CDS Mathematics · Percentage, Profit and Loss

Successive Percentage Change

Two percentage changes combine by multiplying their factors, never by adding the percentages.

All Percentage, Profit and Loss notes CDS Mathematics strategy

Why this matters

Twelve PYQs, none HARD. Each is a product: area from length and breadth, collection from rent and rooms, a salary raised then cut, price times consumption. Turn each change into a factor such as 1.2 or 0.9, multiply, and read the answer off the product.

Concept 1 of 2: Multiplying the factors

20\%

rise multiplies by

1.2

; a

10\%

fall multiplies by

0.9

. When a quantity is a product of two things, or changes twice in a row, the factors multiply.

Definition

Change of $r\%$ $\to$ factor $1 + \dfrac{r}{100}$ (negative $r$ for a fall).
Two changes $a\%$ and $b\%$ : net $a + b + \dfrac{ab}{100}$ per cent.
Up $x\%$ then down $x\%$ : always a net FALL of $\dfrac{x^2}{100}\%$ .
Both sides of a rectangle up $200\%$ : each tripled, area $\times 9$ , an increase of $800\%$ .
Yearly growth for $n$ years: factor $\left(1 + \dfrac{r}{100}\right)^n$ .

Net change

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

Worked example

The length of a rectangle rises by

25\%

and its breadth by

20\%

. By what per cent does the area rise?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q10Easy

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

In a hostel the rent per room is increased by 20%. If number of rooms in the hostel is also increased by 20% and the hostel is always full, then what is the percentage change in the total collection at the cash counter ?

Percentages do not add

Rent up

20\%

and rooms up

20\%

raise the collection by

44\%

, not

40\%

. The second change acts on the already-raised amount.

Increased BY versus increased TO

A side 'increased by

200\%

' is three times as long. The area becomes nine times as big, which is an increase of

800\%

Drill 6 more on multiplying the factors

Concept 2 of 2: Price against consumption

Expenditure is price times quantity. If the price factor is

1.25

, the quantity factor must be

\dfrac{1}{1.25} = 0.8

to keep the spend the same. The same holds for length and width at a fixed area.

Definition

Price up $r\%$ , same spend: consumption down $\dfrac{r}{100 + r} \times 100\%$ .
Price down $r\%$ , same spend: consumption up $\dfrac{r}{100 - r} \times 100\%$ .
Fixed area: one side up $r\%$ means the other down $\dfrac{r}{100 + r} \times 100\%$ .
'A cut of $r\%$ buys $q$ kg more for Rs. $M$ ': the saving $\dfrac{r}{100}M$ buys $q$ kg at the NEW price.

Keeping the spend fixed

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

Worked example

The price of rice rises by

20\%

. By what per cent must a family cut its consumption to spend the same?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q21Easy

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

If the price of wheat rises by 25%, then by how much percent must a man reduce his consumption in order to keep his budget the same as before?

Not the same percentage back

25\%

price rise needs only a

20\%

cut in consumption, because the cut is measured on the old, larger consumption.

Drill 4 more on price against consumption

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)

Multiplying the factors
Net change
$a + b + \dfrac{ab}{100}$
Price against consumption
Keeping the spend fixed
$\dfrac{r}{100 + r} \times 100$

Watch out for (3)

Percentages do not add
→ Multiplying the factors
Increased BY versus increased TO
→ Multiplying the factors
Not the same percentage back
→ Price against consumption

Test yourself on Percentage, Profit and Loss

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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