Percentage

Let us recall, from our earlier studies, that the ratio of a number A to another number B (mathematically expressed as A:B or A/B) is actually the division of A by B.

The percentage of a number A with respect to another number B is mathematically expressed as (A/B) and is defined as the ratio between the two numbers multiplied by 100. Therefore,

$(A/B) = A : B \times 100$

The percentage sign is %. For example: 50%.

One-quarter of something(or number) can be expressed as a fraction as 1/4. Therefore, the percentage of 1/4 is 1/4x100 = 25%.
We can extend the same concept to find the other common percentage.

One-half: 1/2 x 100 = 50%
Full: 1 x 100 = 100%
Double: 2 x 100 = 200%.

Example: Calculate the percentage of 79 with respect to 163.

Solution

$(\frac{79}{163}) = 79 \div 163 \times 100 = 48.47%$

Note that we have rounded the result to the nearest hundredth and 79 is almost half of 163 because 48.47% near 50%.

Percentage Increase/Decrease

When a number increases to another number, the amount of increase is given as:

Increase = New Number - Old Number

However, when a number is decreased to another number, the amount of decrease is given as:

Decrease = Old Number - New Number

The percentage increase or decrease of a number is always expressed to the base of the old number.
Therefore:

%Increase = 100 ⋅ (New Number - Old Number) ÷ Old Number

%Decrease = 100 ⋅ (Old Number - New Number) ÷ Old Number

For example, if you had 80 postal stamps and you started collecting more during this month until the total number of postalsma stamps reached 120. The percentage increase in the number of stamps that you currently have is:

$\frac{120 - 80}{80} \times 100 = 50%$

With you having the 120 stamps, you and your friend agreed to trade his Lego game for some of your stamps. Your friend took some of the stamps he liked and when you counted the remaining stamps you found that you are left with 100 stamps. The percentage reduction in the number of stamps can then be calculated as:

$\frac{120 - 100}{120} \times 100 = 16.67%$

Percentage Calculator

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How does the Percentages Help in Real Life

There are two ways in which percentages help us in our every day's problems;

1. We can compare between different quantities since all quantities are referred to the same base quantity which is 100. To explain this let us consider the following example.

Example: Tom started a new grocery store business. During his first month he bought groceries for \$650 and sold them for \$800, while in the second month he bought with \$800 and sold for \$1200. We need to know if Tom is making more profit or not.

Solution:

We cannot directly tell from these numbers if Tom's profit is increasing or not since his spending and gain are different amounts each month. In order to solve this problem we need to refer all the values to a fixed base value which is 100. Let us express the percentage of his profit with respect to his expenditure for the first month:

(800 - 650) ÷ 650 ⋅ 100 = 23.08%

This means that, if Tom spent $100, he would have made a profit of 23.08 in the first month. Now, let us apply the same for the second month: (1200 - 800) ÷ 800 ⋅ 100 = 50% So, for the second month, if Tom spent \$100 he would have made a profit of \$50(becase \$100⋅50% = \$100⋅50÷100=\$50). Now it is clear that Tom's profits are increasing.

2. We can quantify a portion of a bigger quantity knowing the percentage of this portion. To explain this, let us consider the following example:

Example: Cindy wants to buy 8m of water hose for her garden. She went to the store and found that there is a 30m hose reel. However, she noticed that it is written on the reel that 60% already sold. She needs to know if the balance is enough for her or not.

Solution:

The sign says that

$\frac{Sold\ length}{Total\ length} \times 100 = 60%$

$Sold\ length = \frac{60 \times 30}{100} = 18m$

So the balance is 30 - 18 = 12m which is good enough for Cindy.

Examples

1. Ryan likes to collect sports cards for his favorite player. He has 32 cards for baseball players, 25 cards for football player and 47 for basketball players. What is the percentage of the number of cards for each sport in his collection?

Solution:

The total number of cards = 32 + 25 + 47 = 104

Percentage of baseball cards = 32/104 x 100 = 30.8%

Percentage of football cards = 25/104 x 100 = 24%

Percentage of basketball cards = 47/104 x 100 = 45.2%

Notice that if you add all the percentages you will get 100% which represents the total number of cards.

2. You had a math quiz at class. The quiz had 5 questions; three of them have 3 marks each, and the other two have 4 marks each. You managed to correctly solve 2 questions with the 3 marks and one question with the 4 marks. What is the percentage of the marks that you got in this quiz?

Solution:

The total marks = 3x3 + 2x4 = 17 marks

Marks obtained = 2x3 + 4 = 10 marks

Percentage of marks obtained = 10/17 x 100 = 58.8%

3. You are used to buy a video game for $40. Lately, these games were subject to 20% price increase. What is the new price of the video game? Solution: The price increase is 40 x 20/100 = \$8

The new price is 40 + 8 = \\$48

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