4.8 Solving Percents Using Proportions
SWBAT / 1. Calculate answers to percent problems.
2. Use proportions to solve percent application problems.
3. Find and use percents of increase and decrease.
Ø Gabrielle has two scores on her test: 19 points and 79%. There were 10 problems on the test, and she wonders what the total points for the test were.
Ø Colleen wants to keep her heart rate between 60% and 80% of her maximum while exercising. Her maximum heart rate is 180 beats per minute.
Ø Jeremiah’s total family monthly income after college costs and day care deductions is $3,740. The family mortgage and utilities are $1,890 per month. He sees the recommendation that his housing costs should be about 44% of his budget. He wonders how close he is to this recommendation.
Each of the problems above fits one of three types of percentage problems. Yet each can be solved using the same type of proportion.
In this section we study how to use proportions to solve problems involving percentages and how to think through multiple-step problems involving percentages.
Percents, Amounts, Base and Proportions
When we use a percent, we are always making a comparison. We compare how many out of 100 is equal to a part of a whole something. We call this “whole something” the base. We call the “part of the whole” the amount. For example, on a 50% off sale, we pay $30 for a $60 item. Notice that 50 percent is 50 out of 100 which is equal to the amount of $30 when compared to a base of $60. We can write this using the proportion 50100=3060.
Percent problems usually fall into one of three categories; we need to find the percent, the base or the amount while the other two values are known. We can always write a proportion to solve these problems. To do this we let the ratio, P100 , represent the percentage, where P % is a percent.
DEFINITION: For any percent problem we can write the proportion:
P100=AB that is P100=AmountBase where P% is a percent.
We use the variables A, B and P to name the value that is unknown. Use the table below to become familiar with the English phrases that lead to a correct proportion for a problem situation. Some people use the vertical scale to successfully set up proportions.
Phrases / Vertical Scale / Proportion & Solution9 is what percent of 45?
The percent is missing. / / P100=AB , P100=945
45∙P=900, P=20
9 is 20% of 45
15% of $35 is what?
The amount is missing. / / P100=AB , 15100=A35
525=100∙A, A=5.25
15% of $35 is 5.25
42 is 5% of what?
The base is missing. / / P100=AB , 5100=42B
5∙B=4200, B=840
42 is 5% of 840
ü Check Point 1
a. What percent of 75 is 27?
b. 32% of $4,000 is what?
c. 28 out of 70 is what percent?
d. 120 is 1.5% of what number?
Amounts correspond to the percentage and the Base always corresponds to 100 in the percent proportion equation, P→ → → 100 → → →=AmountBase . To keep from mixing these up, we make sure we focus on the distinctions between the values for Amounts and Bases.
Example 1:
a. What percent is 50 out of 75?
Think it through: a. For “50 out of 75,” 50 is the Amount and 75 is the Base.
You are missing the percent, so find P.
Estimate first that 5075≈70100, so about 70%P100=5075 / Write the proportion.
75P=5,000 / Use cross products and multiply.
P= 5,00075 / P is the missing factor, so divide.
P=6623 / Use your calculator.
6623% is the percent / P % is a percent
6623% is about 70%. Our result is close to our estimate so it checks.
ANSWER: 50 out of 75 is 6623%.
b. What percent is 75 out of 50?
Think it through: b. For “75 out of 50,” 75 is the Amount and 50 is the Base.
We are missing the percent.
We use a proportion to verify this.
P100=7550 / Write the proportion.
50P=7,500 / Use cross products and multiply.
P= 7,50050 / P is the missing factor, so divide.
P=150
150% is the percent / Use mental math or your calculator.
P % is the percent.
Our answer is the same solving this problem two ways, so it checks.
ANSWER: 75 out of 50 is 150%.
In Example 1b the Amount was more than the Base, which led to a percent value that was more than 100%.
ü Check Point 2
Bubba is really confused now. He found a 70% off sale for a shrimp trawling net setup. He paid $79.50 and decides to find how much the net would have cost originally. He knows he paid 30%, so he writes: 30100=A$79.5 , but when he uses cross multiplication and solves this correctly, he comes up with $23.85. He knows he should not have to move the decimal point anywhere, and even if he moves it two places left or right, these values are also not correct. How should Bubba have calculated the original price for his shrimp trawling net?
Solving Percent Application Problems
We can take any percent application and rephrase it to determine what the percent, base and amount are. While some people use this strategy most of the time, others prefer to use the vertical scale to translate. Either way, learn both of these strategies just in case your favorite strategy is not working on a certain problem.
In the last check point, Bubba at least recognized the unreasonableness of his answer. Bubba also knew he could use a proportion to solve his percent problem. But Bubba did not identify the Amount and Base of his proportion correctly. It is often useful to concentrate on finding the Base, the amount that is 100%, first.
Example 2: Gabrielle has two scores on her test: 19 points and 79%. There were 10 problems on the test. What point score would have given her 100%?
Think it through:
Understand: We rephrase the problem, “19 is 79% of what number?” We are missing the Base.
Plan: Use a proportion to solve. Draw a vertical scale to make sure we set up the correct proportion.
Solve: The percent is 79%, so P = 79. The amount is 19 and we are looking for the base. Because 19 to 79 is about 20 to 80 and 20 to 80 is the same ratio as 25 to 100, estimate B≈25 using scaling.
79100=19B / Write the proportion.79B = 1900 / Use cross multiplication.
B=1900÷79 / B is a missing factor so divide.
B≈24.0506≈24 / Use a calculator and round to a reasonable answer.
Check: This fits our estimate of B≈25 points, so we accept this answer.
ANSWER: A score of 24 points would have given Gabrielle 100% . (Notice that when working with percents, we often will have to round our results to reasonable answers.)
Example 3: Colleen wants to keep her heart rate above 60% while exercising. Her maximum heart rate is 180 beats per minute. What is this lowest acceptable heart rate?
Think it through:
Understand: We rephrase this problem, “What is 60% of 180?”
Plan: Use a proportion to solve. Draw a vertical scale to make sure we set up the correct proportion.
Solve: The base is 180, the percent is 60% so P = 60, and we are looking for the amount.
Estimate A ≈ 100 because 60/100 is about 100/180.60100=A180 / Write the proportion.
10,800 = 100A / Use cross products and multiply.
A=108 / Divide.
Check: A=108 fits our estimate of A≈100, so accept this answer.
ANSWER: Colleen should keep her heart rate above 108 beats per minute.
Example 4: Jeremiah’s total family monthly income after college costs and day care deductions is $3,740. The family mortgage and utilities are $1,890 per month. How does this amount compare to the 44% guideline for housing expenses?
Think it through:
Understand: We rephrase this question, “$1,890 is what percent of $3,740?”
Plan: Use a proportion to solve. Draw a vertical scale to make sure we set up the correct proportion.
Solve: Estimate P/100 ≈ 50% for ≈1900 compared to ≈3800.
P100=1,8903,740 / Write the proportion.3,740∙P = 189,000 / Use cross products.
P = 189,000÷3,740 / P is a missing factor, so divide.
P≈50.5 / Use a calculator and round.
50.5% is the percent / P % is the percent.
Check: 50.5% fits our estimate of ≈ 50%, so we accept this answer.
ANSWER: Jeremiah’s current housing expenses are about 50.5% which is higher than the 44% that the guidelines recommend. (Jeremiah has asked a good question here, and now he has information to make some well-informed decisions.)
ü Check Point 3
Colleen wants to keep her heart rate below 80% of her maximum heart rate of 180 beats per minute. What is this heart rate?
ü Check Point 4
In 2012, a National Retail Foundation survey reported that the average “back to college” expenses were $907.22 per college student with an average of $216.4 of this money spent on computers and other electronics[1]. What is the percentage of these back-to-college expenses are dollars that were spent on electronics?
In applications, percentages are often more than 100% or even less than 1%. For instance, a nursing assistant (CNA) who earns a degree as a registered nurse (RN) will see his or her wages grow more than 100% with the first pay check as an registered nurse. In 2011 the 205,000 people of Jackson County, Oregon, represented 0.066% of the population of the United States. Learn to think through the mathematics when the percentages are not numbers that we are used to seeing.
Example 5: Suzanne is thinking of taking a job in San Francisco. She earns $2,800 a month in Eugene, Oregon, right now. Using a cost-of-living calculator that she found online[2], Suzanne discovered that she needs to earn $4,257 in San Francisco to have the same “purchasing power” that she has in Eugene. What percentage of her current wage is the amount she would need to earn to have the same purchasing power in San Francisco?
Think it through: Our answer will be more than 100% because $4,257 > $2,800.
Understand: Suzanne is looking for a missing percent.
Plan: Use a proportion to solve. Draw a vertical scale to make sure we set up the correct proportion.
Solve: Using ≈4,500 and ≈3,000 for wages, estimate P≈150 and our answer should be about150%.
P100=4,2572,800 / Write a proportion.2,800∙P=425,700 / Use cross products.
P=425,700÷2,800 / P is a missing factor, so divide.
P≈152 / Use a calculator.
152% is the percent. / P% is the percent.
Check: Because the estimate and the computation agree, we accept this result.
ANSWER: According to the cost-of-living calculator we found online, Suzanne will need to earn about 152% of what she now earns to enjoy the same purchasing power in San Francisco as in Eugene.
ü Check Point 5
Many of today’s registered nurses worked as nursing assistants before going to school to become RNs. While the average entry level salary for a nursing assistant in Oregon is $24,500, the average entry level salary for a registered nurse in Oregon is $67,104 according to the Oregon Employment Department[3].
What percent of the average nursing assistant’s salary in Oregon is the average salary of Oregon registered nurses?
ü Check Point 6
In 2012 the property tax rate for the Applegate District of Josephine County was $8.4390 dollars per thousand dollars of assessed value. A house assessed at $120,000 would have been charged a property tax of $1,012.68 in this tax district. What percent of $120,000 is $1,012.68?
Increases and Decreases with Percents
When wages go up by a certain percentage, the amount we are paid increases proportionally. When the value of a house falls by a certain amount, we can find the percentage that its value has decreased.
When these changes occur, we have more than one way to think about the percentages and proportions involved. For instance Deirdre and Steve work for the same company, both earn $10.80 per hour, and they find out that they are getting a 2.5% raise starting next month. They both calculated their new salaries, but they each used different methods.
Deirdre’s Method / Steve’s Method2.5100=A10.80
A=$0.27 / Deirdre calculates the amount of her increase in wages. / 100+2.5
=102.5% / Steve calculates the total percentage for his new wage.
A=10.80+.27
A=$11.07 / She adds the new amount to the original base wage. / 102.5100=A10.80
A=$11.07 / Steve finds the amount using the percentage of his original wage.
Both Deirdre and Steve need to know what their missing value is. In Deirdre’s case she solved a proportion to find the amount of increase in her hourly wage. Steve solved a proportion to find the new amount of his new wage.
FORMULA:
For problems with percent of increase:
Ø Amount of increase= New Total Amount-Base
Ø Percent of increase 100=Amount of increaseBase (Deirdre’s method)
or…
Ø 100+ Percent of increase 100=New Total AmountBase (Steve’s Method)
For problems with percent of decrease:
Ø Amount of decrease= Base- New Total Amount