How to Solve Time and Work Problems
Solve time and work aptitude problems using the rate and LCM methods — formulas, a worked example, shortcuts and practice questions with answers.
Expected Interview Answer
Time and work problems are solved by converting each worker’s output into a rate (work done per unit time), adding rates when workers combine, and using total work ÷ combined rate to find the time.
If A finishes a job in a days, A’s rate is 1/a of the job per day. Two workers together have a combined rate of 1/a + 1/b, so together they finish in 1/(1/a + 1/b) days. The unitary/LCM method makes this concrete: assume total work equals the LCM of the individual times, compute each worker’s units-per-day, add them, and divide. Rates add; times do not.
- A single rate-based method covers most variations
- The LCM trick avoids messy fractions
- Extends to pipes/cisterns and efficiency problems
AI Mentor Explanation
Think of two batters chasing a target. If one scores at 6 runs per over and the other at 4, together they don’t bat at "5 overs" — you add their scoring rates: 10 runs per over combined, so the chase finishes faster. Time-and-work is the same: never add the times, add the rates. Convert "finishes in a days" to "does 1/a per day", sum the rates, then divide the total work by the combined rate.
Worked Example (LCM method)
A alone
- 10 days
- 30 ÷ 10 = 3 units/day
B alone
- 15 days
- 30 ÷ 15 = 2 units/day
Together
- 5 units/day
- 30 ÷ 5 = 6 days
Step-by-Step Explanation
Step 1
Convert to rates
A finishes in a days → A’s rate = 1/a of the job per day.
Step 2
Use LCM as total work
Let total work = LCM of the individual times to avoid fractions.
Step 3
Find units per day
Each worker’s units/day = total work ÷ their days.
Step 4
Add rates and divide
Combined units/day added; time = total work ÷ combined rate.
What Interviewer Expects
- Rates add, times do not
- The 1/a per-day formulation
- The LCM shortcut to avoid fractions
- Correct handling of combined-work timing
Common Mistakes
- Averaging or adding the individual times
- Forgetting to convert time to a rate
- Mishandling workers who leave partway through
- Sign errors in pipes-and-cisterns (inflow vs outflow)
Best Answer (HR Friendly)
“Convert each worker’s time into a rate — if someone finishes in 10 days, they do one-tenth of the job per day. Add the rates when they work together, then divide the whole job by the combined rate to get the time. Rates add; times don’t.”
Code Example
def combined_time(a, b):
# a, b = days each worker takes alone
combined_rate = 1/a + 1/b
return 1 / combined_rate
print(combined_time(10, 15)) # 6.0 daysFollow-up Questions
- If A and B finish in 6 days together and A alone in 10, how long does B take?
- How do pipes-and-cisterns problems differ (inflow vs outflow)?
- How do you handle a worker who leaves after some days?
- How does efficiency (A is twice as fast as B) change the setup?
MCQ Practice
1. A does a job in 12 days, B in 6 days. Together they take?
Rates: 1/12 + 1/6 = 1/12 + 2/12 = 3/12 = 1/4, so together 4 days.
2. In time-and-work problems, what adds when workers combine?
Work rates (job per unit time) add; the individual times do not.
3. A finishes in 8 days. A’s one-day work is?
Finishing in 8 days means doing 1/8 of the job per day.
Flash Cards
Rate from time? — Finishes in a days → does 1/a of the job per day.
Combined time formula? — 1 ÷ (1/a + 1/b) — add the rates, then invert.
LCM trick? — Let total work = LCM of the times; each does (total ÷ their days) units/day.
Key rule? — Rates add; times do not.