Divide a Quantity in a Ratio

Understand the concept

A ratio compares quantities in a specified order. If red counters and blue counters are in the ratio 3:5, the order means red:blue. Imagine a repeatable group containing three red counters and five blue counters. Scaling that group preserves the ratio.

To split a known total, add the ratio numbers. A 3:5 ratio has 3 + 5 = 8 equal parts. Divide the total by eight to find the value of one part, then multiply by three and five to obtain the two shares. For a ratio a:b and total T, the shares are T × a/(a+b) and T × b/(a+b).

The word “equal” applies to the size of one ratio part, not to the final shares. When one share uses five parts and another uses three, the shares are different sizes even though each part has the same value. After calculating, check both that the shares add to the given total and that their simplified ratio matches the stated ratio.

A part-to-part ratio differs from a part-to-whole fraction. In a 3:5 mixture, red is 3/5 of the blue amount but 3/8 of the total. Confusing these two denominators is a common source of mistakes. Label the numerator and denominator in words before using a fraction.

Ratios can be multiplied or divided by the same nonzero quantity without changing their meaning. Thus 6:10 simplifies to 3:5. Quantities must use compatible units before a ratio is simplified: compare 2 metres with 50 centimetres by converting to 200:50, which simplifies to 4:1. Dividing 2 by 50 before converting the units would describe the wrong relationship.

Worked example

Divide 240 counters into red and blue in the ratio 3:5.

There are 8 parts in total. One part represents 240/8 = 30 counters. Red uses 3 × 30 = 90 counters; blue uses 5 × 30 = 150 counters.

Check the total: 90 + 150 = 240. Check the ratio: 90:150 simplifies by dividing both numbers by 30, giving 3:5. Red is 90/240 = 3/8 of the whole, while red compared with blue is 90/150 = 3/5.

If 30 additional red counters are added without changing the blue count, the new ratio is 120:150 = 4:5. The original ratio no longer applies because only one component changed.

Practice and answer

1. Split 84 units in the ratio 2:5.
Answer: 7 parts, so each part is 12. The shares are 24 and 60.

2. Flour and sugar are in the ratio 4:1. What fraction of their combined amount is sugar?
Answer: 1/(4+1) = 1/5, or 20 percent.

3. Compare 1.5 metres with 30 centimetres as a simplified ratio.
Answer: convert 1.5 metres to 150 centimetres. The ratio is 150:30 = 5:1.

What you will learn

• Split a total using the sum of ratio parts.
• Keep ratio labels and measurement units consistent.
• Check both the total and the simplified ratio.