How to Solve Age Problems Given a Sum and a Difference
Solve age aptitude problems that give a sum and a difference using elimination, with a worked example and practice questions with answers.
Expected Interview Answer
Given the sum S and the difference D of two present ages, the larger age is (S + D)/2 and the smaller age is (S - D)/2, derived by adding and subtracting the two equations rather than substituting one variable into the other.
Let the ages be p and q with p + q = S and p - q = D. Adding the two equations eliminates q, giving 2p = S + D, so p = (S+D)/2; subtracting eliminates p, giving 2q = S - D, so q = (S-D)/2. This elimination approach is faster than substitution because it produces both unknowns from two clean divisions instead of a multi-step algebraic chain. It generalizes to any two-quantity sum-and-difference problem, not just ages, which is why interviewers use it to test whether a candidate recognizes the pattern rather than just memorizing an age-specific formula.
- Two clean formulas replace multi-step substitution
- Elimination avoids fraction-heavy algebra
- The same technique generalizes beyond age problems
- Fast to verify: check both sum and difference at the end
AI Mentor Explanation
Two batters’ combined score is 130 and one scored 20 more than the other. Adding gives twice the higher score, 150, so the higher score is 75; subtracting gives twice the lower score, 110, so it is 55. Elimination avoids naming one score in terms of the other and then substituting, which is exactly the shortcut used for sum-and-difference age problems.
Worked example
Given
- p + q = 58
- p - q = 12
Add equations
- 2p = 70 -> p = 35
Subtract equations
- 2q = 46 -> q = 23
Step-by-Step Explanation
Step 1
Write the two equations
p + q = S (sum) and p - q = D (difference).
Step 2
Add them
2p = S + D, so p = (S + D) / 2, the larger age.
Step 3
Subtract them
2q = S - D, so q = (S - D) / 2, the smaller age.
Step 4
Verify
Confirm p + q equals S and p - q equals D.
What Interviewer Expects
- Recognizing elimination as faster than substitution here
- Correct formulas p = (S+D)/2 and q = (S-D)/2
- Understanding that this generalizes beyond ages
- A verification step confirming both original conditions
Common Mistakes
- Substituting one variable into the other instead of eliminating
- Forgetting to divide by 2 after adding or subtracting
- Swapping which age is larger when applying the difference
- Not verifying the result against both the sum and the difference
Best Answer (HR Friendly)
“Given a sum and a difference of two ages, I set up p plus q equals the sum and p minus q equals the difference, then add the two equations to cancel q, giving twice the larger age, and subtract them to cancel p, giving twice the smaller age. Dividing each by two gives both ages directly, and I always verify they satisfy both the original sum and difference.”
Follow-up Questions
- How would this change if the difference were given as a percentage instead of an absolute number?
- How do you extend sum-and-difference elimination to three unknowns?
- What if the sum and difference are both given for ages n years in the future?
- How does this elimination method compare in speed to substitution for larger systems?
MCQ Practice
1. The sum of two ages is 70 and their difference is 10. The older age is?
(70+10)/2 = 40.
2. A mother and daughter have ages summing to 66, differing by 26. The daughter’s age is?
(66-26)/2 = 20.
3. Two cousins’ ages sum to 40 and differ by 8. The younger cousin’s age is?
(40-8)/2 = 16.
Flash Cards
Formula for the larger age given sum S and difference D? — p = (S + D) / 2.
Formula for the smaller age? — q = (S - D) / 2.
Why use elimination instead of substitution here? — Adding/subtracting cancels one variable directly, avoiding extra algebra.
How to verify the answer? — Check the two ages sum to S and differ by D.