Percentages are a fundamental part of quantitative aptitude and play a crucial role in various competitive exams and placement tests. Although the basic concept is simple, percentage problems are often combined with other topics to make them more challenging.
The percentage formula can be understood in two equivalent forms, as shown below:
Table with Common Fractions
Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
1/1 | 100% | 1/20 | 5% |
1/2 | 50% | 1/25 | 4% |
1/3 | 33.33% | 1/50 | 2% |
1/4 | 25% | 1/100 | 1% |
1/5 | 20% | 3/4 | 75% |
1/6 | 16.67% | 2/3 | 66.67% |
1/7 | 14.29% | 3/5 | 60% |
1/8 | 12.5% | 5/6 | 83.33% |
1/9 | 11.11% | 7/8 | 87.5% |
1/10 | 10% | 9/8 | 112.5% |
Here are a few of the tricks to find percentages quickly:
Numerator Swapping Trick
For x% of y, we have x% of y = (x/100) × y.
This can also be written as x% of y = x × (y/100).
We can swap the positions of x and y (commutative and associative property of multiplication):
x% of y = y% of x.
Let's say you want to calculate 20% of 50. Using this trick, we can quickly solve it by finding the 50% of 20, that is 10.
This trick is useful for simplifying calculations and understanding that percentage calculations can be swapped between the two numbers.
10% Trick
Finding 10% of any number is easy: simply move the decimal point one place to the left. For example:
- 10% of 250 is 25.0
- 10% of 78 is 7.8
Once you have 10%, you can easily find multiples of 10%:
- 20% is just 10% doubled. So, 20% of 250 is 25.0 * 2 = 50
- 30% is three times 10%. So, 30% of 78 is 7.8 * 3 = 23.4
1% Trick
Similarly, finding 1% of any number involves moving the decimal point two places to the left. This is particularly useful for fine-tuned adjustments.
- 1% of 250 is 2.5
- 1% of 78 is 0.78
To find other percentages:
- 5% is five times 1%. So, 5% of 250 is 2.5 * 5 = 12.5
- 15% is 10% plus 5%. So, 15% of 78 is 7.8 + 3.9 = 11.7
Using Fractions to Simplify Percentages
Percentages are essentially fractions of 100, so some common fractions can make percentage calculations easier. For example:
- 50% is 1/2. So, 50% of 200 is 200 / 2 = 100
- 25% is 1/4. So, 25% of 60 is 60 / 4 = 15
- 75% is 3/4. So, 75% of 80 is 80 * 3 / 4 = 60
Doubling and Halving Strategy
This trick is helpful for percentages like 5%, 20%, and 40%:
- To find 20%, you can find 10% and then double it. For instance, 20% of 150 is 15 (10% of 150) doubled, which is 30.
- For 5%, find 10% and halve it. So, 5% of 300 is 30 (10% of 300) halved, which is 15.
- To find 40%, double 20%. For example, 40% of 50 is 10 (20% of 50) doubled, which is 20.
Quick Estimate Method
For a quick mental estimate, round the numbers to make the math easier:
- To estimate 19% of 47, round 19% to 20% and 47 to 50. Then calculate 20% of 50, which is 10. This gives you an approximate idea.
Increase and Decrease
To quickly determine a percentage increase or decrease:
- Find the percentage (e.g., 20%) of the original amount (e.g., 50) using the methods above (10).
- Add (for increase) or subtract (for decrease) this result from the original number: 50 + 10 = 60 for a 20% increase; 50 10 = 40 for a 20% decrease.
Reverse Percentage Calculation
If you know the final amount after a percentage increase or decrease and want to find the original amount:
- For a 20% increase, the final amount is 120% of the original. Divide the final amount by 1.2. For example, if the final amount is 72, the original is 72 / 1.2 = 60.
- For a 25% decrease, the final amount is 75% of the original. Divide the final amount by 0.75. For example, if the final amount is 45, the original is 45 / 0.75 = 60.
Splitting Percentages
Splitting Percentages is a mental math technique that involves decomposing a percentage into smaller, more easily calculated parts.
For example: To calculate 47% of 9834
Step 1: 47% can be split into 50%, 1% and 3%
Step 2: 50% of 9834 = 4917
Step 3: 1% of 9834 = 98.34 so 3% = 98.34 × 3 =295.02
Step 4: Now , 47% = 50% - 3% = 4917 - 295.02 = 4621.98
so, 4621.98 is the 47% of the 9834.
Quick Formulas
Type 1: What is the the y percent of number A.
To find the y percent of a number A, use the formula:
Required number = (y × A/100)
Example: What is 30% of 500.
Required number = (30 × 500/100) = 150
So, 30% of 500 is 150.
Type 2: Comparison between 2 variables
For questions like one variable what percent of other or vice versa?
A is what percent of B? | B is what percent of A? |
|---|---|
Required percentage = ( A/B × 100) % | Required percentage = ( B/A × 100) % |
Example: Let's say A = 30 and B = 50. Required percentage = (30/50 × 100) = 60% So, 30 is 60% of 50. | Example: Using the same values, A = 30 and B = 50. Required percentage = (50/30 × 100) = 166.67% So, 50 is 166.67% of 30. |
For questions like one variable is how much more or less than the other.
A is what percent less than B? | B is what percent more than A? |
|---|---|
Required percentage = ((B − A)/B × 100) % | Required percentage= ((B − A)/A × 100) % |
Example: Suppose A = 30 and B = 50. Required percentage = ((50 − 30)/50 × 100) = 40% So, 30 is 40% less than 50. | Example: Using the same values, A = 30 and B = 50. Required percentage = ((50 − 30)/30) × 100 = 66.67% So, 50 is 66.67% more than 30. |
Type 3: Product Constant Ratio (decreases/increase) / comparison
If the price of a commodity increases by R%, the required reduction in consumption to maintain the same expenditure is given by the formula:
\left( \frac{R}{100 + R} \times 100 \right) \%
Example: The price of rice increases by 40%. By what percentage should a family reduce its consumption of rice to keep the total expenditure on rice the same?
Solution:
Percentage increase in price (R) = 40%
Substitute R = 40:
= (40/(100 + 40) × 100)%
= (40/140 × 100)%
= (0.2857 × 100)%
= 28.57%
The family should reduce its rice consumption by approximately 28.57% to keep the expenditure the same.
If the price of a commodity decreases by R%, the necessary increase in consumption to keep the expenditure unchanged is given by the formula:
\left( \frac{R}{100 - R} \times 100 \right) \%
Example: The price of milk decreases by 20%. By what percentage should a family increase its consumption of milk to keep the total expenditure the same?
Solution:
Percentage decrease in price (R) = 20%
Substitute R = 20:
= (20/(100 - 20) × 100)%
= (20/80 × 100)%
= (0.25 × 100)%
= 25%
The family should increase its milk consumption by 25% to keep the expenditure unchanged.
Type 4: Successive Change
When a number is first adjusted (increased or decreased) by a% and then further adjusted by b%, the total percentage change in the number is given by the formula:
= (\pm a \pm b \pm \frac{ab}{100}) \%
Note: An increase is represented by a positive sign, while a decrease is represented by a negative sign. The net percentage change, whether an increase or decrease, depends on the sign (positive/negative). This overall effect is also referred to as the successive percentage change.
Example: The price of a gadget increases by 8%, while the demand for it decreases by 5%. What is the net percentage change in the total revenue from the gadget?
Solution:
Percentage increase in price (a) = 8%
Percentage decrease in demand (b) = -5%
Substitute a = 8 and b = −5:
( (8) + (-5) + \frac{(8)(-5)}{100}) \%
8-5+ \frac{-40}{100} \% = 8 − 5 − 0.4
= 2.6%
The net percentage change in total revenue is 2.6%, meaning the total revenue increases by 2.6%
