Percentages is a very important topic with applications in Profit and Loss, Ratio and Proportions, Problems on Simple and Compound Interest.
The introduction and solved question and answers in this section will help you with the following aspects of percentages.
- What is percentage
- How to find percentages
- Percentage Increase and Decrease
- Scaling Factor
We recommend you to go through the introduction and then understand how to solve percentage problems using the question and answers. After completing this topic you can get a fairly good idea on the kind of problems on percentages and you must be able to define your approach to tackle percentage problems.
Introduction to Percentages
What is a Percent?
Percent implies per 100. It is a number or a ratio expressed as a fraction of 100. It is represented by the symbol %.
Example : 20% implies $\frac{20}{100}$.
What does the statement “20% of the students cleared the exam” mean?
=> It means out of every 100 students 20 students have cleared the exam.
To convert x% into fraction we divide x by 100.
To convert a fraction $\frac xy$ into percentage we multiply it with 100.
How to find Percentages?
When solving percentages, identifying the base y in $\frac xy is very important.
For example,
- What percent of B is A $\to \frac AB$ x 100
In this case, we want to express A as a percentage of B. Hence base is B.
- A is what percent more than B $\to \frac {A-B}{B}$ x 100
Here,The difference in A and B gives the value by which A is greater than B.The reference point is B. Hence B is the base. And to find the percentage by which A is greater than B, divide the difference by the base.
- Percentage by which A is less than B is given by $\frac{B-A}{B}$ x 100
- If A is $x$% more than B, then B is $\frac{x}{100+x}$ % less than A.
- If A is $x$% less than B, then B is $\frac{x}{100-x}$ % more than A.
Percentage Increase and Decrease
- If price increases by $x$% and expenditure remains the same, then the consumption decreases by $\frac{x}{100+x}$ x 100
- If price decreases by $x$% and expenditure remains the same, then the consumption increases by $\frac{x}{100-x}$ x 100
Scaling Factor.
- If A is increased by $x$%, then the resulting value is obtained by multiplying A by (1+$\frac{x}{100}$). (1+$\frac{x}{100}$) is called the scaling factor.
Example:
If A is increased by 20%. Then its value after increase is A(1+$\frac{20}{100}$) => A x 1.2
- If A is decreased by x%, then the resulting value is obtained by multiplying A by (1-$\frac{x}{100}$).
Example:
If A is decreased by 10%. Then its value after increase is A(1-$\frac{10}{100}$) => A x .9
Problems
Percentage - Exercise - Blog
20 questions
- What percent of numbers between 1 and 100 are prime numbers?
- 10%
- 15%
- 21%
- 25%
Show the answer
Answer: 25%
There are 25 prime numbers from 1 through 100.
Hence percentage number of prime number from 1 to 100 is
$\frac{25}{100}$ x 100 = 25%
- From a group of 100 students 40% do not play Cricket and from the remaining students 70% are batsmen and the rest bowlers. What percent of students from the group are bowlers?
- 18%
- 36%
- 42%
- 60%
Show the answer
Answer: 18%
40% of the students from the group of 100 students do not play cricket. Hence the no of students who do play cricket is 60% of 100 = 60.
Number of batsmen = $\frac{70}{100}$ x 60 = 42
Number of bowlers = 60 - 42= 18
Percent of bowlers from the group of 100 students = $\frac{18}{100}$ x 100 = 18%
- If A’s income in 2014 is 5 lakhs and in 2015 is 6 lakhs. What is the percentage rise in his income?
- 16.67%
- 20%
- 14.48%
- 25%
Show the answer
Answer: 20%
Percentage rise = $\frac{\text{Income in 2015 - Income in 2014}}{\text{Income in 2014}}$ x 100
= $\frac{6 - 5}{5}$ x 100
= 20%
- If a mixture contains milk and water in the ratio 3:2. What percentage of the mixture is Milk?
- 16.67
- 66.67%
- 10%
- 60%
Show the answer
Answer: 60%
If a mixture contains A and B in the ratio m : n, then percentage of A in the mixture is given by $\frac{m}{(m+n)}$ x 100.
Therefore, percentage of milk in the mixture = $\frac{3}{3+2}$ x 100 = 60%
- From a class of 50 students if 35 students cleared the exam. What % of students did not clear the exam?
- 50%
- 20%
- 30%
- 15%
Show the answer
Answer: 30%
Percentage of students who did not clear the exam = Number of students who did not clear the exam/ Number of students in the class x 100
Number of students who did not clear the exam = 50-35 = 15.
Total number of students in the class = 50.
Percentage of students who did not clear the exam =$\frac{15}{50}$ x 100 = 30%.
- In a test, if 16 questions which account for 30% of the total number of questions are on numbers. Find the number of questions in the test.
- 15
- 30
- 50
- 60
Show the answer
Answer: 50
Let the number of questions in the test be $x$.
Then $\frac{30}{100}$ x $x$ = 15.
$x$ = 50.
- If A = 50 and B = 30. By what percent is A greater than B?
- 66.67%
- 15%
- 30%
- 28.28%
Show the answer
Answer: 66.67%
Percentage by which A is greater than B is given by $\frac{A - B}{B}$ x 100.
= $\frac{50 - 30}{30}$ x 100 = 66.67%
- If A’s income increases by 10% by what percent should his consumption decrease such that the expenditure remains constant.
- 10%
- 11.11%
- 9.09%
- 10%
Show the answer
Answer: 9.09%
If A’s income increases by r%, then the percent decrease in consumption if expenditure remains constant is $\frac {r}{100 + r}$ x 100 %
= $\frac{10}{100+10}$ x 100 % = 9.09 %
A’s consumption decreases by 9.09% for the corresponding 10% rise in his income when the expenditure remains constant.
- If A’s salary is 20% less than B’s salary. By what percent is B’s salary more than A’s?
- 20%
- 25%
- 10%
- 15%
Show the answer
Answer: 25%
If A’s salary is r% less than B, then the B’s salary is $\frac{r}{100 - r}$ x 100% more than A.
= $\frac{20}{100 - 20}$ x 100%
= 25%
B’s salary is 25% more than A’s
- If the length of the Rectangle increases by 15% by what percent does the breadth decrease so as to maintain the same Area.
- 13.04%
- 12.5%
- 10.04%
- 10.5%
Show the answer
Answer: 13.04%
If A x B = k, then when A increases by r%, B decreases by $\frac{r}{100+ r}$ x 100% so that k remains unchanged.
Since length x breadth = Area and given that area does not change.
For a 15% increase in length, the breadth decreases by $\frac{15}{100 + 15}$ x 100% = 13.04% approx.
- An employee wrongly enters his salary as 6 lakhs instead of 5 lakhs. What is the percentage error?
- 16.67%
- 15%
- 20%
- 25%
Show the answer
Answer: 20%
Percentage error = $\frac{\text{Incorrect Salary - Actual salary}}{\text{Actual salary}}$ x 100 %
= $\frac{6 - 5}{5}$ x 100%
= 20%
- If price of apple increases by 25%, i can buy 5 apples less for Rs. 100. Find the original price of apple.
- Rs. 3
- Rs. 4
- Rs. 5
- Rs. 6
Show the answer
Answer: Rs. 4
Let the original price of apple be $x$.
Price of the apple after 25% increase = $x$ x (1+$\frac{25}{100})$
= 1.25$x$
Given
$\frac{100}{x} -\frac{100}{1.25x}$ = 5
$\frac{100}{x} (1 - \frac{4}{5})$ = 5
$x$ = 4.
- A weighs 30% more than B, who weighs twice as much as C. What percent of C does A weigh?
- 60%
- 160%
- 260%
- 360%
Show the answer
Answer: 260%
Let C’s weight be $x$.
B’s weight = 2$x$.
A’s weight = 2$x$ x (1+$\frac{30}{100}$) = 2.6$x$
A’s weight as a percent of C = $\frac{2.6x}{x}$ x 100%
= 260%
- If A is 40% of B, then A is what percent of A+B?
- 14.14%
- 16.67%
- 20.20%
- 28.57%
Show the answer
Answer: 28.57%
If B is $x$, then A is .4$x$.
A+B is 1.4$x$
A as a percent of A+B = $\frac{.4x}{1.4x}$ x 100 %
= 28.57%
- A gives 50% of the amount he has to B, 20% of the remaining to C and 30% of the rest to D. What percent of initial amount is left with A?
- 7%
- 14%
- 21%
- 28%
Show the answer
Answer: 28%
Let the amount with A be Rs. 100. Then,
Amount given to B is 50% of the amount from A= Rs. 50
Remaining amount with A = Rs. 50
Amount C gets is 20% of the amount left with A =$\frac{20}{100}$ x 50 = Rs. 10.
Amount left with A = Rs. 40.
Amount D gets is 30% of the amount left with A = $\frac{30}{100}$ x 40 = Rs.12
Amount left with A = 40 - 12 = Rs. 28
Percentage of the initial amount left with A = $\frac{28}{100}$ = 28%.
- If A’s salary in 2013 is 5 lakhs p.a and he gets an increment of 10% and 20% in 2014 and 2015 respectively, what is his salary after 2015?
- 2657.16
- 2560.16
- 2357.16
- 2617.16
Show the answer
Answer: 2657.16
A’s salary after successive increments = 2013 (1 + $\frac{10}{100}$)(1 + $\frac{20}{100}$)
= 2013 x 1.1 x 1.2
= 2657.16
- If the length of a rectangle increases by 25% and breadth decreases by 10% what is the percentage change in the area of the rectangle?
- 10%
- 12.5%
- 15%
- 17.5%
Show the answer
Answer: 12.5%
Let the area of the rectangle be $x$.
Area of the rectangle after 25% increase in length and 10% decrease in breadth = $x$ x (1 + $\frac{25}{100}$) x (1 - $\frac{10}{100}$)
= $x$ x (1.25)(.9) = 1.125$x$
Percentage change in area = $\frac{1.125x - x}{x}$ x 100 = 12.5%
- If the radius of a circle increases by 10%. What is the % change in the area of the Rectangle?
- 11.11%
- 9.09%
- 21%
- 28,.28%
Show the answer
Answer: 21%
Let the original area of the circle be $x$. Since area of a circle is directly proportional to square of radius.
Increase in area for a 10% increase in radius = $x$(1 + $\frac{10}{100})^2$ = 1.21$x$
Percentage change in area of the circle = $\frac{1.21x - x}{x}$ x 100 = 21%
- If the price of a commodity increases by 10% and its consumption decreases by 10%, what is the percentage change in expenditure?
- 1%
- 9%
- 10%
- 11%
Show the answer
Answer: 1%
Expenditure = Price x Consumption
If Expenditure is $x$, then new expenditure = $x$ (1 + $\frac{10}{100}$)(1 - $\frac{10}{100}$)
= .99$x$
Percentage change in expenditure = $\frac{x - .99x}{x}$ x 100
= .01$x$ x 100
= 1 %
- A spent 40% of the amount he had on shopping and 30% of the remaining on food. He is now left with Rs. 720. Find the initial amount with A.
- Rs. 7200
- Rs. 2700
- Rs. 2000
- Rs. 4000
Show the answer
Answer: Rs. 2000
Let the amount with A initially be $x$. Then,
The amount left with A after shopping = $x$ - $\frac{40}{100}x$ = .6$x$
Amount spent on food = 30/100 x .6$x$ = .18$x$
Remaining amount with A = .6$x$ - .18$x$ = .42$x$.
Given,
.42$x$ = 840
$x$ = 2000
Initial amount with A = Rs. 2000
