Percents

Objective: To solve percent problems using proportions. To solve percent problems using the percent equation.

Objectives

• I can find a percent using the percent proportion.

• I can find a percent using the percent equation.

• I can find a part or a base.

• I can use the simple interest formula.

Vocabulary

• Percents are useful because they standardize comparisons to a common base of 100.

• You can solve problems involving percents using either proportions or the percent equation, which are closely related.

• If you write a percent as a fraction, you can use a proportion to solve a percent problem.

Vocabulary

• The Percent Proportion:• You can represent “a is p percent of b” using the percent proportion shown

below. In the proportion, b is the base, and a is a part of base b.

• Algebra: 𝑎

𝑏=

𝑝𝑒𝑟𝑐𝑒𝑛𝑡 %

100, where b ≠ 0

• Example: What percent of 75 is 30?30

75=

𝑝

100

•𝑖𝑠

𝑜𝑓=

%

100

Finding a Percent Using the Percent Proportion

1. What percent of 56 is 42?

2. What percent of 90 is 54?

3. What percent of 70 is 21?

4. What percent of 30 is 15?

5. What percent of 100 is 45?

Practice

Find each percent.1. What percent of 75 is 15?

2. What percent of 16 is 10?

3. What percent of 48 is 20?

4. What percent of 15 is75?

5. What percent of 32 is 40?

6. What percent of 88 is 88?

Vocabulary

• You have used the percent proportion 𝑎

𝑏=

𝑝

100to find

a percent.

• When you write 𝑝

100as p% and solve for a, you get the

equation 𝑎 = 𝑝% × 𝑏.

• This equation is called the percent equation.

• You can use either the percent equation or the percent proportion to solve any percent problem.

Vocabulary

• The Percent Equation• You can represent “a is p percent of b” using the percent

equation shown below. In the equation, a is a part of the base b.

• Algebra: a = p% × b, where b ≠ 0

• Example: What percent of 60 is 25? 25 = p% x 60

Finding a Percent Using the Percent Equation

1. What percent of 40 is 2.5?

2. What percent of 84 is 63?

3. What percent of 50 is 60?

4. What percent of 25 is 50?

5. What percent of 15 is 25?

Practice

Find each percent.1. What percent of 70 is 40?

2. What percent of 10 is 7?

3. What percent of 150 is 65?

4. What percent of 80 is 35?

5. What percent of 75 is 15?

6. What percent of 240 is 60?

Finding a Part

• A dress shirt that normally costs $38.50 is on sale for 30% off. What is the sale price of the shirt?

• A family sells a car to a dealership for 60% less than they paid for it. They paid $9000 for the car. For what price did they sell the car?

• A tennis racket normally costs $65. The tennis racket is on sale for 20% off. What is the sale price of the tennis racket?

Practice

Find each part.

1. What is 25% of 144?

2. What is 12.5% of 104?

3. What is 125% of 12.8?

4. What is 63% of 150?

5. What is 150% of 63?

6. What is 1% of 1?

7. A beauty salon buys bottles of styling gel for $4.50 per bottle and marks up the price by 40%. For what price does the salon sell each bottle?

Finding a Base

• 125% of what number is 17.5?

• 30% of what number is 12.5?

• 75% of what number is 36?

• 35% of what number is 80?

Practice

Find each base.1. 20% of what number is 80?

2. 60% of what number is 13.5?

3. 150% of what number is 34?

4. 80% of what number is 20?

5. 160% of what number is 200?

6. 1% of what number is 1?

Vocabulary• A common application of percents is simple interest, which is

interest you earn on only the principal in an account.

• Simple Interest Formula:

• The simple interest formula is given below, where I is the interest, P is the prinicpal, r is the annual interest rate written as a decimal, and t is the time in years.

• Algebra: I = Prt

• Example: If you invest $50 at a simple interest rate of 3.5% per year for 3 years, the interest you earn is I = 50(0.035)(3) = $5.25.

Vocabulary

• When you solve problems involving percents, it is helpful to know fraction equivalents for common percents.

• You can use the fractions to check your answers for reasonableness.

• Here are some common percents represented as fractions.

1% =1

1005% =

1

2010% =

1

1020% =

1

525% =

1

4

33. 3% =1

350% =

1

266. 6% =

2

375% =

3

4100% = 1

Using the Simple Interest Formula

• You deposited $840 in a savings account that earns a simple interest rate of 4.5% per year. You want to keep the money in the account for 4 years. How much interest will you earn? Check your answer for reasonableness.

• You deposited $125 in a savings account that earns a simple interest rate of 1.75% per year. You earned a total of $8.75 in interest. For how long was your money in the account?

Practice

• You deposit $1200 in a savings account that earns simple interest at a rate of 3% per year. How much interest will you have earned after 3 years?

• You deposit $150 in savings account that earns simple interest at a rate of 5.5% per year. How much interest will you have earned after 4 years?

Solving Percent Problems

Problem Type Example Proportion Equation

Find a percent. What percent of 6.3 is 3.5? 3.5

6.3=

𝑝

100

3.5 = p% x 6.3

Find a part. What is 32% of 125? 𝑎

125=

32

100

a = 32% x 125

Find a base. 25% of what number is 11? 11

𝑏=

25

100

11 = 25% x b

Change Expressed as a Percent

Objective: To find percent change.

Objectives

• I can find a percent decrease or percent increase.

• I can find percent error.

• I can find minimum and maximum dimensions.

• I can find the greatest possible percent error.

Vocabulary

• A percent change expresses an amount of change as a percent of an original amount.

• You can find a percent change when you know the original amount and how much it has changed.

• If a new amount is greater than the original amount, the percent change is called a percent increase.

• If the new amount is less than the original amount, the percent change is called a percent decrease.

Vocabulary

Percent Change:

• Percent change is the ratio of the amount of change to the original amount.

𝑝𝑒𝑟𝑐𝑒𝑛𝑡, 𝑝% =𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑟 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒

𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡

• Amount of increase = new amount – original amount

• Amount of decrease = original amount – new amount

A common example of finding a percent decrease is finding a percent discount.

Finding a Percent Decrease

• A coat is on sale. The original price of the coat is $82. The sale price is $74.50. What is the discount expressed as a percent change?

• The average monthly precipitation for Chicago, Illinois, peaks in June at 4.1 inches. The average monthly precipitation in December is 2.8 inches. What is the percent decrease from June to December?

Finding a Percent Increase

• A store buys an electric guitar for $295. The store then marks up the price of the guitar to $340. What is the markup expressed as a percent change?

• In one year, the toll for passenger cars to use a tunnel rose from $3 to $3.50. What was the percent increase?

Practice

Tell whether each percent change is an increase or decrease. Then find the percent change. Round to the nearest percent.

1. Original: 12 New: 18

2. Original: 40.2 New 38.6

3. Original: 9 New: 6

4. Original: 7.5 New: 9.5

5. Original: 15 New: 14

6. Original: 2008 New: 1975

7. Original: 14,500 New: 22,320

8. Original: 195.20 New: 215.25

9. Original: 1325.60 New: 1685.60

Vocabulary

• You can use percents to compare estimated or measured values to actual or exact values.

• Relative Error:

• Relative error is the ratio of the absolute value of the difference of a measured (or estimated) value and an actual value compared to the actual value.

• 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑒𝑟𝑟𝑜𝑟 =𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑜𝑟 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 −𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒

𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒

• When relative error is expressed as a percent, it is called percent error.

Finding Percent Error

• A decorator estimates that a rectangular rug is 5 ft by 8 ft. The rug is actually 4 ftby 8 ft. What is the percent error in the estimated area?

• You think that the distance between your house and a friend’s house is 5.5 mi. The actual distance is 4.75 mi. What is the percent error in your estimation?

Practice

Find the percent error in each estimation. Round to the nearest percent.

1. You estimate that your friend’s little brother is about 8 years old. He is actually 6.5 years old.

2. You estimate that your school is about 45 ft tall. Your school is actually 52 fttall.

Vocabulary

• Sometimes we know the actual measurements of something. Sometimes we don’t know the actual measurements, but we know how precise they can be.

• Think of it this way, when we use a ruler, how precise are we?

• We could be off a little, meaning a fraction of a unit is off.

• The most any measurement can be off by is one half of the unit used in measuring.

Finding Minimum and Maximum Dimensions

• You are framing a poster and measure the length of the poster as 18.5 inch, to the nearest half inch. What are the minimum and maximum possible length of the poster?

• A student’s height is measured as 66 inches to the nearest inch. What are the student’s minimum and maximum possible heights?

Practice

A measurement is given. Find the minimum and maximum possible measurements.

1. A doctor measures a patient’s weight as 162 pounds to the nearest pound.

2. An ostrich egg has a mass of 1.1 kg to the nearest of a kg.

3. The length of an onion cell is 0.4 mm to the nearest tenth of a mm.

Finding the Greatest Possible Percent Error

The diagram shows the dimensions of a gift box to the nearest inch. What is the greatest possible percent error in calculating the volume of the gift box?

If the gift box’s dimensions were measured to the nearest half inch, how

would the greatest possible error be affected?

5 in

6 in12 in

Practice

The table below shows the measured dimensions of a prism and the minimum and maximum possible dimensions based on the greatest possible error. What is the greatest possible percent error in finding the volume of the prism?

Dimensions Length Width Height

Measured 10 6 4

Minimum 9.5 5.5 3.5

Maximum 10.5 6.5 4.5

Practice

The side lengths of the rectangle have been measured to the nearest half of a meter. What is the greatest possible percent error in finding the area of the rectangle?

7.5 m

18.5 m