Solve Any Time and Work Problem in Under 30 Seconds

Time and Work problems have exactly four patterns in campus placement tests, and each one maps to a three-step LCM template that executes in under 30 seconds.

The companion article Time and Work Aptitude Questions with Solutions covers every question type with full derivations. This article is the speed layer: identify the type in 5 seconds, apply the LCM template in 25 seconds, answer and move on.

Why LCM Beats Fractions Every Time

The fraction method is correct but slow. A worker finishing a job in 12 days completes 1/12 of the job per day. Adding 1/12 and 1/18 requires finding a common denominator mid-calculation, under timed conditions.

The LCM shortcut converts all rates to whole numbers before any calculation starts. Three steps:

• Step 1: Find the LCM of all individual completion times. That LCM is the total work, in units.
• Step 2: Each worker’s rate in units per day equals LCM divided by that worker’s individual time.
• Step 3: Sum the rates (or subtract for workers who drain or remove), then divide total work by net rate.

Applied to the 12-day and 18-day example:

• LCM(12, 18) = 36. Assign 36 units as total work.
• A’s rate: 36 / 12 = 3 units/day. B’s rate: 36 / 18 = 2 units/day.
• Combined rate: 3 + 2 = 5 units/day.
• Time together: 36 / 5 = 7.2 days.
• Verify: 1/12 + 1/18 = 3/36 + 2/36 = 5/36 per day. Time = 36/5 = 7.2 ✓

Fractions appear nowhere in the working. Khan Academy’s LCM review covers the calculation mechanics for LCM quickly across two- and three-number cases.

The Reciprocal Trap

The single most common setup error: computing individual time / LCM instead of LCM / individual time for the rate. In the example above, A’s rate is 36 / 12 = 3, not 12 / 36. The immediate check: rate multiplied by individual time must equal total work (3 x 12 = 36 ✓). If the check fails, the rate is inverted.

This confusion between “work done in 1 day” (the rate, a small number) and “time taken” (the larger number of days) is the most common reason students get the right method but the wrong answer.

Four Question-Type Patterns: Recognise in 5 Seconds

Identifying the question type before touching a pen saves 10 to 15 seconds. The four patterns that cover campus placement rounds:

Type	Recognition cue	Template
Combined rate	”A and B work together”	Add all rates; divide total work by sum
Find unknown rate	Combined time known; one worker’s time known	Combined rate minus known rate = unknown rate; invert
Departure problem	”After n days, one worker leaves”	Work done (joint phase) + work done (solo phase) = total work
Men-days change	”More workers added” or workforce reduced	Man-days = constant; scale time inversely

The Campus Placement Evaluation Test and most campus aptitude rounds draw Time and Work questions from exactly these four patterns. Drilling one example of each type until recognition is automatic reduces per-question time more than any formula shortcut does.

Five Speed Drills: Setup to Answer

IndiaBix’s Time and Work section has more than 100 problems for volume drilling once the template is internalised. These five drills model sub-30-second execution.

Drill 1: Combined Rate (Two Workers)

• A completes a task in 15 days; B in 10 days. Time working together?
• LCM(15, 10) = 30. Set 30 as total work.
• A’s rate: 30 / 15 = 2 units/day. B’s rate: 30 / 10 = 3 units/day.
• Combined rate: 2 + 3 = 5 units/day.
• Time: 30 / 5 = 6 days.
• Verify: 1/15 + 1/10 = 2/30 + 3/30 = 5/30. Invert: 6 ✓

Drill 2: Combined Rate (Three Workers)

• A: 15 days. B: 10 days. C: 30 days. All three working together?
• LCM(15, 10, 30) = 30. A’s rate: 2, B’s rate: 3, C’s rate: 1 unit/day.
• Combined rate: 2 + 3 + 1 = 6 units/day.
• Time: 30 / 6 = 5 days.
• Verify: 1/15 + 1/10 + 1/30 = 2/30 + 3/30 + 1/30 = 6/30. Invert: 5 ✓

Drill 3: Find Unknown Rate

• A and B together complete a task in 6 days. A alone takes 10 days. How long for B alone?
• LCM(6, 10) = 30. Set 30 as total work.
• Combined rate: 30 / 6 = 5 units/day. A’s rate: 30 / 10 = 3 units/day.
• B’s rate: 5 - 3 = 2 units/day.
• B alone: 30 / 2 = 15 days.
• Verify: 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6. Combined time = 6 ✓

Drill 4: Departure Problem

• A completes a task in 12 days; B in 20 days. They work together for 3 days, then A leaves. How many more days does B need?
• LCM(12, 20) = 60. A’s rate: 60 / 12 = 5 units/day. B’s rate: 60 / 20 = 3 units/day.
• Work done in 3 days together: 3 x (5 + 3) = 24 units.
• Remaining work: 60 - 24 = 36 units. B’s rate: 3 units/day.
• B takes: 36 / 3 = 12 more days.
• Verify: 3 x (1/12 + 1/20) + 12 x (1/20) = 3 x 8/60 + 12/20 = 24/60 + 36/60 = 60/60 = 1 ✓

Drill 5: Men-Days (Workforce Scaling)

• 12 workers complete a project in 10 days. How long for 8 workers, assuming same daily output per worker?
• Total work = 12 x 10 = 120 man-days.
• With 8 workers: 120 / 8 = 15 days.
• Verify: inverse proportion holds — 12 x 10 = 8 x 15 = 120 ✓

Pipes and Cisterns: Same Template, Different Signs

Pipes and cisterns problems follow the identical LCM template. One rule changes: emptying (outlet) pipes contribute negative rate. Filling pipes add; draining pipes subtract.

Template for pipes and cisterns:

• Set LCM of all fill and drain times as total capacity, in units.
• Each filling pipe’s contribution = capacity / fill time (add to net rate).
• Each draining pipe’s contribution = capacity / drain time (subtract from net rate).
• Time to fill = total capacity / net rate.

Drill 6: Mixed Fill and Drain

• Pipe A fills a tank in 6 hours. Pipe B fills it in 4 hours. Pipe C drains it in 12 hours. All three open together. Time to fill?
• LCM(6, 4, 12) = 12. Set 12 units as tank capacity.
• A fills: 12 / 6 = 2 units/hr. B fills: 12 / 4 = 3 units/hr. C drains: 12 / 12 = 1 unit/hr.
• Net rate: 2 + 3 - 1 = 4 units/hr.
• Time: 12 / 4 = 3 hours.
• Verify: 1/6 + 1/4 - 1/12 = 2/12 + 3/12 - 1/12 = 4/12 = 1/3 per hour. Time = 3 ✓

The Sign Error Trap in Pipes

Before running the LCM template, classify each pipe as filling (+) or draining (-) and write the sign explicitly next to the pipe. This takes 2 seconds and eliminates the most common wrong answer in cistern problems, which is adding the drain rate instead of subtracting it.

Three Error Traps to Clear Before Test Day

These three patterns account for the majority of wrong answers in Time and Work questions under timed conditions.

Trap 1: Reciprocal Slip

Rate = LCM / individual time, not individual time / LCM. If total work is 60 and a worker takes 12 days, the rate is 5 units/day, not 1/5 of a unit. Check immediately: rate x individual time must equal total work. If it does not, the rate is inverted. Two seconds on this check prevents the most frequent setup error.

Trap 2: Departure Sign Error

In a departure problem, work done before someone leaves is positive, completed work. It is added to work done after. It is never subtracted from total work. Write this line explicitly before calculating:

• Work done so far = combined rate x days worked together.

Students who get departure problems wrong most often jump directly to subtraction. The explicit “work done so far” line forces the correct direction.

Trap 3: LCM Base Assignment

The LCM of all individual times is the total work, not the smallest number in the problem, not the largest, not any convenient round number. Any other base reintroduces fractions into the working. If the problem’s numbers do not yield a clean LCM quickly, switching to the fraction method for that specific problem is faster than forcing a large LCM calculation.

For quantitative rounds in tests like AMCAT and Mu Sigma’s MuApt, multiple-choice options often include the result of one of these three traps as a decoy. Knowing which trap each wrong option represents is as useful as knowing the correct method.

The six drills above work because each question type reduces to one template: identify the pattern, assign the LCM, execute in whole numbers. TinkerLLM at ₹299 brings the same single-template architecture to LLM fundamentals: token math, prompt structures, and retrieval patterns that handle the majority of real-world tasks in a sandbox you can run without a GPU. If the 30-second discipline from these drills is already built, applying the same habit to AI fundamentals takes one week of practice, not one semester.