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 we say a man made a profit of 20 percent we mean to say that he gained Rs. 20 for every hundred rupees he invested in the business.
Example: 25% = 25/100
What is 200% of 30?
200% × 30 = (200 / 100) × 30 = 60.
What is 13% of 98?
13% × 98 = (13 / 100) × 98 = 12.74.
60% of all university students are male. There are 2400 male students. How many students are in the university?
2400 = 60% × X, therefore X = (2400 / (60 / 100)) = 4000.
There are 300 cats in the village, and 75 of them are black. What is the percentage of black cats in that village?
75 = X% × 300 = (X / 100) × 300, so X = (75 / 300) × 100 = 25, and therefore X% = 25%.
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 = X% × 4125 = (X / 100) × 4125, so X = (495 / 4125) × 100 = 12, and therefore X% = 12%.
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.
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 + 0.1 X = 1.1X
- X increased by 1% would become X + 0.01 X = 1.01X
- X decreased by 10% would become X – 0.1X = 0.9X
- X decreased by 1% would become X – 0.01 X = 0.99X
- X increased by 200% would become X + 2X = 3X
- X decreased by 300% would become X – 3X = −2X
Successive Percentage Change
The population of a city increases by 10% in one year and again increases by 10% in the next year, then what is the net increase in the population in two years. The very common answer is 20% which is wrong. Why?
If Original population = P
After 1st year = 1.1P
After 2nd year = 1.21P
The population increases by 21% of the original value.
This successive change in the percentage can be calculated in the shortcut way as:
Consider a product of two quantities A = a x b.
If a & b change (increase or decrease) by a certain percentage say x & y respectively, then the overall percentage change in their product is given by the formula:
(x + y + xy/100)%