Work and Wages: Formulas, Ratios and Wage Distribution

Work and Wages questions relate payment to the amount of work completed by individuals or groups. Wages are distributed in the ratio of work done, provided the payment rate is the same. The required work can be calculated using the number of workers, days, working hours and individual efficiency. Ratio methods make these calculations shorter.

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What is Work and Wages?

Work and Wages problems distribute a total wage according to the work completed by each person or group. When the rate of payment is uniform, the wage ratio equals the work ratio.

For a person, work is proportional to efficiency × time. If the number of workers, working days and daily hours are included, total work is proportional to workers × days × hours × efficiency. Thus, for two workers or groups, their wage ratio is calculated from the ratio of these work quantities. If the total wage is ₹W and the work ratio is a:b, the shares are ₹Wa/(a+b) and ₹Wb/(a+b).

Work and Wages Formula & Tricks

Important Formulas

Work calculation
Work ∝ Number of workers × Days × Hours per day × Efficiency

Use this relation when workers, time or efficiency differ. The product gives the relative amount of work completed.

Wage ratio
Wage A : Wage B = Work A : Work B

This applies when both persons or groups receive payment at the same rate per unit of work.

Individual wage share
A's wage = Total wage × A's work / Total work

Express the individual work as a fraction of the combined work and multiply by the total wage.

Work using rate
Work = Rate of work × Time

If a person completes 1/x of a work in one day, the rate is 1/x work per day and the work in t days is t/x.

Wage difference
Difference in wages = Total wage × Difference in work units / Total work units

Use this when the question gives the total wage and asks for the difference between two shares.

Quick Tricks

Cancel common factors before forming the ratio

In a work ratio, common numbers of days or hours cancel directly. Compare only the factors that differ.

Example: A works 6 days for 8 hours and B works 4 days for 6 hours, with equal efficiency. Their work ratio is 6×8 : 4×6 = 48:24 = 2:1.
Convert every worker’s contribution into units

When efficiencies or times differ, calculate each person’s product of efficiency, days and hours. Use these products as work units.

Example: A has efficiency 3 and works 4 days; B has efficiency 2 and works 6 days. Their work ratio is 3×4 : 2×6 = 12:12 = 1:1, so equal wages are due.
Use total wage parts

For a work ratio a:b:c, divide the total wage by a+b+c to find one part, then multiply by each ratio term.

Example: If work is in the ratio 2:3:5 and the total wage is ₹2,000, one part is ₹200. The shares are ₹400, ₹600 and ₹1,000.

Work and Wages Concepts

Wages in the ratio of work done

When the payment rate is the same, wages are divided in the same ratio as the work completed.

If A completes x units and B completes y units, then A's wage:B's wage = x:y. The total wage is divided into x+y equal parts. This rule does not require both people to work for the same number of days; their actual work quantities are compared.

Example: A and B complete work in the ratio 3:2 and receive ₹1,500. A's share is 1500×3/5 = ₹900, and B's share is 1500×2/5 = ₹600.

Work ratio using workers, days and hours

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

For groups A and B, the work ratio is (workers in A × days in A × hours in A):(workers in B × days in B × hours in B). If individual efficiencies differ, multiply each group’s product by its efficiency factor.

Example: 4 workers complete 6 days of 5 hours each, while 3 workers complete 8 days of 4 hours each. The work ratio is 4×6×5 : 3×8×4 = 120:96 = 5:4.

Wage distribution with different efficiencies

When workers have different efficiencies, each worker’s wage share is proportional to efficiency × time worked.

If A and B work for t₁ and t₂ days with efficiencies e₁ and e₂, their wage ratio is e₁t₁:e₂t₂, assuming equal daily working hours and payment per unit of work. Include hours as an additional factor when daily hours differ.

Example: A's efficiency is 2 units per day and B's is 3 units per day. A works 6 days and B works 4 days. Their work ratio is 2×6:3×4 = 12:12 = 1:1, so they receive equal wages.

Wages for partial work and unfinished work

A person who completes only a fraction of the assigned work receives the same fraction of the wage when payment is proportional to work.

If the complete work has wage W and a worker completes m/n of it, the wage is W×m/n. For several workers, calculate each completed fraction or work unit first, then distribute the total payment in that ratio.

Example: A completes 3/5 of a job and B completes 2/5. If the total payment is ₹2,500, A receives ₹1,500 and B receives ₹1,000.

Work and Wages Video Lessons

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Work and Wages: Proportional Payment

Learn how to divide wages fairly in proportion to the work completed, using time, efficiency, and contribution to calculate each worker’s share.

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Practice Work and Wages Questions

Practise published questions related to this topic.

1A shopkeeper offers the following four schemes. A) Two successive discounts of 19% and 25% B) Buy 7, get 6 free C) Single discount of 50% D) Two successive discounts of 3% and 39% Which scheme is the best for the shopkeeper?→ 239P6 is divisible by 3. Which of the following cannot be the value of P?→ 3A shopkeeper marks an article 40% above its cost price. He then allows two successive discounts of 18% and 5% on the marked price. Find the profit percentage.→ 4A wholesaler allows a scheme discount of 25% on an article whose marked price is ₹2,400. The retailer is required to pay the remaining amount in full. What is the net price paid by the retailer?→ 5If on a sale, there is 40% discount on the marked price of ₹1,200, but the sale is done at ₹520 only, then how much additional discount did the customer get?→ 6A person sells an article at the loss of 40 percent. Cost price and selling price both are decreased by 50 percent. What will be the new loss percentage?→ 7What is the smallest natural number that should be added to 9054 such that a remainder of 6 is left when the resulting number is divided by each of the numbers 28, 24, 56 and 29?→ 8The profit earned on selling an article for Rs. 1200 is 29 times of the loss incurred on selling the same article for Rs. 600. What is the profit percentage on this article at the selling price of Rs. 1200?→ 9A bag was sold for ₹880 at a 12% loss. What was the cost price?→ 10A trader initially marks all his goods 60% above the cost price. He sells half of the goods at a discount of 25% on the marked price. For the remaining half, he revises the marked price to 80% above the cost price and then offers a discount of 30%. Find the overall profit percentage earned by the trader.→

Work and Wages Quick Quiz

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Quick Revision Notes

Work and Wages Quick Revision

Use these rules to calculate work-based payment shares accurately.

  • Wages are proportional to work done when the payment rate per unit of work is the same.
  • Work is proportional to workers × days × hours × efficiency.
  • For ratio a:b and total wage W, shares are Wa/(a+b) and Wb/(a+b).
  • For different efficiencies, multiply efficiency by the time worked before forming the wage ratio.
  • For work ratio a:b:c, divide the total wage into a+b+c parts.
  • Common factors in workers, days or hours can be cancelled while simplifying the ratio.

Work and Wages FAQs

What is the basic work and wages formula?

Wage share = Total wage × Individual work / Total work. If work is in the ratio a:b, the shares are Total wage × a/(a+b) and Total wage × b/(a+b).

How are wages divided if A and B work in the ratio 4:3?

Their wages are divided in the ratio 4:3. If the total wage is ₹2,100, A receives ₹1,200 and B receives ₹900.

How do days and hours affect the work-wage ratio?

For equal-efficiency workers, work ratio = workers × days × hours. For one worker, working twice as many hours for the same number of days produces twice the work and twice the wage.

How are wages calculated when efficiencies are different?

Multiply each worker’s efficiency by the time worked. For example, efficiency 2 for 5 days and efficiency 3 for 4 days give work units 10 and 12, so the wage ratio is 5:6.

A and B receive wages in the ratio 5:7, and the total wage is ₹3,600. What is A's share?

The total parts are 12. A's share = ₹3,600 × 5/12 = ₹1,500; B's share is ₹2,100.

What is the wage difference if the work ratio is 7:5 and the total wage is ₹2,400?

The difference is 2 parts out of 12. Wage difference = ₹2,400 × 2/12 = ₹400.

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