Adding changes the ratio

The idea

Adding the same number to, or subtracting it from, both terms of a ratio changes the ratio, and the new ratio is not necessarily proportional to the original.

Neelima and her mother

When Neelima was 3, her mother was 10 times her age, so 30. The ratio of their ages was 3 : 30, which is 1 : 10.

9 years later Neelima is 12 and her mother is 39. Is her mother still 10 times as old?

• Yes, still 10 times
• No, fewer than 10 times
• No, more than 10 times

Let the years pass

Each year adds 1 to both ages. See the ratio. Then switch to multiply and compare.

Adding changes the ratio

Adding the same number to, or subtracting it from, both terms of a ratio changes the ratio, and the new ratio is not necessarily proportional to the original.

Add 9 to both: 3 : 30 became 12 : 39, which is 4 : 13, not 1 : 10.

Multiply both by 4: 3 : 30 becomes 12 : 120, which is still 1 : 10.

Notes

Adding the same number to, or subtracting it from, both terms of a ratio changes the ratio, and the new ratio is not necessarily proportional to the original.

Check yourself

Six years later Neelima is 9 and her mother is 36. Her mother is how many times as old as Neelima?

Answer: 4

36 ÷ 9 = 4. The ratio 9 : 36 is 1 : 4, no longer 1 : 10.

Which ratio is in proportion with 3 : 30?

Ravi says: "I added 3 to both terms of 2 : 5, so 5 : 8 is the same ratio." Is he right?

Take 2 away from both terms of 6 : 10. Is the new ratio in proportion with 6 : 10?

• 4 : 31. That adds 1 to both terms. Adding changes the ratio: 4 : 31 is already in simplest form.
• 6 : 33. That adds 3 to both terms. 6 : 33 reduces to 2 : 11, not 1 : 10.
• 6 : 60 — correct. Yes! Both terms are multiplied by 2, and 6 : 60 reduces to 1 : 10.
• 13 : 40. That adds 10 to both terms. 13 : 40 is already in simplest form.
• Yes, the same number was added to both terms. Adding the same number does not keep a ratio. Check the simplest forms.
• No: 2 : 5 and 5 : 8 are different ratios — correct. Yes! Both are already in simplest form, and they are different.
• Yes, both terms went up. Both terms going up is not enough. They must be multiplied by the same factor.
• Yes, because the same number was taken away. Subtracting the same number does not keep a ratio.
• Yes, because 6 and 10 are both even. Both numbers being even has nothing to do with it.
• No: 4 : 8 is 1 : 2, but 6 : 10 is 3 : 5 — correct. Yes! 4 : 8 reduces to 1 : 2 and 6 : 10 reduces to 3 : 5.