Finding the percentage of an amount

Introduction

In this lesson, you will learn how to do calculations using percentages.

These videos illustrate the lesson material below. Watching the videos is optional.

Finding the Percentage: Real Life Example

Percentages are used a lot in life. For example, you see them in tithing, taxes, and store discounts. In order to do calculations with percentages, you must first convert them into decimals.

Example 1
Imagine going to a store where prices are fixed, meaning they cannot be negotiated. In the store, there is a shirt that costs 20 of whatever currency you use. Even though the price is fixed, today there is a 25% off sale on this shirt.

You will answer two questions in this example. These two questions go hand in hand, which is an expression that means they work together.

How much is 25% of 20?
What is the new price of the shirt?

The answer to the first question will tell you how much of a discount you are getting through the 25% off sale. Remember that in math the word ‘of’ means multiplication or times.

When working with percentages, you can’t use the number itself as a percent, meaning you can’t just use 25 to represent 25%. ‘Percent’ means out of 100, so you need to use the decimal form that shows this number as 25 out of 100.

\begin{align*} 25\% = \frac{25}{100} = 0.25 \end{align*}

0.25 is the same as 25%. To answer the first question, multiply 0.25 and 20.

\begin{align*} 25\% \times 20 = 0.25 \times 20 \end{align*}

Use Excel as a calculator to solve $0.25\times\ 20$:

In cell A1, type the equal sign (=).
Type the first number (0.25).
Type the star key (*).
Type the second number (20).
Press Enter to calculate the equation (5).

The image shows the formula =0.25*20 in cell A1 of Excel.

Figure 1

How much is 25% of 20? The answer is 5.
\begin{align*} 0.25 \times 20 = 5 \end{align*}

If the discount is 5, what is the new price of the shirt?

There are two ways to solve this problem.

Example 1: Method A

The first method is to find the discount, like you did in the previous problem, and then subtract the discount amount from the original amount: \begin{align*} 20 - 5 = 15 \end{align*}

The sale price of the shirt is 15.

Example 1: Method B

The second option is to find the percentage that remains rather than the percentage that the discount removes.

You know that the discount is 25% off of the total cost of the shirt. The total cost can be represented by 100%. To calculate the remaining cost of the shirt, subtract the discount percentage from the total percentage, which tells you that the new price of the shirt is 75% of the original price.

\begin{align*}100\% - 25\% = 75\%\end{align*}

Knowing this, you can solve 75% of 20. Set up this equation the same way as before:

\begin{align*} 75\% \times 20 = \frac{75}{100} \times 20 = 0.75 \times 20 \end{align*}

Use Excel as a calculator to solve $0.75\times 20$:

In cell A1, type the equal sign (=).
Type the first number (0.75).
Type the star key (*).
Type the second number (20).
Press Enter to calculate the equation (15).

The image shows the formula =0.75*20 in cell A1 of Excel.

Figure 2

Using this method for solving the problem gives you the same answer as before: 15.

Adding a Percent Amount

Many places in the world have a sales tax. This is sometimes referred to as a VAT, or value added tax. This tax is an extra percentage that is paid on top of the amount of whatever is being purchased.

Example 2

Consider a shirt that costs 20 units of local currency, and imagine that you live in a place with 7% sales tax or VAT. How much money would you need to purchase the shirt?

First, you need to find out what 7% of 20 is so you can find out how much you need to pay in tax:

\begin{align*} 7\% \times 20 = \frac{7}{100} \times 20 = 0.07 \times 20 \end{align*}

Use Excel as a calculator to solve $0.07\times20$ which will give you the amount of tax you owe:

In cell A1, type the equal sign (=).
Type the first number (0.07).
Type the star key (*).
Type the second number (20).
Press Enter to calculate the equation (1.40).

The image shows the formula =0.07*20 in cell A1 of Excel.

Figure 3

If all you wanted to know was how much tax you owed on this amount, 1.40 would be the answer. However, if you want to know the final sales price of the shirt, you will need to add the original amount of the shirt (20) with the sales tax (1.40):

\begin{align*}\text{Original Amount + Sales Tax} &= \text{Final Sales Price}\\20 + 1.40 & = 21.40\\\end{align*}

Finding a Percent versus Finding an Amount

You have learned that in order to solve for a percentage, you take an amount and divide it by the total. When you want to solve for the amount of something, you use the percent as a decimal, and multiply it by the original amount or the total. Here are some practice problems using these principles:

Example 3
What percent of an hour is 45 minutes? In this case, the amount is 45 minutes. The total is 1 hour, or 60 minutes.

\begin{align*}Percentage&=\frac{Amount}{Total}\\\\Percentage &= \frac{45}{60}\\\\Percentage&= 0.75\\\\Percentage &= 75\%\\\\\end{align*}

Using a calculator, you will find that $45\div60=0.75$. In percent terms, it equals 75%, which is the answer for this equation.

Example 4
How long in minutes is 75% of an hour? In this case, you need to multiply the original amount of 60 minutes by the percentage you’ve been given.

\begin{align*}Amount &=\% \times Total\\\\Amount &=75\% \times 60\\\\Amount &=0.75 \times 60\\\\Amount &= 45\\\\\end{align*}

$0.75\times60=45$, so 45 minutes is 75% of an hour.

Things to Remember

To solve for a percent, begin with an amount and divide by the total:
- $Percent = \large\frac{Amount}{Total}$
To solve for an amount, take the percentage as a decimal and multiply by the total:
- $Amount = Percent \times Total$
To find the total cost, do the following:
- First, find the percentage of the original amount (% as a decimal × original amount)
- Second, when adding the percentage (example: taxes), add the percentage amount to the original amount
  - Or, when subtracting the percentage (example: sale), subtract the percentage amount from the original amount.
When working with percentages, you can’t use the number itself as a percent. You must convert the percent into a decimal.
Pay attention to what the question is asking! Is it asking for how much something will cost after the percentage is added or subtracted, or is it asking how much money the percentage is? Total amount or percentage amount?

Practice Problems

How many hours equal 15% of a day? (
Video Solution

x

Solution: 3.6 hours
Details:

(Applying Percentages in Practical Situations #2 (01:24 mins) | Transcript)

| Transcript)
What is 35% of 200? (
Solution

x

Solution: 70
Details:
(This question applies the rule for finding an amount: % as a decimal × total)

Turn 35% into a decimal by removing the % symbol and moving the decimal point two places to the left.

$35\% = {\color{red}0.35}$

Multiply 0.35 by 200.

${\color{red}0.35} \times 200$

The answer is 70.00 or just 70, by dropping the zeros after the decimal point.

$70.00 = {\color{red}70}$

)
If a computer costs $850 and it is on sale for 15% off, how much has the price been reduced? (
Solution

x

Solution: $127.50
Details:
15% of $850 is: $0.15 \times $850 = $127.50$.
This is the amount that the price has been reduced.

)
If a dress costs $150 and it is on sale for 20% off, what is the sale price of the dress? (
Video Solution

x

Solution: $120
Details:

(Applying Percentages in Practical Situations #5 (02:01 mins) | Transcript)

| Transcript)
If a bicycle costs $500 and the sales tax is 6%, what is the total cost of the bicycle with tax included? (
Solution

x

Solution: $530
Details:
The price of the bicycle is $500 and the sales tax is 6%.

The price of the bicycle represents 100% of the price.

Combine 100% of the bicycle price with 6%.

$100\% + 6\% = 106\%$. This new percent of 106% represents the new total cost of the bicycle.

Turn 106% into a decimal by removing the % symbol and moving the decimal to the left 2 places.

$106\% = {\color{red}1.06}$

Use this decimal to multiply by 500 to get the total cost of the bicycle.

$1.06 \times {\color{red}500}$

$530 is the total cost of the bicycle with the tax.

$1.06 \times 500 = {\color{red}530.00}$

)

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