Two numbers are in the ratio 3 : 5. If 9 is added to each, the ratio becomes 3 : 4. Find the numbers.
9 and 15. Let them be 3x and 5x. Then (3x + 9)/(5x + 9) = 3/4, so 12x + 36 = 15x + 27, giving x = 3. The numbers are 9 and 15, and 18 : 24 is indeed 3 : 4.
Explain it simply
A ratio only tells you the proportion, not the actual values, so introduce one unknown multiplier x. Adding the same amount to both numbers changes the ratio, and that change pins down x. Age problems ("after 5 years their ages will be...") work in exactly the same way.
Example
Question: Numbers in ratio 3 : 5. Add 9 to each and the ratio becomes 3 : 4.
- Let the numbers be 3x and 5x.
- (3x + 9) / (5x + 9) = 3 / 4.
- Cross-multiply: 4(3x + 9) = 3(5x + 9), so 12x + 36 = 15x + 27.
- 3x = 9, so x = 3.
- Numbers = 9 and 15.
Check: 9 + 9 = 18, 15 + 9 = 24, and 18 : 24 = 3 : 4.
Quick method from options: the difference stays the same. Original difference 2 parts, new difference 1 part. Make the differences equal: 3 : 5 = 6 : 10 (difference 4) against 3 : 4 = 12 : 16 (difference 4). Each number went up 6 parts = 9, so one part = 1.5 and the numbers are 6 x 1.5 = 9 and 10 x 1.5 = 15.
const x = 3;
console.log([3 * x, 5 * x]); // Output: [9,15]What to say in the interview
"I write the numbers as 3x and 5x, add 9 to each and set the new ratio to 3 over 4. Cross-multiplying gives x equals 3, so the numbers are 9 and 15, which I check."
Common mistakes
- Adding 9 to the ratio terms (3 + 9 : 5 + 9) instead of to the actual numbers.
- Forgetting to multiply x back into the ratio at the end.