Time and Work Problems: The Efficiency Method Explained

"A can do a job in 12 days, B in 18 days. Working together, how long will they take?"

Almost every student meets this question, and almost every student solves it the slow way — with fractions of work per day, a common denominator, and a reciprocal at the end. It works, but it takes two minutes and invites mistakes.

There is a much better way, and it turns most time-and-work questions into a few seconds of mental arithmetic.

The problem with the standard method

The textbook approach says: A does 1/12 of the job per day, B does 1/18. Together they do 1/12 + 1/18 = 3/36 + 2/36 = 5/36 per day. So they take 36/5 = 7.2 days.

Correct — but you have handled three fractions and a reciprocal. Under exam pressure, that is where errors creep in.

The efficiency method

Instead of fractions of a job, give the job a total size and give each person an efficiency.

Take the LCM of the given times as the total work.

For A = 12 days and B = 18 days, LCM(12, 18) = 36 units of work.

• A finishes 36 units in 12 days → A's efficiency = 3 units/day
• B finishes 36 units in 18 days → B's efficiency = 2 units/day
• Together = 5 units/day
• Time = 36 ÷ 5 = 7.2 days

Same answer, but every number is a whole number. No fractions at all. Once a student sees this, they rarely go back.

Worked examples

Three people together

A in 10 days, B in 15 days, C in 30 days. Together?

LCM(10, 15, 30) = 30 units.

• A = 3/day, B = 2/day, C = 1/day → total 6/day
• 30 ÷ 6 = 5 days

Someone leaves partway

A and B together can finish in 8 days. A alone takes 12 days. How long does B alone take?

Let total work = LCM(8, 12) = 24 units.

• A + B = 24 ÷ 8 = 3/day
• A = 24 ÷ 12 = 2/day
• So B = 3 − 2 = 1/day → B alone takes 24 ÷ 1 = 24 days

Work done in stages

A can do a job in 20 days. He works 5 days, then B finishes the rest in 12 days. How long would B take alone?

Total = 20 units (using A's time).

• A = 1 unit/day → in 5 days A does 5 units
• Remaining = 15 units, done by B in 12 days → B = 15/12 = 1.25/day
• B alone: 20 ÷ 1.25 = 16 days

Negative efficiency: pipes and cisterns

Pipes-and-cisterns questions are the same topic wearing a different hat. An inlet pipe has positive efficiency; an outlet pipe has negative efficiency.

Pipe A fills a tank in 6 hours, pipe B fills it in 8 hours, pipe C empties it in 12 hours. All open together?

LCM(6, 8, 12) = 24 units.

• A = +4/hr, B = +3/hr, C = −2/hr
• Net = 4 + 3 − 2 = 5/hr
• Time = 24 ÷ 5 = 4.8 hours

The minus sign is the whole trick. Everything else is identical.

The ratio shortcut

There is one more relationship worth knowing:

> If A is n times as efficient as B, then A takes 1/n of the time B takes.

So if A is three times as fast as B, and together they finish in 9 days, then in ratio terms A does 3 parts and B does 1 part per day — 4 parts total. A alone would take 4 × 9 ÷ 3 = 12 days, and B alone 4 × 9 = 36 days.

Quick reference

How to practise this

Do not start with mixed questions. Build it in order:

1. Two-person "together" questions until the LCM step is automatic
2. Find-the-missing-person questions
3. Pipes and cisterns with one outlet
4. Staged work questions, which combine everything

Fifteen minutes a day for two weeks is usually enough to make this topic feel routine. The arithmetic underneath — LCM, division, ratios — needs to be quick, which is where mental maths practice pays off.

Frequently asked questions

Which class is this topic for? It appears from Class 7 in most boards and is a staple of every competitive exam from NTSE to SSC and banking.

Is the efficiency method acceptable in board exams? Yes. It is standard mathematics, not a trick — you are simply choosing a convenient unit of work. Show the LCM step and the working is complete.

What if the times do not have a neat LCM? Use the product of the two numbers instead. For A = 7 and B = 11, take total work = 77. The method is unchanged.

Why do students find this topic hard? Because the standard method buries a simple idea under fraction arithmetic. Change the unit and the difficulty largely disappears.

Does this connect to ratio and proportion? Directly. Efficiency is a ratio, and most harder questions in this topic are really ratio questions in disguise.

Make the arithmetic automatic first

Every shortcut here assumes LCMs, division and ratios come instantly. When they do not, the method still feels slow.

That foundation is what our Quantitative Aptitude course builds, from Class 6 upward.

Book a free demo class — a real lesson, and an honest assessment of where your child is.

Cover image: Heidy Garcia — CC BY 4.0.