Time and Work LCM Method: Tricks & Formulas

📑 On this page

⚡ The short answer

Time and work runs on rates, not days. A job done in n days means 1/n of it a day. The LCM method makes that whole numbers: job = LCM(times) units, find each worker's units per day, add, divide. Two people together: a × b ÷ (a + b). The trap: never average the times.

🎬 The 30-second story

The goodie bag disaster

“I can pack all 240 bags in 8 hours. Help?”

🎁

Zoya

The fest needs 240 goodie bags packed by tonight.

“I'd take 24 alone. Together? Average of 8 and 24. Sixteen hours!”

🧮

Aarav

Aarav, confident as ever.

“Your help makes me slower?”

⏳

Zoya

Something has gone very wrong.

“Zoya packs 30 an hour, you pack 10. That's 40 an hour.”

🛍️

Meher

Meher counts bags, not hours.

Tara's maths: Rates add; times don't. 240 bags at 40 an hour = 6 hours. Any helper, however slow, makes the job faster, so the answer had to be under Zoya's 8.

Why do you add work rates instead of averaging days?

A time says how long the whole job takes, not how much gets done each hour. What piles up when two people work together is output: 30 bags an hour plus 10. So turn times into rates, add, then convert back.

⚔️ Side by side

Aarav's method vs the real one

❌Averaging the times

what Aarav did

•(8 + 24) ÷ 2 = 16 hours
•Slower than Zoya alone

✅Adding the rates

what the job actually does

•30 + 10 = 40 bags an hour
•240 ÷ 40 = 6 hours

Test	Averaging the times	Adding the rates
Answer for 8 h and 24 h	16 hours	6 hours
Passes the sanity check?	No	Yes: under 8

📌 The sanity band that kills options

Two people together finish faster than the faster one alone, but never faster than half that time (two Zoyas). For 8 and 24 hours, the answer sits between 4 and 8. Anything outside is dead.

What is the LCM method in time and work?

Fractions like 1/8 + 1/24 are miserable under pressure. The LCM method pretends the job has a size every worker's time divides into exactly: the LCM of their times. Then every rate is a whole number.

🧮 The formula

The one relationship behind every question

⚙️

Rate

units per hour

⏱️

Time

hours worked

📦

Work

units done

Try it on three people. Meher, Rohan and Kabir are editing the fest aftermovie; alone they would need 12, 20 and 30 hours.

1
Size the job
LCM(12, 20, 30) = 60 units.
2
Rate each worker
60 ÷ 12 = 5, 60 ÷ 20 = 3, 60 ÷ 30 = 2 units an hour.
3
Combine
5 + 3 + 2 = 10 units an hour. Anyone undoing work is subtracted.
4
Convert back
60 ÷ 10 = 6 hours.

📊 The data

The aftermovie: 60 units, split by speed

Meher · alone: 12 hours

5 units/hr

Rohan · alone: 20 hours

3 units/hr

Kabir · alone: 30 hours

2 units/hr

All three · 60 ÷ 10 = 6 hours

10 units/hr

Meher carries half the job. Rates stack like these bars; times never do.

🎛️ Try it yourself — change any number

Pack the goodie bags yourself

Zoya alone (hours) Aarav alone (hours) Meher alone (hours; 0 if she's busy)

Together

Work done per unit time

16.67%

1/8 + 1/24 = 0.1667 → 6

As set: 6 hours, 16.67% of the job an hour. Add Meher at 12: 4 hours. Make C −12 (someone unpacking): 12 hours. That minus sign is pipes and cisterns.

How do you solve time and work questions step by step?

✍️ Worked example Warm-up ⏱ 30 sec

Two people, one stack of answer sheets

Tara can check a batch of mock-test answer sheets in 9 hours. Rohan, new to the job, takes 18. Together, how long do they take?

1
Total work = LCM(9, 18) = 18 units.
2
Tara does 2 units an hour, Rohan 1.
3
Together: 3 an hour, so 18 ÷ 3 = 6 hours. Inside the 4.5–9 band. ✓

Answer: 6 hours

⚡ Product over sum

Two workers: a × b ÷ (a + b) = 162 ÷ 27 = 6.

✍️ Worked example CLAT level ⏱ 60 sec

Someone leaves halfway

Kabir can set up the fest stage alone in 16 hours; Aarav would take 24. They start together, but Aarav leaves after 6 hours for cricket practice. How many more hours does Kabir need?

1
Total = LCM(16, 24) = 48 units. Kabir does 3 an hour, Aarav 2.
2
First 6 hours together: 6 × 5 = 30 units.
3
Left: 18 units for Kabir alone: 18 ÷ 3 = 6 hours.

Answer: 6 more hours (12 hours in all)

⚡ Phase route

Reread the question: "how many more" (6) and "total" (12) will both be options.

Efficiency questions flip the method. "Zoya is 50% more efficient than Meher" means their rates are 3 : 2, so use those as units per day. If together they finish in 12 days, the job is 5 × 12 = 60 units, and Meher alone needs 60 ÷ 2 = 30 days.

How do man-days and M1 × D1 = M2 × D2 work?

When every worker is equally good, a job has a size in person-days: people × days. A campus clean-up that 6 volunteers finish in 20 days is a 120 person-day job, whoever does it. More people, fewer days, same product: inverse proportion.

Volunteers	Days	Person-days
4	30	120
6	20	120
10	12	120
12	10	120

→ scroll to see more

Add hours a day and amount of work: M₁ × D₁ × H₁ ÷ W₁ = M₂ × D₂ × H₂ ÷ W₂. People, days and hours share the load; more work needs more of them.

✍️ Worked example CLAT level ⏱ 45 sec

More boards, fewer volunteers, longer shifts

12 volunteers working 5 hours a day paint 60 stall boards in 4 days. How many days will 8 volunteers working 6 hours a day take to paint 96 boards?

1
Effort per board: 12 × 5 × 4 ÷ 60 = 4 volunteer-hours.
2
96 boards need 384 volunteer-hours.
3
The new team supplies 8 × 6 = 48 a day: 384 ÷ 48 = 8 days.

Answer: 8 days

⚡ One-line formula

(12 × 5 × 4) ÷ 60 = (8 × 6 × D) ÷ 96, so 4 = D/2 and D = 8.

⚠️ Person-days only work for identical workers

If men and women, or a pro and a newbie, work at different speeds, convert everyone into one currency first. CLAT 2018 did exactly this: two mixed teams give you an equation, and it tells you how many women equal one man.

What are the fastest time and work shortcuts?

✓Two workers: a × b ÷ (a + b). Three or more: LCM units.
✓Efficiency is the inverse of time. Times 2 : 3 means efficiencies 3 : 2.
✓Together in T, one alone in a: the other = a × T ÷ (a − T). Story check: 8 × 6 ÷ 2 = 24. ✓
✓Pair sums: add the A + B, B + C and C + A rates and halve.
✓Join or leave: count units per phase.

And because CLAT loves money: pay follows work, not hours on the clock. If Zoya and Aarav share ₹1,200 for the bags, split it by what each packed in those 6 hours.

🟦 Picture the percent

Who packed the 240 bags, and who gets paid what

3 : 1

₹900 to Zoya, ₹300 to Aarav

Zoya: 180 bagsAarav: 60 bags

Both worked all 6 hours, so the split is their rates, 30 : 10.

💡 Pipes and cisterns: same engine, one minus sign

A tap filling a tank is a worker; a leak undoes the job, so give it a negative rate. The full method, with both CLAT questions, is on our pipes and cisterns page.

How does time and work hide in a CLAT caselet?

Since 2020, CLAT wraps arithmetic in a paragraph: machines, shifts, breakdowns. A print shop takes an order for 3,600 fest flyers. Machine P alone would print them in 6 hours, Q in 9 and R in 18. No LCM needed: the passage gives the job's real size, so rates come out in flyers per hour.

📊 The data

Flyers per hour, straight from the passage

P · 3,600 ÷ 6

600 /hr

Q · 3,600 ÷ 9

400 /hr

R · 3,600 ÷ 18

200 /hr

All three · 3,600 ÷ 1,200 = 3 hours

1,200 /hr

Turn every "takes x hours" into "does y per hour" before you read a single question.

🧭 Decision flow

Before you calculate: three checks

❓ Is every worker's time or rate given, or buildable ("twice as fast as Q")?

No →

Cannot be determined: a group's total rate doesn't give each member's.

Yes ↓

❓ Does everyone work for the whole job?

No →

Split into phases, then add the units.

Yes ↓

❓ Is anyone undoing work?

Yes →

Subtract their rate: pipes and cisterns logic.

No →

Add the rates. Time = total ÷ combined rate.

Which time and work traps cost the most marks?

Most wrong answers here are reading errors, and the options are built to catch them.

The passage says	What it means	The trap option
"A leaves after 4 days"	Both work 4 days, then B alone	Treating it as 4 days before the end
"A leaves 4 days before completion"	A works T − 4 days; solve for T	Assuming A worked 4 days
"How many more days?"	Only the last phase	The total time
"Paid ₹X for the job"	Split by rate × days worked	Equal split, or by speed alone

→ scroll to see more

How CLAT asked it

📝 Asked in CLAT 2018 · Q143 Men and women, person-days

Twelve men and 5 women can complete a work in 2 days whereas 4 men and 3 women can complete the same work in 5 days. In how many days can 8 men complete the work?

Reveal answer + reasoning

Answer: B

Team 1: 24 man-days + 10 woman-days. Team 2: 20 man-days + 15 woman-days. So 24M + 10W = 20M + 15W, giving 4 man-days = 5 woman-days: a woman does 0.8 of a man's work.
Job = 24 + 10 × 0.8 = 32 man-days (Team 2: 20 + 15 × 0.8 = 32 ✓). Eight men need 32 ÷ 8 = 4 days.
Eliminate first: Team 2 is 4 + 2.4 = 6.4 men strong and needs 5 days. Eight men are stronger, so the answer is under 5: C and D die at once. About 60 seconds.

Why the others fail

A — Counts only the men's 24 man-days in Team 1 and drops the women's 8: 24 ÷ 8 = 3.
C — Copies Team 2's 5 days, though 8 men out-work 6.4 men.
D — Slower than Team 2 with a stronger team.

🧪 CLAT-style practice (not a past question) CLAT-style caselet: a machine breaks down

A print shop takes an order for 3,600 fest flyers. Machine P alone can print all of them in 6 hours, machine Q in 9 hours and machine R in 18 hours. P and Q start together at 9:00 am. At 11:00 am, P breaks down, and R is switched on to work alongside Q until the order is complete.

At what time is the order complete?

A12:36 pm

B1:40 pm

C3:00 pm

D12:20 pm

Reveal answer + reasoning

Answer: B

Rates: P 600, Q 400, R 200 an hour. From 9 to 11, P + Q print 2,000, leaving 1,600. Then Q + R print 600 an hour: 1,600 ÷ 600 = 2 hours 40 minutes. Done at 1:40 pm. About 75 seconds.

Why the others fail

A — Assumes R is as fast as the broken P: 1,600 ÷ 1,000 = 96 minutes.
C — Forgets R: Q alone needs 4 hours.
D — Keeps P running after it broke: 1,600 ÷ 1,200 = 80 minutes.

Can you solve these time and work questions?

💥 Myth vs fact

Myth vs fact

🙅 Myth

Two people together take the average of their times.

Tap to bust it →

✅ Fact

Add the rates. 8 and 24 hours give 6, not 16.

🙅 Myth

Double the people, half the time. Always.

Tap to bust it →

✅ Fact

Only if the new people work at the same rate.

🙅 Myth

Wages are split by speed.

Tap to bust it →

✅ Fact

By work done: rate × days worked. A fast worker who leaves early can earn less.

🙅 Myth

"Leaves after 3 days" = "leaves 3 days before the end".

Tap to bust it →

✅ Fact

The first fixes the start; the second fixes the end and needs an equation.

🎯 Quick quiz

Five time and work checks

0 / 5

Meher can edit a reel in 6 hours; Rohan takes 12.

Q1. Working together, how long do they take?

Aarav and Zoya can clear the fest registrations in 12 days, Zoya and Kabir in 15, Kabir and Aarav in 20.

Q2. How long would all three take together?

Aarav works three times as fast as Rohan. Together they assemble a robotics kit in 9 hours.

Q3. How long would Rohan take alone?

15 volunteers can clean the campus in 8 days. After 2 days, 5 more join.

Q4. How many more days will the clean-up take?

Kabir and two helpers finish a shop makeover in 6 days. Kabir alone would take 15 days.

Q5. How long would one helper take alone?

🎯 Time and work in a nutshell

Work piles up as rates: n days means 1/n of the job a day.
LCM method: job = LCM of the times, so every rate is whole.
Together time: between half the faster time and the faster time.
Wages follow rate × days worked.

📎 Formula card

Rate = 1/n a day; work = rate × time
Two together: a × b ÷ (a + b)
Together T, one alone a → other alone = a × T ÷ (a − T)
Efficiency ratio = inverse of the time ratio
M₁D₁H₁ ÷ W₁ = M₂D₂H₂ ÷ W₂ (identical workers)
Pipes: fill +1/a, leak −1/b

Turn the method into a reflex

Timed profit, interest, speed and work questions, with full working.

Practise profit, interest & TSW drills

Frequently asked questions

What is the LCM method in time and work?

Take the total work as the LCM of the times given, so every rate becomes a whole number. If Meher takes 12 hours, Rohan 20 and Kabir 30, call the job 60 units: they do 5, 3 and 2 units an hour. Together that is 10 an hour, so 6 hours, with no fractions anywhere.

What is the formula when two people work together?

If A takes a days and B takes b days, together they take a × b ÷ (a + b) days. For 8 and 24 hours: 192 ÷ 32 = 6 hours. It comes from adding the rates 1/a and 1/b. For three or more people, add the rates or use LCM units instead.

Why can't you average the times in time and work?

Because output adds up, not time. Averaging 8 and 24 hours gives 16, slower than the faster person alone, which is impossible: a helper, however slow, can only speed the job up. Convert each time to a rate, add the rates, then convert back.

How is time and work related to pipes and cisterns?

They are the same topic. A pipe filling a tank is a worker with a positive rate; a leak or outlet has a negative rate. Add the rates with their signs and divide the job by the net rate. If the net rate is zero or negative, the tank never fills.