Time & Work

Overview

Time & Work problems appear on both CAT and GMAT. They involve people or machines working on a task, with questions asking how long tasks take when workers combine, quit midway, or work at different efficiencies. The core idea is simple — rate = 1/time — and every variant follows from this one concept.

On CAT, 1–3 Time & Work questions appear per exam, often with a twist (worker quits, efficiency changes, pipes). On GMAT, these appear as Problem Solving questions in the 600–750 difficulty band.

Core Concept: Rate = 1/Time

If A can complete a job in X hours (or days), then in one hour A does 1/X of the job.

The three-step method:

1. Find each person/machine's rate: rate = 1/(time to complete alone)
2. Add rates to get the combined rate
3. Combined time = 1/(combined rate)

Example: Jack paints a wall in 3 hrs; John in 5 hrs.

• Jack's rate = 1/3/hr; John's rate = 1/5/hr
• Combined rate = 1/3 + 1/5 = 8/15 per hr
• Time together = 15/8 hrs

Key Formulas

Two workers together:

Time = ab / (a + b), where a and b are their individual times

Three workers together:

Time = abc / (ab + bc + ca), where a, b, c are their individual times

These are just the algebra of 1/a + 1/b = 1/T solved for T. Don't memorize blind — re-derive in 10 seconds if you forget.

Man-Days (Work Units)

Total work = Number of workers × Days taken

If 8 men do a job in 12 days, total work = 96 man-days. This is a fixed quantity — use it to compare different workforce combinations.

Equivalency: If 1 man-day = k woman-days, convert before combining.

Example: 8 men × 12 days = 96 man-days; 20 women × 10 days = 200 woman-days. Same job → 96 man-days = 200 woman-days → 1 man-day = 25/12 woman-days.

Machines: N machines × T days = Work Units

If n identical machines do job in t days, total work = n × t machine-days.

• Want fewer days? Need more machines. n₁t₁ = n₂t₂
• Want same job in t₂ days: n₂ = n₁t₁/t₂

Pipes and Cisterns

• Fill pipe adds to tank: rate = +1/t per minute (fills 1 tank in t minutes)
• Drain pipe removes from tank: rate = −1/t per minute

Combine net rate and find total time to fill (net rate > 0) or empty (net rate < 0).

Cycling pipes: If pipes operate alternately in cycles, find net fill per cycle, then check the last partial cycle to determine exact fill time.

Sequential / Partial Work

When workers join or leave mid-task:

1. Calculate work done in each phase separately
2. Track remaining work after each phase
3. If a worker quits with d days remaining, the remaining workers must finish the residual work

Example (Jose & Jane): Jose alone takes x days; Jane alone takes y days; together they take 20 days (xy/(x+y)=20 → xy=20(x+y)). Jose does first half, Jane does second half; total = x/2 + y/2 = 45 days (x+y=90). Solving: x=60, y=30.

Efficiency Variations

If worker B is k times as efficient as A:

• B's time = A's time / k
• If A takes t hours, B takes t/k hours

Rate scaling: If B's speed increases by p%, B's new rate = (1+p/100) × original rate.

Common Mistakes

• Adding times instead of rates: "A in 3 hrs, B in 5 hrs → together 4 hrs" is wrong. Add 1/3 + 1/5, not 3 + 5.
• Forgetting to find remaining work before computing solo completion time
• In pipes, misidentifying which pipe fills and which drains
• Misreading "together complete in T hours" — that means 1/T is their combined rate

Exam Tips

• Assign the total job = 1 (or a convenient integer like LCM of the times, to avoid fractions)
• "LCM method": if A takes 4 days, B takes 6 days — set total job = 12 units. A does 3 units/day, B does 2 units/day, together 5 units/day → 12/5 days. Avoids fractions throughout.
• For "worker quits k days before the end" problems: let T = total time; set up equation counting A's work for (T−k) days + B's work for T days = 1 job.
• For identical machines: n × t = constant. To reduce time, increase machines proportionally.