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AFCAT Numerical Ability · Profit and Loss

Two Articles at One Price, and Costs with Overheads

When two articles sell at the same price, one at a gain and one at a loss, find each cost price separately and compare the totals; equal percentages always give a loss of x²/100 per cent.

All Profit and Loss notes

Why this matters

This is where AFCAT asks most: seven of the chapter's thirteen past-year questions are on this page, from 2016, 2018 and 2019. Five of them are two articles sold at one price with an equal gain and loss, and the trap is answering 'no gain, no loss'.

Concept 1 of 3: Equal gain and loss on equal selling prices

The article sold at a loss cost more than the one sold at a gain, so the loss is taken on the bigger cost. The gain and the loss never cancel: with the same percentage xx on both, the seller always loses.

Definition

  • Same selling price, a gain of x%x\% on one and a loss of x%x\% on the other: overall loss % =x2100= \dfrac{x^2}{100}.
  • The loss in rupees: the costs are SP1+x/100\dfrac{SP}{1 + x/100} and SP1−x/100\dfrac{SP}{1 - x/100}; subtract the total takings from their sum.
  • The rule needs equal selling prices. Two articles with equal cost prices, one at x%x\% gain and one at x%x\% loss, give no gain and no loss.

Equal gain and loss

Loss%=x2100\text{Loss}\% = \dfrac{x^2}{100}

Worked example

Two phones are sold for ₹6,000 each, one at a gain of 25%25\% and the other at a loss of 25%25\%. Find the overall loss per cent and the loss in rupees.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (I) 2019 — Numerical Ability · 02-205Easy

Example 1 · Profit and Loss · Combined Transactions

Two horses are sold at ₹1200 each, the first one at a profit of 20% and the second one at a 20% loss. The overall profit/loss in percentage is

Not 'no gain, no loss'

Equal percentages on equal selling prices always lose x2100%\tfrac{x^2}{100}\%. The option 'no profit, no loss' is the one set for students who let the two cancel.

Equal costs are a different question

If two articles cost the same and one gains x%x\% while the other loses x%x\%, the result is no gain, no loss. Check which price the question says is equal.
Drill 18 more on equal gain and loss on equal selling prices

Concept 2 of 3: One selling price, different percentages

With different percentages there is no shortcut rule. Divide the common selling price by each multiplier to get the two costs, add them, and compare with twice the selling price.

Definition

  • CP1=SP÷(1+a100)CP_1 = SP \div \left(1 + \tfrac{a}{100}\right) and CP2=SP÷(1−b100)CP_2 = SP \div \left(1 - \tfrac{b}{100}\right).
  • Result =2×SP−(CP1+CP2)= 2 \times SP - (CP_1 + CP_2); a percentage is on the total cost.
  • 16.67%16.67\% means 16\tfrac 16 and 33.33%33.33\% means 13\tfrac 13: divide by 76\tfrac 76, 56\tfrac 56, 43\tfrac 43 or 23\tfrac 23 as fractions.
  • Some pairs balance exactly, such as a 25%25\% gain with a 1623%16\tfrac 23\% loss, or a 50%50\% gain with a 25%25\% loss.

Total cost of two articles at one price

CP1+CP2=SP1+a100+SP1−b100CP_1 + CP_2 = \dfrac{SP}{1 + \frac{a}{100}} + \dfrac{SP}{1 - \frac{b}{100}}

Worked example

Two machines are sold at ₹5,600 each, one at a gain of 40%40\% and the other at a loss of 20%20\%. Find the overall gain or loss.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Profit and Loss · Combined Transactions

A shopkeeper sells two cycles at Rs. 27002700 each. He makes 35%35\% profit on one and 10%10\% loss on the other. Find his overall gain or loss percentage.

Do not average the percentages

A 50%50\% gain and a 25%25\% loss on equal selling prices is not a 12.5%12.5\% gain. The costs are different, and here they balance exactly.

The percentage is on the total cost

A gain of ₹200 on costs of 4000 and 7000 is 20011000\tfrac{200}{11000}, about 1.8%1.8\%, not 20011200\tfrac{200}{11200}.
Drill 35 more on one selling price, different percentages

Concept 3 of 3: Overheads, repairs and broken stock

Anything the seller spends before selling, on transport, repairs or overheads, belongs to the cost price. Stock he cannot sell still counts in the cost but brings nothing in.

Definition

  • Cost price = purchase price + overheads (repairs, transport, labour).
  • Profit % =total takings−total costtotal cost×100= \dfrac{\text{total takings} - \text{total cost}}{\text{total cost}} \times 100.
  • Broken or lost items: the full purchase is the cost, and only the unbroken ones are sold.
  • Items sold in groups at different prices: add up the takings group by group.

Cost with overheads

CP=purchase price+overheadsCP = \text{purchase price} + \text{overheads}

Worked example

A man buys a used bike for ₹18,500, spends ₹1,500 on repairs and sells it for ₹23,000. Find his profit per cent.
Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (II) 2019 — Numerical Ability · 02-201Moderate

Example 3 · Profit and Loss · Combined Transactions

Ramesh bought 10 cycles for ₹500 each. He spent ₹2000 on the repair of all the cycles. He sold five of them for ₹750 each and the remaining for ₹550 each. Then, the total gain or loss % is

Overheads are part of the cost

The repair money goes into the cost price, so the percentage is taken on purchase plus repairs. Leaving it out overstates the profit.

Broken items still cost money

If 12 of 60 cups break, the cost is still that of all 60; only 48 bring money in.
Drill 3 more on overheads, repairs and broken stock

Summary — formulas & gotchas at a glance

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Formulas (3)

Watch out for (6)

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