plain parts.
Stopping too soon. 12 : 18 divided by 2 is 6 : 9, but 6 and 9 still share the factor 3, so 6 : 9 is not yet in its simplest form. Before you stop, check that the only common factor left is 1.