Percentages

Lesson 2.05

After completing this lesson, you will be able to say: • I

can

define a percentage as a ratio of a number to 100.

• I

can

calculate a percent of a number as a rate per 100.

• I

can

solve a variety of problems involving percentages.

What is a Percentage?

Percentage

Percentage

A part-to-whole ratio that compares a number to 100; percentages are written with the percent symbol (%).

• When the ratio becomes a percentage, the second term is always 100.

• When you record a percentage, write the first term in the ratio, followed by the percent sign (%). • The table below will show how you can take a ratio from a situation, create a ratio, and write the percent: In words 26 out of 100 26 for every 100 Written as a Ratio 26:100 26 to 100 26 100 Written as a percent 26%

In the world of 100 people, there are 8 seniors. How is this expressed as a ratio and a percentage?

In words Written as a Ratio Written as a percent

Check your work In the world of 100 people, there are 8 seniors. How is this expressed as a ratio and a percentage?

In words 8 out of 100 8 for every 100 Written as a Ratio 8:100 8 to 100 8 100 Written as a percent 8%

Finding Percentages

What if the part-to-whole ratio was not out of 100?

Fractions

also represent part-to-whole relationships, which means that

all fractions can be written as a percent

.

For this reason, it will be best to write all ratios in fraction form when working with percentages.

Finding Percentages

Sofia went to the bakery and bought 20 cookies. Out of the 20 cookies, she has 5 sugar cookies. What is the percentage of sugar cookies?

Determine the part to whole ratio

In this scenario, you are already given the part to whole ratio. It is 5 out of 5 the 20 cookies, or 20

Ratio Reasoning let's use a visual for the ratio reasoning

C C C C C C C C C C C C C C C C C C C C 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% Since you know that 5 are sugar cookies, the first 5 are colored

If you match up the 5 cookies in the top row with the percentages in the second 1 row, you can see that there are 2 boxes shaded.

2 If each box is 10%, then 2 1 2

10% + 10% + 5% = 25%

means: So if 5 out of 20 cookies are sugar, that means 25% of them are sugar cookies. In other words, if Sofia had 100 cookies, 25 of them would be sugar cookies.

• Using equivalent fractions

Step 1 2 3 Fraction 5 20 = 100 100 ÷ 20 = 5 5 𝑥 5 20 𝑥 5 = 25 100 Explanation To set up the equivalent ratios, the first fraction will be the ratio of sugar cookies to total number of cookies. Then set it equal to another ratio, with a denominator of 100. Since you are trying

to find the percent, the ratio needs to be out of 100.

Next determine which number times 20 would give you 100.

Multiply both terms by the value.

So every 5 out of 20 is equal to 25 out of 100. This means 25% of the cookies were sugar cookies.

Try It

A cookie jar has 20 cookies: 6 oatmeal cookies, 9 chocolate chip cookies, and 5 sugar cookies. Determine the percentage for oatmeal cookies. Explanation 2 3 Step 1 Fraction

Check your work

A cookie jar has 20 cookies: 6 oatmeal cookies, 9 chocolate chip cookies, and 5 sugar cookies. Determine the percentage for oatmeal cookies. Determine the percentage for chocolate chip cookies.

Step 1 2 3 Fraction 6 20 100 ÷ 20 = 5 6 𝑥 5 20 𝑥 5 = = 100 30 100 Explanation To set up the equivalent ratios, the first fraction will be the ratio of sugar cookies to total number of cookies. Then set it equal to another ratio, with a denominator of 100. Since you are trying

to find the percent, the ratio needs to be out of 100.

Next determine which number times 20 would give you 100.

Multiply both terms by the value.

The percent of oatmeal cookies is 30%.

Using Percentages

What if you know the percent but need to determine the part or the whole?

In the morning, the local grocery story had 130 tubs of ice cream. By the end of the day, they sold 55%. How many tubs of ice cream did they sell? What is 55% of 130?

Method 1: Since 55% of 130 tells you part of a whole, to find the answer you multiply the percent (35%) by the total amount (130). Change the percent to a decimal. To change the percent to a decimal, divide by 100.

55% ÷100 = 0.55

Multiply the decimal by the whole, which is 130.

0.55 × 130 = 71.5

Therefore, 55% of 130 is 71.5.

Method 2:

Leave the percent over 100 and multiply by 130. This will allow you to multiply two fractions. 55 100

𝑥 130 =

55 100

𝑥

130 1

=

7150 100

= 71.5

The last step of dividing by 100 is very important and must not be left out since you are finding the part.

Method 3:

Use equivalent ratios. The part-to-whole fraction must equal the percentage part (55%) to the whole percent of 100%. You do not know the part, but you know the whole is 130.

55 = 130 100 Determine the equivalent ratio. Reverse the order of the ratios and scale up. You can scale up because you know that 100 × 1.3 = 130 (find this by dividing 130 by 100).

55 𝑥 1.3

71.5

= 100 𝑥 1.3

130 The missing amount from the fraction is 71.5

Now that you completed this lesson, you should be able to say: • I

can

determine the appropriate conversion factor when converting units.

• I

can

convert units between metric and customary systems.