7.4: Percentages
- Page ID
- 186590
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Motivating young learners to engage with percentages is essential because percentages are all around them—on sales tags, test scores, nutrition labels, and sports statistics. When students see how percentages connect to their daily lives and future aspirations, the math becomes relevant and meaningful. Effective teachers build on this relevance by designing lessons that are engaging and connected to students' interests, helping them experience success and joy in learning (NCTM, 2020).
Percents as Decimals and Fractions
Percent ( from the Latin word per centum meaning ”per hundred”) is denoted by n% = \(\frac{n}{100}\) where the symbol % indicates percent. The "cent" part of the word percent means "hundred". It is important to know that percents are an alternative representation for fractions and decimals; and be able to convert between all forms, Figure 7.4.1.
Figure \(\PageIndex{1}\):Percents are alternative representations of fractions and decimals
Case 1: Percents to Fractions
Use the definition of percent. For example, 23% = \(\frac{23}{100}\) by the meaning of percent.
Case 2: Percents to Decimals
To convert a percent directly to a decimal, you drop the % symbol and move the number's decimal point two places to the left. This shortcut works because dividing a number by 100 (as indicated by the "per hundred" definition) is equivalent to moving the decimal point two places to the left.
23% = 0.23, 135% = 1.35, 0.56% = 0.0056
Case 3: Decimals to Percents
To convert a decimal to a percent, you move the decimal point two places to the right and then add the % sign to the resulting number. This process essentially reverses the steps used to convert a percent to a decimal (case 2), where the decimal point is moved two places to the left. Also, fractions can be converted to decimals (using a calculator or long division), and then case 3 can be applied.
Case 4: Fractions to Percents
Some fractions that have terminating decimals can be converted to percents by expressing the fraction with a denominator of 100.
a) Write \(\frac{3}{5}\) as a decimal and percent.
b) Write 0.52 as a fraction and percent.
c) Write 216% as a decimal and fraction.
- Answer
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a) 3/5=60/100, 0.60, 60% b) 52/100=13/25. 52% c) 216/100=54/25, 2.16
Solving Percent Problems
Solving percent problems involves understanding the "per hundred" concept and its connection to fractions and decimals, recognizing the common problem types, and employing various methods including grids, proportions, equations, and mental math strategies. Effective teaching supports students in developing conceptual understanding, connects problems to relevant contexts, addresses language and context barriers, and encourages exploration of multiple solution strategies.
Percent Equations As Proportions
Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.
For example, and we can simplify Since the equation shows a percent equal to an equivalent ratio, we call it a percent proportion.
\[\frac{amount}{base}\nonumber\text{=}\frac{percent}{100}\nonumber\]
\[\frac{3}{5}\nonumber\text{=}\frac{60}{100}\nonumber\]
Percent Proportion
Translate to a proportion. What number is 75% of 90 ?
\[\frac{n}{90}\nonumber\text{=}\frac{75}{100}\nonumber\]
Translate to a proportion.
a) What number is 40% of 85?
b) 19 is 25% of what number?
c) What percent of 27 is 9 ?
- Answer
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a) \(\frac{n}{85}\text{=}\frac{40}{100}\)
100n = 3400
n=34
b) \(\frac{19}{n}\text{=}\frac{25}{100}\)
1900 = 25n
n=76
c) \(\frac{9}{27}\text{=}\frac{n}{100}\)
900=27n
n≈33.33
We can use a visual representation to help us solve percent problems using bar drawings and proportions, start by representing the whole amount as a bar divided into sections representing percentages. Then, use the given information to determine the value of each section Figure 7.4.2. Technically, we can use the drawings to set up equivalent proportions to solve for n. For example, we can write \(\frac{n}{85}\text{=}\frac{40}{100}\), or \(\frac{85}{n}\text{=}\frac{100}{40}\), or \(\frac{n}{40}\text{=}\frac{85}{100}\), or \(\frac{40}{n}\text{=}\frac{100}{85}\). This flexibility the drawing provides may make it easier to set up and solve a problem.
Now that we have written percent equations as proportions and represented in a visual, we are ready to solve the equations.
a) 125% of 25 is what number?
b) 6.5% of what number is $1.56 ?
c) What percent of 72 is 9 ?
- Answer
-
Figure \(\PageIndex{3}\): Visual for each problem.
a)
\(\frac{n}{25}\text{=}\frac{125}{100}\)
100n=3125
n=31.25
125% of 25 is 31.25.
b)
\(\frac{n}{1.56}\text{=}\frac{100}{6.5}\)
n=(1.56x100)/6.5
n=24
6.5% of $24 is $1.56.
c)
\(\frac{9}{72}\text{=}\frac{n}{100}\)
n=900/72=12.5
12.5% of 72 is 9.
Algebra Approach to Percent Equations
The following equations illustrate these three forms, where x represents an unknown and takes the place of percent, base, or amount depending on what is given and what is unknown in the problem.
\begin{align*}
\text{(a)} \quad & p\% \text{ of } n \text{ is } x & \quad & \left( \frac{p}{100} \right) n = x \\
\text{(b)} \quad & p\% \text{ of } x \text{ is } a & \quad & \left( \frac{p}{100} \right) x = a \\
\text{(c)} \quad & x\% \text{ of } n \text{ is } a & \quad & \left( \frac{x}{100} \right) n = a
\end{align*}
Once we have obtained one of these three equations, the solution, x, can be found. In equation (a), we multiply p/100 and n. In equations (b) and (c) we solve for the missing factor x.
Use the equation approach to solve.
a) 125% of 25 is what number?
b) 6.5% of what number is $1.56 ?
c) What percent of 72 is 9 ?
- Answer
-
a) 1.25x25=31.25
b) 0.065x = 1.56, x = 1.56/0.065=24
c) \(\frac{x}{100}\cdot\)72=9, x = (900/72) = 12.5
Now that we have the strategies to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we'll solve involve everyday situations, you can rely on your own experience.
Dejohn and his girlfriend enjoyed a dinner at a restaurant, and the bill was $68.50. They want to leave an 18% tip. If the tip will be 18% of the total bill, how much should the tip be?
- Answer
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t=12.33
If we approximate the bill to $70 and the percent to 20%, we would have a tip of $14. So a tip of $12.33 seems reasonable.
NOTE: you can also use other methods to calculate the tip. 18/100=n/68.50
The label on Masao's breakfast cereal said that one serving of cereal provides 85 milligrams (mg) of potassium, which is 2% of the recommended daily amount. What is the total recommended daily amount of potassium?
- Answer
-
a=4250,
NOTE: you can also use other methods to calculate the amount. 2/100=85/n, OR 1% = 42.5mg, so 100%=4250 mg
Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was 480 calories, and had 240 calories of fat. What percent of the total calories in each brownie comes from fat?
- Answer
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p=0.5, or 50%
NOTE: you can also use other methods to calculate the percentage. n/100 = 240/480
Wonder, Play, Grow
Work with your peers to complete your assigned percentage task. It may involve some creative thinking and multiple steps to complete, but you got this. Just start playing with the numbers and see what works and doesn't work.
Option 1: Concert tickets originally cost $100. Resellers list them for $250. Additionally, resale sites charge a 15% service fee on the sale price.
- What is the total price a buyer pays after the service fee?
- What is the percentage increase from the original $100 price to the final price?
- How much profit (as a percent) does the reseller make before fees?
Option 2: A refrigerator was on sale at the appliance store for 20% off. As a loyalty member, Marcus received a coupon from the store for an additional 30% off any current price in the store. If he uses the coupon to buy the refrigerator, the price would be $487.20 before taxes. What was the original price?
Option 3:
A workers' union negotiates a 3-year contract with these annual raises:
- 3% raise in Year 1
- 4% raise in Year 2
- 5% raise in Year 3
The base salary at the start of Year 1 is $50,000.
- What will the worker's salary be at the end of Year 3?
- What is the total percent increase from the original $50,000 salary?
- Why is the total increase not equal to 12%?
Do not look at the answers until you have some agreed upon solutions that seem reasonable.
- Answer
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Option 1: $287.50, 187.5%, 150% Option 2: $870 Option 3: $56238, 12.476%, Year 2's 4% applies to the new, higher salary after Year 1's raise. Year 3's 5% applies to an even higher salary, percent increases compound, they build on a new amount.
Mental Math and Estimation Using Fraction Equivalents
Since many common percentages are easily expressed as simple fractions, it’s often faster to calculate percentages mentally by using their fraction equivalents (see Figure 7.4.4). Just like with proportions, you can also estimate percentages by choosing fractions that are close and easy to work with.
Figure \(\PageIndex{4}\): Fraction Percent Equivalents
Find the following percents mentally, using fraction equivalents.
a) 25% of 44
b) 75% of 24
c) 38% of 50
d) 35% of 20
- Answer
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a) 1/4 x 44 = 11 b) 3/4 x 24 = 18 c) 38% of 50 is the same as 50% of 38, 1/2 x 38 = 19 d) 35% of 20 is the same as 20% of 35, 1.5 x 35 = 7
Estimate the following percents mentally, using fraction equivalents.
a) 48% of 73
b) 32% of 95
c) 123% of 59
d) 81% of 24
- Answer
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a) 50% of 72 is 36 b) 33 1/3% of 96 is 32 c) 100% of 60 and 25% of 60 is 60 + 15, 75 d) 25% of 80 is 20
*You may have other and better ways to estimate. :)
Before doing from retrieval practice below, let's learn more from Dr. Howie Hua.
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Use what you know about percentages and fractions to help you calculate each percentage.