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Percentages

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The word 'percentage' is used regularly in the media to describe changes in the interest rate, income tax, VAT, exam results, success rate etc. They are very much part of our lives.

Per cent means out of 100 (divide by) in mathematical language. It is a number as a fraction of 100. Denoted using the percent sign "%", and abbreviated as 'pct'. In Latin, per centum means 'by the hundred'. With the help of percentages, it is easy to express how large or small one quantity is relative to the other. Ratio can be expressed as a percentage and percentages are also used to express numbers between zero and one. To change a fraction to a percentage, divide the numerator by the denominator and multiply by 100%.

Percentage Increase

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Percentage increase is the way to express a change in percentage of a number with respect to the reference number. Percentage increase is useful in a variety of situations. For instance, increase in expenses when tracking home budget, tax returns etc.
Mathematically, percentage increase is expressed as

Percentage Increase = $\frac{\text{Increase}}{\text{Original price}}$ * 100%
A change of x percent in a quantity results in a final amount that is 100 + x percent of the original amount. It is the increase as a percentage of the starting value.

Solved Example

Question: Price of a pen increases from $\$$20 to $\$$30. Express this increase as a percentage of the original price.
Solution:
Original price of pen = $\$$20
New price of pen = $\$$30
Increase  =  New price of pen - Original price of pen 
= $\$$30 - $\$$20
= $\$$10
Formula to find the percentage increase is given as
Percentage Increase = $\frac{\text{Increase}}{\text{Original price}}$ $\times$ 100%
= $\frac{10}{20}$ $\times$ 100%
= 50%
Therefore, there is 50% increase in the original price.

Percentage Decrease

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Percentage decrease is the difference between the two numbers you wish to compare. Decrease is found by finding the difference between the original number and the new number. The formula to find percentage difference is given as
Percentage decrease = $\frac{\text{Decrease}}{\text{Original Number}}$ * 100%

Solved Example

Question: Price of a calculator decreases from $\$$35 to $\$$30. Express the decrease as a percentage of the original number.
Solution:
First, we can find the difference

Decrease = Old price - New price
= $\$$35 - $\$$30
= 5

The formula to find the percentage decrease is given as
Percentage decrease = $\frac{\text{Decrease}}{\text{Original Number}}$ * 100%
 = $\frac{5}{35}$ * 100%
= 14.28%

The formula for percentage increase and decrease are given as follows:
Percentage Increase
Percentage Increase = $\frac{\text{Increase}}{\text{Original Price}}$ * 100%

Percentage Decrease
Percentage decrease = $\frac{\text{Decrease}}{\text{Original Number}}$ * 100%
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How to Calculate Percentage?

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Steps for calculating percentage change is given below:
  1. Subtract the new given value from the original value.
  2. Divide the result from the original value.
  3. Multiply the obtained result by 100% to find the percentage change (Increase/Decrease).

Percentages Problems

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Given below are some of the problems on percentages.

Solved Examples

Question 1: In April, Mark worked a total of 52 hours for 6 days. In May, he worked for 60 hours. By what percentage did Mark's working hours increase in May?
Solution:
Given: In April, Mark worked for 52 hours.
In May, Mark worked for 60 hours.
Increase = Mark's work hour in May - Mark's work hour in April
= 60 - 52
= 8

The formula to find the percentage increase is
Percentage Increase = $\frac{\text{Increase}}{\text{Original Price}}$ $\times$ 100%

= $\frac{8}{52}$ $\times$ 100%

= 15.38%
Therefore, there is 15.38% increase in the Mark's working hours in May.

Question 2: Price of a whole sale fruits decreases from $\$$105 to $\$$80. Express the decrease as a percentage of the original number.
Solution:
First, we can find the difference

Decrease = Old price - New price
= $\$$105 - $\$$80
= 25

The formula to find the percentage decrease is given as
Percentage decrease = $\frac{\text{Decrease}}{\text{Original Number}}$ $\times$ 100%

= $\frac{25}{105}$ $\times$ 100%

= 23.81%