# Percentages

The terms percent means for every hundred.** Percentage** is a way of expressing a number as a fraction of 100. It is denoted by using the percent sign, %. For example, when a person made a profit of 20 percent, it means that the person gained Rs. 20 for every hundred rupees invested in the business. Scoring 60 percent marks means out of every 100 marks the candidate scored 60 marks.

**1. Convert a fraction into a percent**

To convert any fraction p/q to rate percent, multiply it by 100 and put % sign.

**Example:** What percentage is equivalent to 3/5?

3/5 × 100 = 60%

**2. Convert a percent into a fraction**

To convert a percent into a fraction, drop the percent sign and divide the number by 100.

**3. Find a percentage of a given number**

x% of given number (N) = x/100 × N

**Example:** 75% of 400 = ?

75% of 400 = 75/100 × 400

= 300

**Example:** Find a number whose 4% is 72

Let the required number be n

Then, 4% of n = 72

⇒ 4/100 × n = 72

⇒ n = 100/4 × 72 = 1800

**Example:** What percent of 25 kg is 3.5 kg?

Let p% of 25 kg be 3.5 kg.

Then, p% of 25 kg = 3.5 kg

⇒ p/100 × 25 = 3.5

⇒ p = (3.5 × 100)/25

= 14

Hence, 3.5 kg is 14% of 25 kg

### Examples

**Example 1: **60% of all university students are male. There are 2400 male students. How many students are in the university?

Let n be the total number of students in the university.

2400 = 60% × n

Therefore, n = (2400 × 100)/60

= 4000

**Example 2: **There are 300 cats in the village, and 75 of them are black. What is the percentage of black cats in that village?

75 = p% × 300 = (p/100) × 300

p = (75/300) × 100 = 25

Therefore, p% = 25%

**Example 3:** The number of students at the university increased to 4620, compared to last year's 4125, an absolute increase of 495 students. What is the percent increase?

495 = p% × 4125 = (p/100) × 4125

p = (495/4125) × 100 = 12

Therefore p% = 12%

**Example:** In an examination, 42% students failed in Mathematics and 52% failed in Science. If 17% failed in both the subjects, find the percentage of those who passed in both the subjects?

% of students who failed in Mathematics only = (42 - 17)% = 25%

% of students who failed in Science only = (52 - 17)% = 35%

% of students who failed in both the subjects = 17%

% of students who passed in both the subjects = 100% - (25% + 35% + 17%)

= 23%

### Constant Product Rule

You can apply this rule when you have two parameters whose product is constant. In other words, when two parameters are inversely proportional to each other. For examples,

- Time × Speed = Distance
- Price × Consumption = Expenditure
- Length × Breadth = Area

The rule states that 1/x increase in one of the parameters will result in a 1/(x+1) decrease in the other parameter.

Let's understand with the help of example. Suppose speed increases by 25% (or 1/4) and distance is constant, time required will decrease by 1/(4+1) or 1/5 or 20%.

### Percentage and Fraction Equivalents

If someone asks you to represent 50% in fractions then what will you do? Certainly, you will come with the answer 1/2. What this value actually represents? This is nothing but the fractional equivalent of the given percentage. From CAT point of view it is very important to know the fractional equivalent of the percentages.

### Multiplying Factor

While dealing with percentage increase or decrease, 10% increase is 1.1 and that of 15% decrease is 0.85.

- X increased by 10% would become X + 0.1X = 1.1X
- X increased by 1% would become X + 0.01X = 1.01X
- X decreased by 10% would become X - 0.1X = 0.9X
- X decreased by 1% would become X - 0.01X = 0.99X
- X increased by 200% would become X + 2X = 3X
- X decreased by 300% would become X - 3X = -2X

### Successive Percentage Change

If a number is changed (increased or decreased) successively by x% and y%, then net % change is given by **(x + y + xy/100)%**.

**Example:** If salary of a person is first increased by 15% and thereafter decreased by 12%, what is the net change in his salary?

Here x = 15 and y = -12

The net % change in the salary

= 15 - 12 - (15×12/100)

= 1.2%

Since the sign is +ve, the salary of the person increases by 1.2%.

**Example:** The population of a town is decreased by 25% and 40% in two successive years. What percent population is decreased after two years?

Here x = -25 and y = -40

The net % change in population

= -25 - 40 + (25×40/100)

= -55%

Since the sign is -ve, there is decrease in population after two years by 55%.