Time and Work: Formulas, Tricks and Solved Questions

Time and Work deals with the rate at which a person or group completes a task. This chapter uses work units, daily efficiency, combined rates, and proportional reasoning to calculate completion time. Learn the standard formulas, efficiency relationships, alternate-work methods, and common shortcuts used in Time and Work questions.

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What is Time and Work?

Time and Work is the study of the relationship between the amount of work completed, the time taken, and the efficiency or rate of a worker.

If a person completes a work in T days, the work completed in one day is 1/T of the total work. If the total work is represented by W units and the daily rate is R units, then time = W ÷ R. For example, a worker completing a task in 8 days completes 1/8 of the task per day.

Time and Work Formula & Tricks

Important Formulas

Basic work-rate formula
Work = Time × Rate

If a worker's rate is R units per day and the worker works for T days, the completed work is R × T units.

One-day work
One-day work = 1/T

A person who completes the whole work in T days completes 1/T of the work in one day.

Combined work rate
Combined rate = 1/A + 1/B

If A and B complete a work alone in A and B days, their combined daily rate is 1/A + 1/B.

Time taken together
T = AB/(A + B)

When two persons independently complete a work in A and B days, they complete it together in AB/(A + B) days.

Efficiency and time
Efficiency ∝ 1/Time; E₁/E₂ = T₂/T₁

For the same work, efficiency is inversely proportional to time. A person who takes less time has greater efficiency.

Work and wages
Wages ratio = Work contribution ratio

When workers are paid according to their contribution, their wages are divided in the ratio of the work completed by each worker.

Quick Tricks

Use the LCM as total work

When completion times are whole numbers, take their LCM as the total work. The daily efficiency becomes an integer, which avoids repeated fractions.

Example: A completes a task in 12 days and B in 18 days. Take total work as LCM(12, 18) = 36 units. Their efficiencies are 3 and 2 units per day, so together they complete 5 units per day and take 36/5 = 7 1/5 days.
Convert efficiency into a ratio

For the same work, efficiency and time are inversely related. If A is twice as efficient as B, A takes half the time taken by B.

Example: If A and B have efficiency ratio 3:2, their time ratio is 2:3. If B takes 15 days, A takes 10 days.
Subtract completed work

In work-sharing questions, calculate the work completed by the workers who leave, then subtract it from the total work before finding the remaining time.

Example: A completes 1/10 of a task per day and works for 3 days. The completed work is 3/10, so the remaining work is 7/10.
Handle alternate days with a cycle

For workers working on alternate days, calculate the work completed in one complete cycle and then handle any remaining work separately.

Example: A completes 1/6 and B completes 1/8 of a task per day. In a two-day cycle they complete 1/6 + 1/8 = 7/24 of the work.

Time and Work Concepts

Individual Work Rates

A worker's work rate is the fraction or number of work units completed in one unit of time.

If A completes a task in 15 days, A's one-day work is 1/15. If total work is taken as 60 units, A's daily rate is 60/15 = 4 units. The selected total work can be any positive value, but using a common multiple usually simplifies calculations.

Example: A completes a work in 20 days. In 5 days, A completes 5 × 1/20 = 1/4 of the work.

Combined Work by Two or More Workers

When workers operate together, their rates are added because their completed work is added during the same time period.

If A takes 10 days and B takes 15 days, their combined rate is 1/10 + 1/15 = 1/6 of the work per day. Therefore, they complete the work together in 6 days. For more workers, add all individual rates: 1/T = 1/T₁ + 1/T₂ + 1/T₃ + ...

Example: A, B and C complete a task alone in 12, 18 and 36 days. Their combined rate is 1/12 + 1/18 + 1/36 = 1/6, so they finish it in 6 days.

Efficiency and Time Ratio

For a fixed amount of work, a worker's efficiency is inversely proportional to the time taken.

If the efficiency ratio of A and B is m:n, their time ratio is n:m. If A is 25% more efficient than B, then Eₐ:Eᵦ = 125:100 = 5:4, so Tₐ:Tᵦ = 4:5. Thus, if B takes 20 days, A takes 16 days.

Example: A and B have efficiency ratio 4:3. Their time ratio is 3:4. If A takes 15 days, B takes 20 days.

Work Done by Men, Days and Hours

For workers with equal efficiency, work is proportional to the product of the number of workers, working days, and working hours per day.

The relation is M₁D₁H₁ = M₂D₂H₂ for the same work, where M is the number of workers, D is days, and H is hours per day. If efficiencies differ, include efficiency: M₁D₁H₁E₁ = M₂D₂H₂E₂.

Example: 8 workers complete a task in 15 days by working 6 hours daily. The same task with 12 workers working 5 hours daily takes (8 × 15 × 6)/(12 × 5) = 12 days.

Alternate-Day Work

In alternate-day work, calculate each worker's rate and add the work completed during one repeating cycle.

If A works on the first day and B on the second day, one cycle consists of two days. Calculate the work done in each cycle, multiply by the number of complete cycles, and then calculate the remaining work using the rate of the worker assigned to the next day.

Example: A completes 1/4 of a task daily and B completes 1/6 daily. In two days they complete 1/4 + 1/6 = 5/12. After two cycles, 5/6 is complete; the next A-day completes the remaining 1/6 in 2/3 of a day.

Time and Work Video Lessons

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Time and Work Concepts

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Practice Time and Work Questions

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1If the √18 is expressed as a√2 , what is the value of a?→ 2A shopkeeper declares the following three schemes of discounts. Which scheme is the most beneficial to a customer? A. Buy 42 items and get 47 of the same items B. Two successive discounts of 16% and 36% C. Two successive equal discounts of 46%→ 3A shopkeeper marks an item at 1000 and allows a 20% discount. What is the selling price?→ 4The marked price of an item is 10% higher than the cost price and a discount of 10% is given on the marked price. What is the percentage gain or loss to the business owner in this sale?→ 5The marked price of an article is Rs. 1000. It is sold at a discount of 30 percent. If the cost price of the article is Rs. 520, then what is the value of profit?→ 6If the selling price of a bed is 6 times the initial selling price, the profit is 9 times the initial profit. Find the initial profit percentage.→ 7√(0.1225/0.25) = ?→ 8A product is marked at ₹500 with a 20% discount. What is the selling price?→ 9What is the smallest number which when increased by 30 is exactly divisible by 6, 10 and 12?→ 10A shopkeeper bought an item for ₹960 and sold it at a 25% profit. What was the selling price?→

Time and Work Quick Quiz

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Quick Revision Notes

Time and Work Revision Points

Use these rules to revise the main calculation methods and relationships.

  • If a person completes work in T days, one-day work is 1/T.
  • Work = Time × Rate.
  • For workers working together, add their daily rates.
  • For two workers taking A and B days alone, time together is AB/(A + B).
  • Efficiency and time are inversely proportional for the same work.
  • If efficiency ratio is m:n, the time ratio is n:m.
  • For equal-efficiency workers, M × D × H remains constant for the same work.
  • In alternate-day questions, calculate work done in one complete cycle before handling the remainder.

Time and Work FAQs

What is the formula for one day's work?

If a person completes a work in T days, one day's work is 1/T of the total work. For example, a 12-day worker completes 1/12 of the work per day.

How do you find the time taken by two people working together?

Add their individual rates: combined rate = 1/A + 1/B. Therefore, time together = AB/(A + B). If A takes 8 days and B takes 12 days, the time is 8 × 12/(8 + 12) = 24/5 = 4.8 days.

If A is 50% more efficient than B, what is the ratio of their times?

Their efficiency ratio is 150:100 = 3:2. Since time is inversely proportional to efficiency, their time ratio A:B is 2:3.

How is the LCM method used in Time and Work?

Take the LCM of the individual completion times as total work. For completion times 10 and 15 days, total work is 30 units, so the daily rates are 3 and 2 units.

What is the formula for workers, days and hours?

For equal-efficiency workers doing the same work, M₁D₁H₁ = M₂D₂H₂. If efficiency differs, use M₁D₁H₁E₁ = M₂D₂H₂E₂.

How are wages divided among workers?

Wages are divided in the ratio of work contributions when payment is based on work done. Work contribution equals rate × time, so two workers need not receive equal wages if their rates or working times differ.

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