Why Adding the Same Number to a Ratio Changes It

Does adding the same number preserve a ratio?

Usually, no. Adding the same nonzero number to both parts of a ratio changes it unless the two original parts are equal, assuming the denominators remain nonzero. Multiplying both parts by the same nonzero number does preserve the ratio. For example, 2:3 equals 4:6, but not 4:5.

Why multiplication works but addition does not

Treat a:b as the fraction a/b. Multiplying numerator and denominator by k gives ka/kb = a/b when k and b are nonzero. By contrast, (a+c)/(b+c) = a/b requires b(a+c) = a(b+c). After cancelling ab, this becomes c(b−a) = 0. Thus c must be zero or a must equal b, with valid denominators.

For positive unequal quantities, adding a positive amount moves the ratio closer to 1:1; it does not make them equal. Their difference stays unchanged.

Worked example: two quantities increase

Two quantities are in the ratio 2:3. After adding 4 to each, their ratio becomes 3:4. Find the original quantities.

Write them as 2k and 3k. Then (2k+4)/(3k+4) = 3/4. Cross-multiplying gives 8k+16 = 9k+12, so k = 4. The original quantities are 8 and 12.

Check both conditions: 8:12 simplifies to 2:3. After adding 4, 12:16 simplifies to 3:4. Satisfying only the new ratio is not enough; the original ratio must also work.

Worked example: subtracting an equal amount

Two positive amounts are in the ratio 3:5. Subtracting 4 from each gives 1:2. Let the original amounts be 3k and 5k. Then 2(3k−4) = 5k−4, so k = 4. The amounts are 12 and 20. After subtraction they become 8 and 16, which is 1:2.

Practice

• Original ratio 4:7; adding 6 to each gives 5:8. Answer: 24 and 42. Check 30:48 = 5:8.
• Original ratio 2:5; adding 3 to each gives 1:2. Answer: 6 and 15. Check 9:18 = 1:2.

Use one shared multiplier; choosing separate multipliers loses the original ratio condition. See dividing an amount in a ratio for the related starting method. You can also ask for an attempt check in the paid study space.