Ten years ago, A was half as old as B. If the ratio of their present ages is 3 : 4, what is the sum of their present ages?
SHORT ANSWER
35 years. Present ages 3x and 4x. Ten years ago: 3x - 10 = (4x - 10)/2, so 6x - 20 = 4x - 10 and x = 5. Ages are 15 and 20.
Explain it simply
Write present ages using the ratio with one multiplier, then move both ages back in time by the same 10 years. The second condition gives an equation in that single multiplier. Always plug the result back in to check both conditions, because ratio equations are easy to set up upside down.
Example
Question: Present ages 3 : 4. Ten years ago A was half of B. Sum of present ages?
- A = 3x, B = 4x.
- Ten years ago: 3x - 10 = (4x - 10) / 2.
- 6x - 20 = 4x - 10, so 2x = 10, x = 5.
- A = 15, B = 20. Sum = 35.
Check: ten years ago A was 5 and B was 10. 5 is half of 10. Present ratio 15 : 20 = 3 : 4.
js
const x = 5;
console.log([3 * x - 10, 4 * x - 10]); // Output: [5,10]
console.log(7 * x); // Output: 35What to say in the interview
"I write the ages as 3x and 4x, take 10 off each, and set A's past age to half of B's. That gives x equals 5, so 15 and 20, and the sum is 35."
Common mistakes
- Applying the ratio 3 : 4 to the ages ten years ago.
- Writing 2(4x - 10) = 3x - 10, which says B was half of A.