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CDS Mathematics · Formula sheet

Percentage, Profit and Loss formulas

7 formulas and 10 common traps for CDS Mathematics Percentage, Profit and Loss, grouped by subtopic.

Full notes with worked examples

Percentages of a Quantity

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Percentage of a base

Percentage

AB×100\dfrac{A}{B} \times 100

More than and less than

Reversing a percentage

r100+r×100\dfrac{r}{100 + r} \times 100

Common traps

The square root of a percentage

64%=0.6464\% = 0.64, and 0.64=0.8=80%\sqrt{0.64} = 0.8 = 80\%. Taking the root of 6464 alone gives 8%8\%, which is wrong by a factor of ten.

20% more does not reverse to 20% less

If XX is 20%20\% more than YY, YY is 1623%16\tfrac23\% less than XX, not 20%20\%. The base changed from YY to XX.

Successive Percentage Change

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Multiplying the factors

Net change

a+b+ab100a + b + \dfrac{ab}{100}

Price against consumption

Keeping the spend fixed

r100+r×100\dfrac{r}{100 + r} \times 100

Common traps

Percentages do not add

Rent up 20%20\% and rooms up 20%20\% raise the collection by 44%44\%, not 40%40\%. The second change acts on the already-raised amount.

Increased BY versus increased TO

A side 'increased by 200%200\%' is three times as long. The area becomes nine times as big, which is an increase of 800%800\%.

Not the same percentage back

A 25%25\% price rise needs only a 20%20\% cut in consumption, because the cut is measured on the old, larger consumption.

Profit and Loss

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Cost price, selling price and the two-scenario trick

Selling price

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

Profit hidden in quantities

False weight

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

Common traps

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\%.

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.

Discounts and Marked Price

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Marked price, discount and profit

Successive discounts

1−(1−d1100)(1−d2100)1 - \left(1 - \tfrac{d_1}{100}\right)\left(1 - \tfrac{d_2}{100}\right)

Common traps

Discounts do not add

Discounts of 20%20\%, 10%10\% and 5%5\% come to 31.6%31.6\%, not 35%35\%. Each is taken off the price left by the one before.

Two bases in one question

The discount is a percentage of the MARKED price; the profit is a percentage of the COST. Keep them apart: CP 100100, MP 120120, 10%10\% off gives SP 108108, an 8%8\% profit.

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