How to Solve Age Problems Given a Ratio and a Sum
Solve age aptitude problems that give a ratio and a sum using the x = S/(a+b) shortcut, with a worked example and practice questions.
Expected Interview Answer
When a problem gives both a ratio and a sum of two present ages, assign the ratio terms a common multiplier x, express the sum as one linear equation in x, solve for x, and multiply back to get each actual age.
If two ages are in ratio a:b and their sum is S, write the ages as ax and bx, so ax + bx = S gives x = S/(a+b) directly. This single-equation setup is faster than naming two separate variables because the ratio already encodes the relationship between the ages, leaving only one unknown to solve for. Once x is found, both ages fall out by substitution, and a quick check — do the two ages actually sum to S and reduce to the given ratio — catches arithmetic slips before they compound into a wrong final answer. The same multiplier trick extends cleanly to three or more people sharing a ratio.
- Reduces two unknowns to one variable instantly
- x = S/(a+b) is a reusable one-line formula
- Extends directly to three-term ratios
- Built-in sum-and-ratio check catches mistakes early
AI Mentor Explanation
Two opening batters split a partnership total of 90 runs in the ratio 5:4. Writing their scores as 5x and 4x turns the sum condition into 9x = 90, so x = 10, giving 50 and 40 runs. The ratio fixes the shape of the split instantly, and the sum pins down the actual scale, exactly how age-ratio-and-sum problems are solved in one pass.
Worked example
Set up ratio
- Ages = 5x and 3x
Sum equation
- 5x + 3x = 64
- 8x = 64 -> x = 8
Final ages
- 40 and 24
Step-by-Step Explanation
Step 1
Assign the multiplier
Write the two ages as ax and bx using the given ratio a:b.
Step 2
Form the sum equation
ax + bx = S, where S is the given total of the two ages.
Step 3
Solve for x
x = S / (a + b), a single division.
Step 4
Recover and verify
Multiply back to get both ages, then confirm they sum to S and reduce to a:b.
What Interviewer Expects
- Correct common-multiplier setup from the ratio
- x = S/(a+b) derived, not just recalled
- Verification that the final ages satisfy both the ratio and the sum
- Ability to extend the method to a three-term ratio
Common Mistakes
- Adding the ratio terms incorrectly before dividing into the sum
- Forgetting to multiply x back into both ratio terms
- Mixing up which ratio term belongs to which person
- Not verifying the final ages actually reduce to the stated ratio
Best Answer (HR Friendly)
“Whenever a problem gives a ratio and a sum for two ages, I write the ages as ax and bx using the ratio, then the sum becomes one simple equation, ax plus bx equals the total, which solves for x in one step. Multiplying x back into a and b gives both actual ages, and I always double check they add up correctly and match the original ratio before finalizing.”
Follow-up Questions
- How would you extend this method to three ages sharing one ratio?
- What changes if the problem gives a difference instead of a sum?
- How do you handle a ratio given as a fraction like 1.5:1 instead of whole numbers?
- How would you verify your answer without resolving the whole problem?
MCQ Practice
1. Two present ages are in ratio 7:5 and sum to 96. The older age is?
x = 96/12 = 8. Older age = 7x = 56.
2. The ages of a father and son are in ratio 9:2 and their sum is 55. The son’s age is?
x = 55/11 = 5. Son = 2x = 10.
3. Three friends’ ages are in ratio 2:3:4 and sum to 54. The middle friend’s age is?
x = 54/9 = 6. Middle = 3x = 18.
Flash Cards
Setup for a ratio a:b with sum S? — Ages = ax and bx, so ax + bx = S.
Formula for x? — x = S / (a + b).
How to get final ages? — Multiply x back into each ratio term.
Quick check for correctness? — Ages must sum to S and reduce to the given ratio.