CDS Mathematics · Percentage, Profit and Loss

Percentages of a Quantity

A percentage is a fraction with denominator 100; every question turns on which quantity is the base, the 100%.

All Percentage, Profit and Loss notes CDS Mathematics strategy

Why this matters

Fourteen PYQs, half of them EASY. Pass marks, populations split by sex and literacy, and 'X is what per cent of Y' are all one move: write each percentage as a fraction of its own base. The 'more than / less than' pair is the one that catches people, because the two use different bases.

Concept 1 of 2: Percentage of a base

x\%

of something is

\dfrac{x}{100}

of it. Name the base first — the whole exam, the males, the sound mangoes — and the rest is one multiplication or division.

Definition

$x\%$ of $B$ $= \dfrac{x}{100}B$ ; $A$ as a percentage of $B$ $= \dfrac AB \times 100$ .
Pass mark problems: score $\pm$ the margin $=$ pass mark $= p\%$ of the maximum.
Percentages of percentages multiply: $30\%$ of $\tfrac13$ of the population.
A split population: males $\cdot\, y\%$ $+$ females $\cdot\, z\%$ $=$ total $\cdot\, x\%$ .
Convert units before comparing: $1$ quintal $= 100$ kg, $1$ tonne $= 1000$ kg.

Percentage

\dfrac{A}{B} \times 100

Worked example

The pass mark is

35\%

. A candidate scores

62

and fails by

8

marks. Find the maximum marks.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q27Easy

Example 1 · Percentage, Profit and Loss · Percentage

A student has to secure 40% of marks to pass an examination. He gets only 45 marks and fails by 5 marks. The maximum marks are

The square root of a percentage

64\% = 0.64

, and

\sqrt{0.64} = 0.8 = 80\%

. Taking the root of

64

alone gives

8\%

, which is wrong by a factor of ten.

Drill 10 more on percentage of a base

Concept 2 of 2: More than and less than

X

20\%

more than

Y

' measures from

Y

. 'Then

Y

is less than

X

by …' measures from

X

, a bigger base, so the percentage is smaller.

Definition

$X$ is $r\%$ more than $Y$ : $X = \left(1 + \dfrac{r}{100}\right)Y$ .
Then $Y$ is less than $X$ by $\dfrac{r}{100 + r} \times 100\%$ .
$X$ is $r\%$ less than $Y$ : $Y$ is more than $X$ by $\dfrac{r}{100 - r} \times 100\%$ .
Chains: write each as a multiple of one quantity, then divide.

Reversing a percentage

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

Worked example

A

's salary is

25\%

more than

B

's. By what per cent is

B

's less than

A

's?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (I) 2020 — Elementary Mathematics · Q46Easy

Example 2 · Percentage, Profit and Loss · Percentage

If the annual income of

X

is 20% more than that of

Y

, then the income of

Y

is less than that of

X

p

%. What is the value of

p

20% more does not reverse to 20% less

X

20\%

more than

Y

Y

16\tfrac23\%

less than

X

, not

20\%

. The base changed from

Y

X

Drill 2 more on more than and less than

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)

Percentage of a base
Percentage
$\dfrac{A}{B} \times 100$
More than and less than
Reversing a percentage
$\dfrac{r}{100 + r} \times 100$

Watch out for (2)

The square root of a percentage
→ Percentage of a base
20% more does not reverse to 20% less
→ More than and less than

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.

Take the Percentage, Profit and Loss chapter test Drill all 14 questions