Percentages Lesson

What is a Percent?

We previously learned that decimals and fractions are used to describe parts of a whole. A percent is another commonly used method to describe part of a whole. A percent is a fraction whose denominator is 100. The literal meaning of percent is "parts per hundred" or parts (number in the numerator) per (division or fraction bar) 100 (denominator). When we encounter a percent, we see the "%" symbol placed behind the number. As an example, suppose we had 100 boxes with 30 shaded yellow and 70 shaded blue: In our above example, we can see that 30 boxes are shaded yellow out of a total of 100. This can be written with a fraction as: $$\frac{30}{100}$$ To write this as a percentage we take the top number of the fraction (30) followed by the "%" symbol: $$30\%$$ Similarly, we know there are 70 blue boxes. We can show this with a fraction as: $$\frac{70}{100}$$ To write this as a percentage we take the top number of the fraction (70) followed by the "%" symbol: $$70\%$$ We can say that 30 percent (30%) of the boxes are yellow and 70 percent (70%) of the boxes are blue.

Converting a Fraction with a Denominator of 100 to a Percent

When converting to a percentage, the easy scenario occurs when we have a fraction with a denominator of 100. In this case, we can write the number in the numerator, followed by the "%" symbol. Let's try a few examples.
Example 1: Convert each into a percentage. $$\frac{9}{100}$$ We write the numerator (9) followed by the percentage symbol "%":
$$\frac{9}{100}=9\%$$ Example 2: Convert each into a percentage. $$\frac{84}{100}$$ We write the numerator (84) followed by the percentage symbol "%":
$$\frac{84}{100}=84\%$$ We don't have to start with a denominator of 100 to use a percentage. We previously learned that proportions were equivalent ratios/fractions. In this case, we are looking to convert our fraction into an equivalent fraction where 100 is the denominator. Let's take a look at a few examples.
Example 3: Convert each into a percentage. $$\frac{17}{25}$$ The denominator is 25, what can we multiply by 25 that would result in a product of 100?
? x 25 = 100
Use a related division statement:
100 ÷ 25 = 4
We will create an equivalent fraction where 100 is the denominator by multiplying the numerator and denominator by 4: $$\frac{17}{25}\cdot \frac{4}{4}=\frac{68}{100}$$ We can then convert this into a percentage: $$\frac{68}{100}=68\%$$ We can place this in terms of our original fraction: $$\frac{17}{25}=68\%$$ Example 4: Convert each into a percentage. $$\frac{15}{50}$$ The denominator is 50, what can we multiply by 50 that would result in a product of 100?
? x 50 = 100
Use a related division statement:
100 ÷ 50 = 2
We will create an equivalent fraction where 100 is the denominator by multiplying the numerator and denominator by 2: $$\frac{15}{50}\cdot \frac{2}{2}=\frac{30}{100}$$ We can then convert this into a percentage: $$\frac{30}{100}=30\%$$ We can place this in terms of our original fraction: $$\frac{15}{50}=30\%$$

Converting Decimals into Percents

In most cases, it will not be practical to convert a fraction into a percentage using an equivalent fraction where the denominator is 100. Let's suppose we saw: $$\frac{6}{13}$$ We would need to multiply 6 and 13 by the fraction 100/13 in order to get a denominator of 100. This procedure would be extremely tedious and fortunately, there is an easier way. Let's begin by talking about how to convert a decimal into a percent:

• Move the decimal point two places to the right
• Add the percentage "%" symbol to the end of the number

We can reverse this process to go from a percent back to a decimal:

• Delete the percentage symbol "%"
• Move the decimal point two places to the left

Let's look at a few examples.
Example 5: Convert each decimal into a percent.
0.005
We move our decimal point two places to the right:
0.005 » 0.5
Place a percentage symbol "%" at the end of the number:
0.5 » 0.5%
Our answer:
0.005 = 0.5%
Example 6: Convert each decimal into a percent.
0.98
We move our decimal point two places to the right:
0.98 » 98.0
Place a percentage symbol "%" at the end of the number:
98 » 98%
Our answer:
0.98 = 98%
Example 7: Convert each percent into a decimal.
34%
We delete our percentage symbol:
34% » 34
We move our decimal point two places to the left:
34.0 » 0.34
Our answer:
34% = 0.34
Now, let's return to the situation where we have a fraction that does not easily transform into an equivalent fraction where 100 is the denominator. In this case, we will divide the numerator by the denominator and obtain a decimal form for the number. We can then transform our decimal into a percentage. Suppose we ran into 9/15 and wanted this as a percentage. We could divide 9 by 15 and get our decimal form:
9 ÷ 15 = 0.6
We can then transform this decimal into a percentage:
0.6 » 60%
When we compare this to our other method, we can see how tedious the process would be:
$$\require{cancel}\frac{9 \cdot \frac{100}{15}}{15 \cdot \frac{100}{15}}=\frac{\cancel{9}3 \cdot \frac{\cancel{100}20}{\cancel{15}}}{\cancel{15}\cdot \frac{100}{\cancel{15}}}=\frac{60}{100}$$ $$\frac{60}{100}=60\%$$ Let's try another example.
Example 8: Convert each fraction into a percentage. $$\frac{5}{8}$$ Divide 5 by 8:
5 ÷ 8 = 0.625
Convert to a percentage:
0.625 » 62.5%
Our answer: $$\frac{5}{8}=62.5\%$$

Multiplying with Percents

In some cases, we may need to find a certain percentage of something. For example, we may have 500 containers and want to ship 10% of the containers. We can find the number of containers to ship by multiplying the number 500 by 10%. In most cases, the easiest way to do this is by converting to a decimal and then performing the multiplication:
500 x 0.10 = 50
10% of 500 is 50
We would ship 50 containers.
Let's take a look at a few examples.
Example 9: Find 20% of 300 and 1425.
300
Multiply 300 by .2:
300 x 0.20 = 60
20% of 300 is 60
1425
Multiply 1425 by .2:
1425 x 0.2 = 285
20% of 1425 is 285
Example 10: Solve the following word problem
A government department spends $11,000 each year on supplies. Due to budget cuts, they are asked to decrease their spending by 22%. Given this decrease, what is the new budget for supplies?
First, let's find 22% of 11,000:
0.22 x 11,000 = 2420
This means the department will need to decrease its spending by $2420. We can subtract the initial spending ($11,000) minus the decrease ($2420):
11,000 - 2420 = 8580
The department's new budget for supplies will be $8580.

Percents More than 100

In some cases, we may see percents that are larger than 100. We may see statements such as: sales are up 120% or XYZ company's stock is up 340%. Having a percentage larger than 100 depends on the context of what's being discussed. Suppose we all sit for a math exam with a total of 100 questions. In this particular case, we cannot say we answered 130% of the questions since at most we can answer 100 questions or 100% of the questions. As another example, suppose we heard that a particular toothpaste was favored by 190% of the people surveyed. This would be another example where a percentage over 100 does not make logical sense. If every person surveyed favored the toothpaste, the result would be 100%. We generally will see percentages over 100 when we talk about increases. We may see a stock index start with a value of 6000. If the value doubles, we could say it increased by 100%. This means it went from 6000 to 12,000. Now, what if our stock index increased from 6000 to 18,000? In this case, we could say we had an increase of 200%.

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