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CDS Mathematics · Percentage, Profit and Loss

Profit and Loss

Profit and loss are percentages of the cost price: SP = CP × (1 + p/100) for a profit of p%.

All Percentage, Profit and Loss notesCDS Mathematics strategy

Why this matters

Seventeen PYQs, one of them HARD. Twelve are the cost-price equation in some form — two scenarios, two articles, a changed cost. The other five hide the profit in a QUANTITY: a false weight, goods lost in transit, the cost of 100 equal to the price of 80.

Concept 1 of 2: Cost price, selling price and the two-scenario trick

Everything is measured against the cost price. Set CP =100= 100 (or CC) and each sentence becomes a multiple of it; two scenarios then differ by a known percentage of the same CP.

Definition

  • Profit p%p\%: SP=CP(1+p100)\text{SP} = \text{CP}\left(1 + \dfrac{p}{100}\right). Loss ℓ%\ell\%: SP=CP(1−ℓ100)\text{SP} = \text{CP}\left(1 - \dfrac{\ell}{100}\right).
  • Gain g%g\% instead of loss ℓ%\ell\% earns (g+ℓ)%(g + \ell)\% of CP more.
  • Two articles: total profit == sum of the separate profits, measured on the TOTAL cost.
  • Two items sold at the SAME price, one at r%r\% profit and one at r%r\% loss: always a net loss of r2100%\dfrac{r^2}{100}\%.
  • A changed cost with the same SP: recompute profit on the NEW cost.

Selling price

SP=CP(1+p100)\text{SP} = \text{CP}\left(1 + \dfrac{p}{100}\right)

Worked example

Selling at a 10%10\% gain instead of a 5%5\% loss earns Rs. 4545 more. Find the cost price.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q35Moderate

Example 1 · Percentage, Profit and Loss · Profit and Loss

If an article is sold at a gain of 6% instead of a loss of 6%, the seller gets Rs. 6 more. What is the cost price of the article ?

Profit is a percentage of COST

A profit of Rs. 2525 on a selling price of Rs. 125125 is 25%25\%, not 20%20\%. Dividing by the selling price is the most common slip.

The new cost is the new base

When the cost rises and the selling price stays, the new profit percentage is measured on the NEW cost: from 132132 on 120120, it is 10%10\%, not 12%12\%.
Drill 11 more on cost price, selling price and the two-scenario trick

Concept 2 of 2: Profit hidden in quantities

When a trader sells at cost but gives short weight, or when some goods are lost, the money per unit is unchanged but the number of units changes. Compare what is paid for with what is handed over.

Definition

  • CP of aa items == SP of bb items: profit =a−bb×100%= \dfrac{a - b}{b} \times 100\%.
  • A false weight of ww g passed as 10001000 g, sold at cost: profit =1000−ww×100%= \dfrac{1000 - w}{w} \times 100\%.
  • Some goods lost: the whole cost is spread over the goods that remain.
  • The profit is on what the trader actually GAVE, so ww is the denominator.

False weight

1000−ww×100\dfrac{1000 - w}{w} \times 100

Worked example

A grocer sells at cost price but uses a 960960 g weight for 11 kg. Find the profit percentage.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q36Easy

Example 2 · Percentage, Profit and Loss · Profit and Loss

The cost price of 100 mangoes is equal to the selling price of 80 mangoes. What is the profit percentage?

Divide by the smaller count

With CP of 100100 equal to SP of 8080, the profit is 2080=25%\dfrac{20}{80} = 25\%, not 20100=20%\dfrac{20}{100} = 20\%. The base is what the trader actually gave.
Drill 4 more on profit hidden in quantities

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (3)

Test yourself on Percentage, Profit and Loss

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