# Ratios, Rates, and Percents

### Unit 4 - Mrs. Jenkins Math

Here are our Unit 4 Vocabulary Words

## Ratios

*Textbook Pages 149-152*

**I understand and can write ratios to make comparisons in real-world contexts.**

**Essential Questions: **What are ratios? How can we use ratios to make real-world comparisons?

## Read about Ratios

(From PurpleMath.com)

Proportions are built from ratios. A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20.

Notice that, in the expression "the ratio of men to women", "men" came first. This order is very important, and must be respected: whichever word came first, its number must come first. If the expression had been "the ratio of women to men", then the numbers would have been "20 to15".

Expressing the ratio of men to women as "15 to20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:

odds notation: 15** :** 20

fractional notation: 15/20

You should be able to recognize all three notations; you will probably be expected to know them for your test.

Given a pair of numbers, you should be able to write down the ratios. For example:

- There are 16 ducks and 9 geese in a certain park. Express the ratio of
__ducks to geese__in all three formats. **16:9, 16/9, and 16 to 9**

- Consider the above park. Express the ratio of
__geese to ducks__in all three formats. **9:16, 9/16, and 9 to 16**

The numbers were the same in each of the above exercises, but the *order* in which they were listed differed, varying according to the order in which the elements of the ratio were expressed. In ratios, order is very important.

Let's return to the 15 men and 20 women in our original group. I had expressed the ratio as a fraction, namely, 15/20. This fraction simplifies to 3/4. This means that you can also express the ratio of men to women as 3/4, 3** :** 4, or "3 to 4".

This points out something important about ratios: the numbers used in the ratio might not be the *absolute* measured values. The ratio "15 to 20" refers to the *absolute* numbers of men and women, respectively, in the group of thirty-five people. The simplified ratio "3 to 4" tells you only that, for every three men, there are four women. The simplified ratio also tells you that, in any representative set of seven people (3 + 4 = 7) from this group, three will be men. In other words, the men comprise 3/7 of the people in the group. These relationships and reasoning are what you use to solve many word problems:

**In a certain class, the ratio of passing grades to failing grades is 7 to 5. How many of the 36 students failed the course?**

The ratio, "7 to 5" (or 7** :** 5 or 7/5), tells me that, of every 7 + 5 = 12 students, five failed. That is, 5/12 of the class flunked. Then ( 5/12 )(36) = **15 students failed.**

**In the park mentioned above, the ratio of ducks to geese is 16 to 9. How many of the 300 birds are geese?**

The ratio tells me that, of every 16 + 9 = 25 birds, 9 are geese. That is, 9/25 of the birds are geese. Then there are ( 9/25 )(300) = **108**** geese.**

Generally, ratio problems will just be a matter of stating ratios or simplifying them. For instance:

**Express the ratio in simplest form: $10 to $45**

This exercise wants me to write the ratio as a *simplified* fraction:

.10/45 = **2/9**.

This simplified fraction is the ratio's expression in simplest fractional form. Note that the units (the "dollar" signs) "canceled" on the fraction, since the units, "$", were the same on both values. When both values in a ratio have the same unit, there should generally be no unit on the reduced form.

**Express the ratio in simplest form: 240 miles to 8 gallons**

When I simplify, I get (240 miles) / (8 gallons) = (30 miles) / (1 gallon), or, in more common language, **30 miles per gallon.**

In contrast to the answer to the previous exercise, this exercise's answer did need to have units on it, since the units on the two parts of the ratio, the "miles" and the "gallons", do not "cancel" with each other.

Simplified ratios are equivalent ratios. Equivalent ratios are ratios that name the same comparison. You can find equivalent ratios by either multiplying or dividing both terms in the ratio by the same number.

Conversion factors are simplified ratios, so they might be covered around the same time that you're studying ratios and proportions. For instance, suppose you are asked how many feet long an American football field is. You know that its length is 100 yards. You would then use the relationship of 3 feet to 1yard, and multiply by 3 to get 300 feet.

Ratios are the comparison of one thing to another (miles to gallons, feet to yards, ducks to geese, et cetera). But their true usefulness comes in the setting up and solving of proportions....

Practice Writing Ratios! Make sure you're logged into your Khan Academy!

Practice Identifying Equivalent Ratios! Make sure you are logged into your Khan Academy!

## Rates

*Textbook Pages 155-158*

**Learning Target:** I understand and can apply rates in real world situations.

**Essential Questions: **What are rates? How do we use rates in real world situations?

## Read about Rates

A **ratio** is a comparison of two numbers or measurements. The numbers or measurements being compared are called the **terms** of the ratio. A **rate** is a special ratio in which the two terms are in different units. For example, if a 12-ounce can of corn costs 69¢, the rate is 69¢ for 12 ounces. The first term of the ratio is measured in cents; the second term in ounces.

You can write this rate as 69¢/12 ounces or 69¢:12 ounces. Both expressions mean that you pay 69¢ "for every" 12 ounces of corn.

Rates are used by people every day, such as when they work 40 hours a week or earn interest every year at a bank. When rates are expressed as a quantity of 1, such as 2 feet per second or 5 miles per hour, they are called **unit rates**. If you have a multiple-unit rate such as 120 students for every 3 buses, and want to find the single-unit rate, write a ratio equal to the multiple-unit rate with 1 as the second term.

The unit rate of 120 students for every 3 buses is 40 students per bus. You could also find the unit rate by dividing the first term of the ratio by the second term. When prices are expressed as a quantity of 1, such as $25 per ticket or $0.89 per can, they are called **unit prices**. If you have a multiple-unit price, such as $5.50 for 5 pounds of potatoes, and want to find the single-unit price, divide the multiple-unit price by the number of units.

The unit price of potatoes that cost $5.50 for 5 pounds is $1.10 per pound.

Rates and unit rates are used to solve many real-world problems. Look at the following problem.

- "Tonya works 60 hours every 3 weeks. At that rate, how many hours will she work in 12 weeks?" The problem tells you that Tonya works at the rate of 60 hours every 3 weeks.
- To find the number of hours she will work in 12 weeks, write a ratio equal to 60/3 that has a second term of 12.
- 60/3 = 240/12
- Tonya will work 240 hours in 12 weeks.

You could also solve this problem by first finding the unit rate and multiplying it by 12.

- 60/3 = 20/1
- 20 x 12 = 240

When you find equal ratios, it is important to remember that if you multiply or divide one term of a ratio by a number, then you need to multiply or divide the other term by that same number. Now let's take a look at a problem that involves unit price.

- "A sign in a store says 3 Pens for $2.70. How much would 10 pens cost?"
- To solve the problem, find the unit price of the pens, then multiply by 10.
- $2.70 ÷ 3 = $0.90
- $0.90 10 = $9.00

Finding the cost of one unit first makes it easy to find the cost of multiple units.

Practice finding Unit Rates!

Practice solving these problems by finding the Unit Rate! Make sure you are logged into your Khan Academy!

## Ratio and Rate Reasoning

*Textbook Pages 161-164*

**Learning Target:** I can use ratio and rate reasoning to solve problems by creating tables, double number lines, graphs, and proportions.

**Essential Question: **How can tables, double number lines, graphs, and proportions help me use ratio/rate reasoning to solve problems?

## Extra Notes/Work Page 1 Extra notes/work on using ratio and rate reasoning to solve Your Turn Question 2 on Page 162. | ## Your Turn Question 2 on Page 162 This is the Your Turn question that we solved 4 different ways in class. The different strategies to solve the problem are shown in the extra Notes/Work Pages | ## Extra Notes/Work Page 2 Extra notes/work on using ratio and rate reasoning to solve Your Turn Question 2 on Page 162. |

## Extra Notes/Work Page 1

## Your Turn Question 2 on Page 162

## Read about Equivalent Ratios and Rates

A **rate** is a ratio that compares quantities in different units. The word "per" is used when talking about rates and is sometimes abbreviated with a slash, /.

Some common rates are used all the time, like miles per hour, dollars per gallon, and days per week. Others are more specific to an occasion, like cats per household or chocolate chips per cookie.

Two rates are **equivalent** if they show the same relationship between two identical units of measure.

The same strategies used to find equivalent ratios can be used to find equivalent rates.

Here is an example:

- Two cars are traveling. One car goes 40 miles in 2 hours, and the other goes 80 miles in 4 hours. Determine whether or not the rates are equivalent.
- First, write the rates as fractions. Remember to make sure the terms are the same, in this case, miles per hour.
- 40 miles per 2 hours = 40 miles / 2 hours = 40/2
- 80 miles per 4 hours = 80 miles / 4 hours = 80/4
- Next, simplify both fractions to their lowest terms.
- 40/2 = 40÷2 / 2÷2 = 20 miles / 1 hour
- 80/4 = 80÷4 / 4÷4 = 20 miles / 1 hour
- Then, compare the fractions in their lowest terms. Remember to include the units.
- 20 miles / 1 hour = 20 miles / 1 hour
- The answer is that the rates are equivalent. Both cars traveled at a rate of 20 miles per hour.

Other methods that can be used to determine whether or not rates are equal include changing one or both fractions so that the denominators are equal, and cross multiplying.

Cross multiplication does not yield a rate. Only the numerical values are used as a check.

When using rates, it does not matter which unit is in the numerator and which is in the **denominator**, as long as they match one another.

For example:

- gumdrops per person is not the same as persons per gumdrop
- gumdrops / person ≠ persons / gumdrop

__Guided Practice__

Determine if two machines are winding thread at the same speed. One machine winds at a rate of 5 meters every 3 seconds. The other takes 18 seconds to wind 20 meters.

- First, write the rates as fractions, making sure the units are identical.
- 53 meters / 1 second and 20 meters / 18 seconds
- Next, assume the rates are equal to one another and write an equation.
- 5/3 = 20/18
- Then cross multiply.
- 5 x 18 = 3 x 20
- 90≠60
- The answer is that the rates are not equivalent.

__Example 1__

Are the rates equivalent? 3 feet in 9 seconds and 6 feet in 18 seconds

- First, write the rates as fractions.
- 3/9 and 6/18
- Next, determine a method. Multiply the first fraction times 2 to convert to, and work with, 18ths.
- 3/9 = 3×2 / 9×2 = 6/18
- Then, compare the fractions.
- 6 feet / 18 seconds = 6 feet / 18 seconds
- The answer is that the rates are equivalent.

__Example 2__

Compare the rates: 5 miles in 30 minutes and 42 minutes to go 6 miles

- First, write the rates as fractions being sure to keep the units consistent.
- 5 miles / 30 minutes and 6 miles / 42 minutes
- Next, determine a method. Simplify both rates to their lowest terms.
- 5/30 = 5÷5 / 30÷5 = 1 mile / 6 minutes
- 6/42 = 6÷6 / 42÷6 = 1 mile / 7minutes
- Then, compare the fractions.
- 1/6 ≠ 1/7
- The answer is that the rates are not equal.

__Example 3__

Compare 5 pounds for $20.00 and 8 pounds for $32.00

- First, write the rates as fractions.
- 5 pounds / 20 dollars and 8 pounds / 32 dollars
- Next, determine a method. Simplify each fraction to its lowest terms.
- 5/20 = 5÷5 / 20÷5 = 1 pound / 4 dollars
- 8/32 = 8÷8 / 32÷8 = 1 pound / 4 dollars
- Then, compare the rates.
- 1/4 = 1/4
- The answer is that the rates are equivalent.
- 1 pound / 4 dollars can also be written 4 dollars / 1 pound or $4 per pound.

Practice Comparing Ratios! Make sure you are logged into your Khan Academy!

Practice Finding Equivalent Rates and Unit Rates! Make sure you are logged into your Khan Academy!

Practice Solving Unit Rate Problems! Make sure you are logged into your Khan Academy!

## Read About Scale

A **scale** is a ratio that shows the relationship between the representation of an object and the real measurement of an object. Toy makers use scale ratios to make models of cars and airplanes that are proportional to the real thing. Architects use scale drawings to plan the design of buildings.

A **scale drawing** is a drawing that is used to represent and object that is too large to be drawn in its actual dimensions. If a drawing is to scale, you can use **proportions** to determine the actual dimensions of the scale drawing.

Let’s look at a scale drawing of a sculpture. In the drawing, the sculpture is 4 inches tall. The scale is 1 inch : 4 feet. Its scale can be written as 1 in. / 4 ft. A drawing with this scale tells you that 1 inch on paper is equal to 4 feet in real life. If the drawing is 4 inches tall, use the scale to find the actual height of the sculpture.

- First, create a
**proportion**using equivalent ratios. Remember to write the corresponding units in the numerator and in the denominator. The actual height of the sculpture is represented by the variable*x*. - 1 in. / 4 ft. = 4 in. /
*x*ft. - Then, cross multiply and simplify the equation to find the value of
*x*. - 1 ×
*x*= 4 × 4 *x*= 16- The sculpture will be 16 feet tall.

Let’s make a scale drawing. Use the scale 1 in. = 2 ft. Draw a room that is 8 feet × 12 feet.

- First, write a proportion to find the measurement of the width.
- 1 in. / 2 ft. =
*x*in. / 8 ft. - Next, cross multiply and simplify the equation to find the value of
*x*. - 1 × 8 = 2 ×
*x* - 8 = 2 ×
*x* *x*= 4**The drawing will be 4 inches wide**.- Then, write a proportion to find the measurement of the length.
- 1 in. / 2 ft. =
*x*in. / 12 ft. - Finally, cross multiply and simply the equation to find the value of
*x*. - 1 × 12 = 2 ×
*x* - 12 = 2 ×
*x* *x*= 6**The drawing will be 6 inches long**.- In the drawing room will be 4 inches × 6 inches.

Scale dimensions are also used to figure out the actual dimensions of something. The flower bed design shows that the width of the garden on the drawing is six inches. If the scale is 1 in. = 5 ft, how wide is the actual flower garden?

- First, write a proportion to find the actual measurement of the flower bed.
- 1 in. / 5 ft. = 6 in. /
*x*ft. - Then, cross multiply and simplify the equation to find the value of
*x*. - 1 ×
*x*= 5 × 6 *x*= 30- The actual flower bed is 30 feet wide.

__Guided Practice__

Solve the proportion.

- 7 in. / 70 ft. =
*x*in. / 140 ft. - First, cross multiply and simplify to find the value of
*x*. - 7 × 140 = 70 ×
*x* - 980 = 70 ×
*x* - Divide both sides of the equation by 70.
*x**× 70*÷ 70 =*x*- 980 ÷ 70 = 14
*x*= 14- Or, use mental math to solve for
*x*. - First, look at the relationship between the two given denominators. Think, “The second denominator (140 ft.) is double (2 ×) the first denominator (70 ft.).”
- Then, take the given numerator and multiply by 2. Think, “7 times 2 is 14.”
- The answer is 14 inches.

__Examples__

Solve each of the following proportions.

**Example 1:** 1 in. / 3 ft. = *x* in. / 21 ft.

- First, cross multiply and simplify to find the value of
*x*. - 1 × 21 = 3 ×
*x* - 21 = 3 ×
*x* *x*= 7- The answer is 7 inches.

**Example 2:** 3 in. / 6 ft. = 9 in. / *x* ft.

- First, cross multiply and simplify to find the value of
*x*. - 3 ×
*x*= 6 × 9 - 3 ×
*x*= 54 - Divide both sides of the equation by 3.
*x*× 3 ÷ 3 =*x*- 54 ÷ 3 = 18
*x*= 18- The answer is 18 feet.

**Example 3:** 2 in. / 10 ft. = *x* in. / 120 ft.

- First, cross multiply and simplify to find the value of
*x*. - 2 × 120 = 10 ×
*x* - 240 = 10 ×
*x* - Divide both sides by 10.
*x*× 10 ÷ 10 =*x*- 240 ÷ 10 = 24
*x*= 24- The answer is 24 inches.

Now that you have figured out how to use** proportions **to figure out actual and scale dimensions, we can look at figuring out *scale factors*.

What is a scale factor?

- A
*scale factor*is another name for a scale ratio. When looking for a scale factor, you can look at the relationship between the scale measurement and the actual measurement to determine what scale was used. This scale is called the scale factor.

A fence is actually 16 feet long. If the fence is drawn as four inches, what is the scale factor?

- To figure this out, we need to write a ratio to compare the drawing of the fence to the actual measurement.
- 4 in. / 16 ft.
- Now we want to figure out the scale factor. To do this, we simplify the ratio using the greatest common factor. The greatest common factor of 4 and 16 is 4.
- 4÷4 / 16÷4 = 1 / 4
- The scale factor is 1 in. / 4 ft.

Use this information to simplify and find the following scale factors.

**Example A: **5 in. / 25 ft.

- Solution: 5÷5 / 25÷5 = 1 in. / 5 ft.

**Example B:** 2 in. / 50 ft.

- Solution: 2÷2 / 50÷2 = 1 in. / 25 ft.

**Example C:** 12.5 in. / 25 ft.

- Solution: 12.5÷12.5 / 25÷12.5 = 1 in. / 2 ft.

At the supermarket, Sarah was looking at the cereal boxes as she was putting them away. On the back of one of the boxes, there was a picture of a skyscraper. Underneath the picture was a ratio. It said: **1 inch / 30 feet**

- Sarah wasn't sure what this meant. Do you know?
- Sarah has seen a scale ratio. Inches are being compared to feet. In other words, the scale is saying that for every inch of the building on the cereal box that it represents 30 feet of actual height.
- What about scale factor?
- We can figure out the scale factor if we can simplify this ratio. However, this ratio is already in simplest form so this scale ratio is also the scale factor.

__Guided Practice__

What is the scale factor: 3 / 12

**Answer:**To figure this out, we divide the denominator and the numerator by the same number (GCF). Once the fraction is__simplified__, it will show the scale factor: 1 / 4

Practice Using Scales to Solve Problems!

## Unit Conversions

**Learning Target:**

*I can convert within and between the metric and customary unit systems.*

**Essential Question:** How can proportions help me convert within and between the metric and customary unit systems?

## Read About Converting Within the Customary System

When converting customary units of measure from a larger unit to a smaller unit, create a proportion using the unit rate of the conversion factor to find an equivalent rate. Here is an example to demonstrate this.

- 1 dollar = 100 pennies or
__100 pennies / 1 dollar__

There are 100 pennies in one dollar. The dollar is a larger unit than the penny. You need many pennies to equal one dollar. The same is true when working with units of length, weight and capacity. You need more of a smaller unit to equal a larger unit.

Think back to all of the units of length*,* weight and capacity that you have previously learned about.

Let's look at a conversion problem.

John has a rope that is 10 feet long. How long is his rope in inches?

- Notice, you are converting from feet to inches. A foot is a larger unit than an inch.
- To solve this problem, create a proportion with the unit rate of how many inches in 1 foot and an equivalent rate for 10 feet. This will give you the measurement in inches.
- 1 foot = 12 inches or
__12 inches / 1 foot__ - 12 in. / 1 ft. =
*x*in. / 10 ft. - The number of feet was multiplied by 10 to make 10 ft.; so, the number of inches must in 1 foot must be multiplied by 10 to get an equivalent rate for 10 feet.
- 12 in. × 10 / 1 ft. × 10 = 120 in. / 10 ft.
- The answer is 10 feet is equivalent to 120 inches.

__Guided Practice__

Convert the units of measure.

Jason’s baby brother drank 3 cups of milk. How many fluid ounces did he drink?

- Once again, you are going from a larger unit to a smaller unit.
- To solve this problem, create a proportion with the unit rate of how many ounces in 1 cup and an equivalent rate for 3 cups. This will give you the measurement in ounces.
- 1 cup = 8 fl oz. or
__8 fl oz. / 1 cup__ - 8 fl oz. / 1 cup =
/ 3 cups*x*fl oz. - To change 1 cup to 3 cups we must multiply by three. The same must happen to 8 fl oz.
- 8 fl oz. × 3 / 1 cup × 3 =
__24 fl oz.__/ 3 cups - The answer is 3 cups is equivalent to 24 fluid ounces.

__Examples__

Convert the following units of measure.

**Example 1:** 4 tons = ____ pounds

- First create a proportion with the unit rate of how many pounds that are in 1 ton and find an equivalent rate for 4 tons. This will give you the measurement in pounds.
- 1 ton = 2,000 pounds or
__2,000 lb. / 1 T__ - 2,000 lb. / 1 T =
/ 4 T*x*lb. - The number of tons was multiplied by 4 to get 4 tons total. So, the number of pounds must be multiplied by 4 also.
- 2,000 lb. × 4 / 1 T × 4 =
__8,000 lb.__/ 4 T - The answer is 4 tons is equivalent to 8,000 pounds

**Example 2:** 5 feet = ____ inches

- First create a proportion with the unit rate of how many inches that are in 1 foot and find an equivalent rate for 5 feet. This will give you the measurement in inches.
- 1 foot = 12 inches or
__12 in. / 1 ft.__ - 12 in. / 1 ft. =
/ 5 ft.*x*in. - The 1 foot was multiplied by 5 to get 5 feet. The number of inches needs to also be multiplied by 5.
- 12 in. × 5 / 1 ft. × 5 =
__60 in.__/ 5 ft. - The answer is 5 feet is equivalent to 60 inches

**Example 3:** 8 pints = ____ cups

- First create a proportion with the unit rate of how many cups that are in 1 pint and find an equivalent rate for 8 pints. This will give you the measurement in cups.
- 1 pint = 2 cups or
__2 cups / 1 pint__ - 2 cups / 1 pint =
/ 8 pints*x*cups - The number of pints in the unit rate was multiplied by 8 to get 8 pints. So, the number of cups needs to be multiplied by 8 also.
- 2 cups × 8 / 1 pints × 8 =
__16 cups__/ 8 pints - The answer is 8 pints is equivalent to 16 cups.

To convert from a smaller unit to a larger unit, create a proportion using the unit rate of the conversion factor to find an equivalent rate. Let’s consider pennies.

- 5,000 pennies = ____ dollars

The penny is a smaller unit than the dollar. You need more of a smaller unit to equal a larger unit, create a proportion using the unit rate of the conversion factor to find an equivalent rate. You know that there are 100 pennies in one dollar.

- 100 pennies = 1 dollar or
__100 pennies / 1 dollar__ - Set up the proportion: 100 pennies / 1 dollar = 5,000 pennies /
*x*dollars - To determine what 100 pennies was multiplied to get 5,000 pennies, divide 5,000 by 100 = 50. So, 100 pennies was multiplied by 50 to get 5,000 pennies. That means that the 1 dollar must also be multiplied by 50, which equals 50 dollars.
- 100 pennies × 50 / 1 dollar × 50 = 5,000 pennies /
__50 dollars__ - Therefore, 5,000 pennies is equivalent to 50 dollars.

Apply this to your work with converting measurements. Remember to think about the equivalent units of length, capacity and weight when dividing.

Let's look at a conversion problem: 5,500 pounds = ____ tons

- A pound is smaller than a ton. To solve this problem, create a proportion using the unit rate of the conversion factor to find an equivalent rate.
- 2,000 pounds = 1 ton or
__2,000 lb. / 1 T__ - 2,000 lb. / 1 T = 5,500 lb. /
*x*T - To figure out what 2,000 lb. was multiplied to get 5,500 lb. you must divide 5,500 by 2,000 = 2.75. This means that 2,000 lb. was multiplied by 2.75 to get 5,500 lb. We must then do the same to the 1 T.
- 2,000 lb. × 2.75 / 1 T × 2.75 = 5,500 lb. /
__2.75 T__ - Therefore, 5,500 pounds is equivalent to 2.75 tons.

__Guided Practice__

Convert the unit of measure.

1,250 inches = _____ feet

- To solve this problem, convert inches to feet. Create a proportion using the unit rate of the conversion factor to find an equivalent rate.
- 12 inches = 1 foot or
__12 in. / 1 ft.__ - 12 in. / 1 ft. = 1,250 in. /
*x*ft. - Divide 1,250 by 12 to determine what 12 was multiplied by to get 1,250. 1,250 ÷ 12 = 104.16.
- 12 ft. × 104.16 / 1 ft. × 104.16 = 1,250 in. /
__104.16 ft.__ - The answer is 1,250 inches is equivalent to 104.16 feet.

__Examples__

Convert the following units of measure.

**Example 1:** 84 inches = ____ feet

- Create a proportion using the unit rate of the conversion factor to find an equivalent rate.
- 12 inches = 1 foot or
__12 in. / 1 ft.__ - 12 in. / 1 ft. = 84 in. /
*x*ft. - 84 ÷ 12 = 7
- 12 in. × 7 / 1 ft. × 7 = 84 in. /
__7 ft.__ - The answer is 84 inches is equivalent to 7 feet.

**Example 2: **40 cups = ____ pints

- Create a proportion using the unit rate of the conversion factor to find an equivalent rate.
- 2 cups = 1 pint or
__2 cups / 1 pint__ - 2 cups / 1 pint = 40 cups /
*x*pints - 40 ÷ 2 = 20
- 2 cups × 20 / 1 pint × 20 = 40 cups /
__20 pints__ - The answer is 40 cups is equivalent to 20 pints.

**Example 3:** 800 pounds = ____ tons

- Create a proportion using the unit rate of the conversion factor to find an equivalent rate.
- 2,000 pounds = 1 ton or
__2,000 lb. / 1 T__ - 2,000 lb. / 1 T = 800 lb. /
*x*T - 800 ÷ 2,000 = 0.4
- 2,000 lb. × 0.4 / 1 T × 0.4 = 800 lb. /
__0.4 T__ - The answer is 800 pounds is equivalent to 0.4 tons.

## Read About Converting Within the Metric System

Converting within the Metric System is the same process as converting within the Customary System. You just need to write the conversion factors as unit rates; then, create proportions to find equivalent rates.

Convert the units of measurement: 12,350 mL= ____ L

- First, identify the unit conversion. You are converting a smaller unit to a larger unit. You know that a 1,000 milliliters is one liter, or
__1,000 mL / 1 L__. - Create the proportion using the unit rate.
- 1,000 mL / 1 L = 12,350 mL /
*x*L - To determine what 1,000 was multiplied by to get 12,350 you must divide 12,350 by 1,000 which equals 12.35. So both 1,000 mL and 1 L were multiplied by 12.35 in the new equivalent rate.
- 1,000 mL × 12.35 / 1 L × 12.35 = 12,350 mL /
__12.35 L__ - 12,350 milliliters is equal to 12.35 liters.

__Examples__

Covert the units of measurement.

**Example 1:** 1,340 mL= _____ L

- First, identify the unit conversion. You are converting a smaller unit to a larger unit. 1,000 mL = 1 L or
__1,000 mL / 1 L__ - Create the proportion using the unit rate.
- 1,000 mL / 1 L = 1,340 mL /
*x*L - Then, divide 1,340 mL by 1,000 to determine what 1,000 mL was multiplied by to get 1,340 mL. 1,340 ÷ 1,000 = 1.34
- 1,000 mL × 1.34 / 1 L × 1.34 = 1,340 mL /
__1.34 L__ - 1,340 milliliters is equal to 1.34 liters.

**Example 2:** 66 g = _____ mg

- First, identify the unit conversion. You are converting a larger unit to a smaller unit. 1,000 mg = 1 g or
__1,000 mg / 1 g__ - 1,000 mg / 1 g =
/ 66 g*x*g - Since 1 grams was multiplied by 66 to get 66 grams, the 1,000 milligrams must also be multiplied by 66.
- 1,000 mg × 66 / 1 g × 66 =
__6,600 mg__/ 66 g - 66 grams is equal to 6,600 milligrams.

**Example 3:** 1,123 m = _____ km

- First, identify the unit conversion. You are converting a smaller unit to a larger unit. 1,000 m = 1 km or
__1,000 m / 1 km__ - 1,000 m / 1 km = 1,123 m /
*x*km - 1,123 ÷ 1,000 = 1.123
- 1,000 m × 1.123 / 1 km × 1.123 km = 1,123 m /
__1.123 km__ - 1,123 meters is equal to 1.123 kilometers.

Practice Converting Customary Units! Make sure you are logged into your Khan Academy!

Practice Converting within the Customary Measurement System Word Problems! Make sure you are logged into your Khan Academy!

Practice Converting Metric Units! Make sure you are logged into your Khan Academy!

Practice Converting within the Metric System Word Problems! Make sure you're logged into your Khan Academy!

## Read About Converting Between Measurement Systems

Sometimes it is necessary to convert between customary and metric units of measurements. These measurement conversions will be estimates, because you cannot make an exact measurement when converting between systems of measurement. Let’s look at an example.

Randy ran a 20 kilometer race. How many miles did Randy run?

- First, set up a proportion.
- 1.61 km / 1 mi = 20 km /
*x*mi - Next, cross multiply.
- 1.61 ×
*x*= 1 × 20 - 1.61 ×
*x*= 20 - Then, divide both sides by 1.61 to solve for
*x*. *x*× 1.61 ÷ 1.61 =*x*- 20 ÷ 1.61 = 12.42
- The answer is 12.42.
- Randy ran 12.42 miles.

__Guided Practice__

How many meters are in 67 feet?

- First, set up a proportion.
- 0.305 m / 1 ft. =
/ 67 ft.*x*m - 1 foot was multiplied by 67 to get 67 feet. So, you must do the same to 0.305 meters.
- 0.305 m × 67 / 1 ft. × 67 =
__20.435 m__/ 67 ft. - Therefore 67 feet is approximately 20.435 meters.

__Examples__

**Example 1:** John ran 5 kilometers. How many miles did he run?

- First, set up a proportion.
- 1.61 km / 1 mi = 5 km /
*x*mi - Next, cross multiply.
- 1.61 ×
*x*= 1 × 5 - 1.61 ×
*x*= 5 - Then, divide both sides by 1.61 to solve for
*x*. *x*× 1.61 ÷ 1.61 =*x*- 5 ÷ 1.61 = 3.11
- The answer is 3.11 miles.
- Therefore, 5 kilometers is approximately 3.11 miles.

**Example 2:** Kary measured out 12 inches on a ruler. About how many centimeters would that be?

- First, set up a proportion.
- 2.54 cm / 1 in. =
/ 12 in.*x*cm - 1 inch was multiplied by 12 to get 12 inches. You must multiply the 2.54 cm by 12 also to create an equivalent rate.
- 2.54 cm × 12 / 1 in. × 12 =
__30.48 cm__/ 12 in. - Therefore, 12 inches is 30.48 centimeters.

**Example 3:** Sandy ran 15 meters. About how many feet is that?

- First, set up a proportion.
- 0.305 m / 1 ft. = 15 m /
*x*ft. - Next, cross multiply.
- 0.305 ×
*x*= 1 × 15 - 0.305 ×
*x*= 15 - Then, divide both sides by 0.305 to solve for
*x*. *x*× 0.305 ÷ 0.305 =*x*- 15 ÷ 0.305 = 49.18
- The answer is 49.18.
- Therefore, 15 meters is approximately 49.18 feet.

Practice Converting Between Customary and Metric Unit Systems!

## Percents

**Learning Target:**I understand that a percent is a quantity compared to 100. I can convert between fractions decimals and percents. I can use proportional reasoning to problem solve with percents.

**Essential Questions:** What are percents? How can I convert between decimals, fractions, and percents? How can I solve problems involving percents?

## What are Percents? A percent is a quantity compared to 100. A ratio/rate in which the second quantity is 100. | ## Benchmark Fractions and Percents These are some benchmark fractions and percents that need to be MEMORIZED. Also include 1/5 = 20%. | ## Example of using Benchmark Fractions and Percents In this example you are shown how you can use benchmark fractions and percents to solve problems. You can also use benchmark fractions and percents to determine if your answer is reasonable. |

## What are Percents?

## Benchmark Fractions and Percents

*MEMORIZED*. Also include 1/5 = 20%.

## Read about Percents

**Percent** means the number of parts per 100. You know that 100% of anything is the entire or whole amount. Therefore, percents can tell you what part of the whole you are dealing with. To understand percents, you will set up ratios.

A **ratio** is a comparison of two numbers and that a ratio can be written in three ways. For example: 1 to 2, 1:2, or 1/2. Percents are ratios in which the second number (or the denominator) is 100.

Let’s look at an example.

- If there are 13 red jelly beans and 15 yellow jelly beans in a jar, the ratio of red jelly beans to yellow jelly beans can be written as 13 to 15, 13:15, or 13/15. Each of these ratios is read as “thirteen to fifteen.”

A percent is a type of ratio. You are now going to apply percents directly to ratios. A percent is a ratio that compares a number to 100. Percent means “per hundred” and the symbol for percent is %. 100% represents the ratio 100 to 100 or 1/1. Therefore, the value of 100% is 1.

Let’s look at another example.

- If there are 100 jelly beans in a jar and 19 are black, you can say that 19/100 or 19% of the jelly beans in the jar are black.

Let’s look at one more situation that uses a real-life scenario.

- There were 100 questions on a test and Amanda answered 92 of them correctly. What percent did she answer correctly? What percent did she answer incorrectly?
- Amanda answered 92 out of 100 questions correctly. First, you can write this as the ratio 92/100.
- Next, since the denominator is 100, you can write this ratio in percent form as 92%.
- Then, since there were 100 questions on the test and Amanda answered 92 correctly, you can determine that she answered 8/100, or 8 out of 100 incorrectly.
- Next, you can write this as the ratio 8/100 and as the percent 8%.
- The answer is Amanda answered 92% of the questions correctly and 8% incorrectly.

__Examples__

**Example 1:** Solve the following problem. Karen ate 12 out of 100 blueberries. What percent of the blueberries did she eat?

- First, you can write this as the ratio 12/100.
- Then, since the denominator is 100, you can write this ratio in percent form as 12%.
- The answer is she ate 12% of the blueberries.

**Example 2:** Solve the following problem. Joey answered 93 questions correctly out of 100 questions on his test. What percent of the questions did he answer correctly and what percent did he answer incorrectly?

- First, you can write this as the ratio 93/100
- Next, since the denominator is 100, you can write this ratio in percent form as 93%.
- Then, since there were 100 questions on the test and Joey answered 93 correctly, you can determine that he answered 7/100, or 7 out of 100 incorrectly.
- Next, you can write this as the ratio 7/100 and as the percent 7%.
- The answer is Joey answered 93% of the questions correctly and 7% incorrectly.

**Example 3:** Solve the following problem. Sarah gathered 25 roses out of 100 flowers. What percent of the flowers were roses?

- First, you can write this as the ratio 25/100.
- Then, since the denominator is 100, you can write this ratio in percent form as 25%.
- The answer is 25% of the flowers were roses.

## Percents as Decimals and Fractions Percents can be written as both fractions and decimals. In this example you are shown how to turn a percent into a fraction and a decimal. | ## Fractions as Decimals and Percents There are certain ways that we can take fractions and make them into decimals and percents. | ## Fractions as Decimals and Percents Example Here is an example problem of how to change fractions into decimals and percents. |

## Percents as Decimals and Fractions

## Fractions as Decimals and Percents

## Read About Converting Between Decimals and Percents

Decimals represent a part of a whole just like fractions and percents do. You are able to write fractions as percents and percents as fractions. You can also convert a decimal to a percent and a percent back to a decimal again. A decimal organizes numbers according to place value. When you write a decimal as a percent, you also keep track of the place value. A percent is a part of a whole out of 100. Well, so is a decimal.

Let’s see if you can make better sense of the connection between a decimal and a percent using the decimal 0.52.

This decimal states that you have __52 hundredths__. The percent sign means **“out of 100”** so you have __52 hundredths or 52 parts out of 100__. Once you see the connection, you can write this and all decimals as a percent. 0.52 = 52%. Both mean that you have 52 parts out of 100. You move the decimal point two places to the right to show hundredths. Percent is “out of 100,” so the **% sign is the same as two decimal places**: 0.52 = 52/100 = 52%.

Let’s look at how you write any percent as a decimal.

- First,
__drop the % symbol__. - Then,
__move the decimal point two places to the left__. - Finally, add zeros to the right of the decimal point as placeholders, if necessary.

Let’s take a look at another example.

- Write 9% as a decimal.
- First,
__drop the % sign__: 9 - Next,
__move the decimal point two places to the left__. Because 9 is only one digit, you have to__add a placeholder zero__to show the two decimal places. - 0.
__09__ - The answer is 9% written as a decimal is 0.09.

Let’s look at one final example.

- Write 35.5% as a decimal.
- First,
__drop the % sign__: 35.5 - Next,
__move the decimal point two places to the left__. - 0.
__35__5 - The answer is written as a decimal, 35.5% is 0.355.

__Guided Practice__

Write 25% as a decimal.

- First,
__drop the % sign__: 25 - Although you can’t see it, the decimal point falls immediately to the right of any whole number. Next,
__move the decimal point two places to the left__. This is the hundredths place. Remember that two places is hundredths.*You just dropped the percent sign, which means out of 100, so your answer has to reflect “out of 100” in a different way.* - 0.
__25__ - The answer is 25% written as a decimal is 0.25.

__Examples__

**Example 1: **What is 35% written as a decimal?

- First,
__drop the % sign__: 35 - Next,
__move the decimal point two places to the left__. - 0.
__35__ - The answer is 35% written as a decimal is 0.35.

**Example 2: **What is 2% written as a decimal?

- First,
__drop the % sign__: 2 - Next,
__move the decimal point two places to the left__. Because 2 is only one digit, you have to__add a placeholder zero__to show the two decimal places. - 0.
__02__ - The answer is 2% written as a decimal is 0.02.

**Example 3: **What is 18.7% written as a decimal?

First, __drop the % sign__.

18.7

Next, __move the decimal point two places to the left__.

0.__18__7

The answer is 18.7% written as a decimal is 0.187.

You can write percents as decimals, and you can also write decimals as percents.

First, move the **decimal** point two places to the right. Add zeros to the right of the decimal point as placeholders, if necessary. Then, write a % symbol after the resulting number.

Let’s apply these steps to an example. Write 0.78 as a percent.

- This decimal is already written in
__hundredths__because there are__two digits after the decimal point__. All you have to do is**move the decimal two places to the right and add a percent sign**. - 0.
__78__= 78% - The answer is 0.78 written as a percent is 78%.

Let’s try another example. Write 0.345 as a percent.

- This decimal is written in
__thousandths__because there are__three digits after the decimal point__. You only need to**move the decimal point two places to the right to make it a percent**, so*you will have a digit left over that remains after the decimal point*. - 0.
__34__5 = 34.5% - The answer is 0.345 written as a percent is 34.5%.

Notice that sometimes **you can have a decimal in a percent**. In this case, it means that you have 34 and one-half percent. **Not all percents are whole percents**.

Let’s try one more example. Write 3.5 as a percent.

- This decimal is written with a
__whole number and five tenths__. You still**move the decimal point to the right two places**,__adding a zero as a placeholder__. Notice that*because you have a whole number in front of the decimal point that the percent will be greater than 100*. - 3.5 = 3.
__50__= 350% - The answer is 3.5 written as a percent is 350%.

__Guided Practice__

Write 0.6 as a percent.

- This decimal is written in
__tenths__because there is only__one digit after the decimal point__. When you**move the decimal two places to the right**, you will therefore__need to add a zero placeholder__. - 0.6 = 0.
__60__= 60% - The answer is 0.6 written as a percent is 60%.

__Examples__

**Example 1:** Write 0.45 as a percent.

- This decimal is already written in
__hundredths__because there are two digits after the decimal point. Therefore, all you have to do is**move the decimal two places to the right**and add a percent sign. - 0.
__45__= 45% - The answer is 0.45 written as a percent is 45%.

**Example 2: **Write 2.5 as a percent.

- This decimal is written with a
__whole number and five tenths__. You still**move the decimal point to the right two places**,__adding a zero as a placeholder__. Notice that*because you have a whole number in front of the decimal point that the percent will be greater than 100*. - 2.5 = 2.
__50__= 250% - The answer is 2.5 written as a percent is 250%.

**Example 3: **Write 0.875 as a percent.

- This decimal is written in
__thousandths__because there are__three digits__after the decimal point. You only need to**move the decimal point two places to the right**to make it a percent, so*you will have a digit left over that remains after the decimal point*. - 0.
__87__5 = 87.5% - The answer is 0.875 written as a percent is 87.5%.

## Read about Converting Between Fractions and Percents

A fraction can be written as a percent if it has a denominator of 100. Sometimes, you will be given a fraction with a denominator of 100 and sometimes you will have to rewrite the fraction to have a denominator of 100 before you can write it as a percent.

For example: 9/100 This fraction is already written with a denominator of 100, so you can just change it to a percent. 9/100 = 9%

A **proportion** is two equal ratios. If a fraction does not have a denominator of 100, you can write a fraction equal to it that does have a denominator of 100 and then solve the proportion.

Let’s look at an example.

- Write 3/5 as a percent.
- First, notice that the denominator is not 100. Therefore, you need to create a new fraction equivalent to this one with a denominator of 100.
- Next, set up a proportion.
- 3/5 =
*x*/100 - Then, you can cross multiply to find the value of
*x*. - 3/5 =
*x*/100 - 3 times 100 = 5 times
*x* - 300 = 5 times
*x* - 300 divided by 5 =
*x*times 5 divided by 5 - 60 =
*x* - 3/5 = 60/100
- Now you have a fraction with a denominator of 100, and you can write it as a percent.The answer is that the fraction is equal to 3/5 = 60%.

To work with an **improper fraction**, you have to think about what improper means. An **improper fraction** is greater than 1, so the percent would be greater than 100%. Sometimes you can have percents that are greater than 100%. Most often they are not, but it is important to understand how to work with a percent that is greater than 100%.

You already know some common fraction equivalents for percents. Think of 25 cents, 50 cents, and 75 cents. 25 cents means 25 cents out of a dollar, or 25% of a dollar. Since a quarter is 25 cents,

- 1/4 = 25%.
- 50 cents means 50 cents out of a dollar, or 50% of a dollar. Since a half dollar is 50 cents, 1/2 = 50%.
- 75 cents means 75 cents out of a dollar, or 75% of a dollar. Since three quarters of a dollar is 75 cents, 3/4 = 75%.

Let’s look at an example with a fraction that doesn‘t convert easily to a percent.

- Write 2/3 as a percent.
- First, set up the proportion.
- 2/3 =
*x*/100 - Next, cross multiply to solve for the value of
*x*. - 3 times
*x*= 2 times 100 - 3 times
*x*= 200 *x*times 3 divided by 3 = 200 divided by 3*x*= 66.6...- Notice that you end up with a decimal and it is a repeating decimal. If you keep dividing, you will keep ending up with 6s. Therefore, you can leave this percent with one decimal place represented.
- So 2/3 = 66.6/100
- The answer is 66.6%.

Sometimes, you will see fractions like this, but you will get used to them and often you can learn the percent equivalents of these fractions by heart.

As a final example, let’s take a look at a real-life word problem.

- James ate three out of ten pieces of pizza. What percent of the pizza did he eat? What percent didn’t he eat?
- First, let’s write a fraction to show the part of the pizza that James did eat.
- 3/10

- Next, you convert that to a fraction out of 100 by setting up a proportion.
- 3/10 =
*x*/100 - Then you can write it as a percent.
- 3 times 100 = 10 times
*x* - 300 = 10 times
*x* - 300 divided by 10 =
*x*times 10 divided by 10 - 30 =
*x* - So, 3/10 = 30/100 = 30%
- The answer is James ate 30% of the pizza, and James did not eat 70% of the pizza.

__Guided Practice__

Write 9/4 as a percent.

- First, you write a proportion with a denominator of 100.
- 9/4 =
*x*/100 - Next, you cross multiply to find the value of
*x*. - 9 times 100 = 4 times
*x* - 900 = 4 times
*x* - 900 divided by 4 =
*x*times 4 divided by 4 - 225 =
*x* - So, 9/4 = 225/100 = 225%
- The answer is 94 is equal to 225%.

__Examples__

**Example 1: **Write 1/4 as a percent.

- First, set up the proportion.
- 1/4 =
*x*/100 - Next, cross multiply to solve for the value of
*x*. - 25 =
*x* - 1/4 = 25/100
- The answer is 25%.

**Example 2:** Write 25 as a percent.

- First, set up the proportion.
- 2/5 =
*x*/100 - Next, cross multiply to solve for the value of
*x*. - 40 =
*x* - 2/5 = 40/100
- The answer is 40%.

**Example 3:** Write 4/40 as a percent.

- First, set up the proportion.
- 4/40 =
*x*/100 - Next, cross multiply to solve for the value of
*x*. - 10 =
*x* - 4/40 = 10/100
- The answer is 10%.

Practice Converting Fractions, Decimals, and Percents! Make sure you are logged into your Khan Academy!

Practice Converting Fractions, Decimals, and Percents! Make sure you are logged into your Khan Academy!

## Comparing Fractions, Decimals, and Percents

To compare and order rational numbers, you should first convert each number to the same form so that they are easier to compare. Usually it will be easier to convert each number to a decimal. Then you can use a number line to help you order the numbers. To order the rational numbers from greatest to least, or least to greatest, you need to use **inequality** signs. The inequality signs are:

- Greater than: >
- Less than: <
- Greater than or equal to: ≥
- Less than or equal to: ≤

Let’s look at an example. Place the following number in order from least to greatest: 1/8, 8%, and 0.8.

- First, convert 1/8 and 8% into decimals.
- 1/8 = 0.125
- 8% = 0.08
- Next, place the three decimals on a number line between 0 and 1. Then, since you have compared the numbers by placing them on a number line, you can order the rational numbers.
- The answer is 8% < 1/8 < 0.8.

Here is another example. Which inequality symbol correctly compares 0.29% to 0.029?

- First, convert the percent into a decimal.
- 0.29% = 0.0029
- Next, compare 0.0029 to 0.029.
- 0.0029 < 0.029
- So, 0.29% < 0.029 .

Remember, **the key to comparing and ordering rational numbers is to be sure that they are all in the same form**. You want to have all fractions, all decimals or all percentages so that your comparisons are accurate. __You may need to convert before you compare!!__

__Guided Practice__

Order the following rational numbers from least to greatest: 0.5%, 0.68 , 3/15

- First, convert them all to the same form. You could use fractions, decimals or percent, but for this situation,
__let’s use percent__. - Since 0.5% is in the percent form, it is done.
- 0.68 = 68%
- 3/15 = 20%
- Next, order the numbers. 0.5% < 20% < 68%. Before writing your final answer be sure to write them as they first appeared.
- So the answer is 0.5% < 3/15 < 0.68.

__Examples__

**Example 1:** Which inequality symbol correctly compares 0.56 to 4/5?

- First, convert them all to the same form. You could use fractions, decimals or percent, but for this situation,
__let’s use fractions__. - Since 4/5 is in the fraction form, it is done.
- 0.56 = 56/100 = 14/25
- Next, get a common denominator so you can compare the fractions.
- 4/5 = 16/25
- 0.56 = 14/25
- Then, order the numbers. Be sure to write them as they first appeared. 0.56 < 4/5
- The answer is 0.56<45.

**Example 2:** Which inequality symbol correctly compares 0.008 to 0.8%?

- First, convert them all to the same form. You could use fractions, decimals or percent, but for this situation,
__let’s use percent__. - Since 0.8% is in the percent form, it is done.
- 0.008 = 0.8%
- Next, order the numbers. Be sure to write them as they first appeared. 0.008 = 0.8%
- The answer is 0.008=0.8%.

Practice Comparing Fractions, Decimals, and Percents

## Solving Percent Problems

**Learning Target:**I can solve real world percent problems using proportional reasoning.

**Essential Question: **How can you use proportional reasoning to solve real world percent problems?

## Example 1 Problem A Example of a proportion being used to find a part when given a whole and a percent. | ## Example 1 Problem B Example of multiplication of the percent written as a fraction to find a part when given a whole and a percent. | ## Example 1 Problem C Example of multiplication of the percent written as a decimal to find a part when given a whole and a percent. |

## Example 1 Problem A

## Example 1 Problem B

## Read About Solving Percent Problems to Find a Part

In previous lessons, you were shown how to convert a decimal to a percent and a percent to a decimal. Thus, if you were asked to **Find 15% of 120**, you would multiply 0.15 by 120, to get an answer of 18. But what would you do if you given this problem: **8 is what percent of 20?** In this problem, the percent is the unknown quantity! We need to figure out how to find this unknown quantity.

Every statement of percent can be expressed verbally as: "*One number is some percent of another number.*" Percent statements will always involve three numbers. For example:

____ is ____ % of ____.

In the problem, **8 is what percent of 20?,** the number 8 is __some percent__ of the number 20.** **Looking at this problem, it is clear that 8 is the __part__ and 20 is the __whole__. Similarly, in the statement, "*One number is some percent of another number.*", the phrase "*one number*" represents the __part__ and "*another number*" represents the __whole__. Thus the statement, "*One number is some percent of another number,*" can be rewritten:

*"One number is some percent of another number"*becomes,*"The part is some percent of the whole."*

From previous lessons we know that the word "is" means __equals__ and the word "of" means __multiply__. Thus, we can rewrite the statement above:

The statement:*"The part is some percent of the whole."*, becomes the equation:

**the part =****some percent****x the whole**

Since a percent is a whose second term is 100, we can use this fact to rewrite the equation above as follows:

*the part = some percent x the whole*becomes:**the part = percent/100 x the whole**

Dividing both sides by __"the whole"__ we get the following proportion:

**PART / WHOLE = PERCENT / 100**

Since percent statements always involve **three numbers**,__ given any two of these numbers__, **we can find the third** __using the proportion above__. Let's look at an example of this.

**Problem 1: **If 8 out of 20 students in a class are boys, what percent of the class is made up of boys?

__Analysis:__In this problem, you are being asked**8 is what percent of 20?**You are given two numbers from the proportion above and asked to find the third. The percent is the unknown quantity in this problem. We need to find this unknown quantity.__Identify:__The phrase**8 is**means that 8 is the part.- The phrase
**what percent**tells us that percent is the unknown quantity. This unknown quantity will be represented by*x*in our proportion. - The phrase
**of 20**means that 20 is the whole. __Substitute:__Now we can substitute these values into our proportion.**PART / WHOLE = PERCENT / 100**becomes 8/20 =*x*/100__Solve:__*x*= 800- Divide both sides by 20 to solve for
*x*and we get:*x*= 40 __Solution:__8 is 40% of 20. Therefore, 40% of the class is made up of boys.

Note that in Problem 1 we did not have to cross multiply to solve the proportion. We could have used equivalent fractions instead (i.e., since 20 multiplied by 5 equals 100, we get that 8 multiplied by 5 equals *x*, so *x* equals 40).

In Problem 1 we were asked **8 is what percent of 20?** and we found the solution by substituting into a proportion. But how would we solve this problem: **18 is 40% of what number?** and how would we solve this problem: **What is 20% of 45? **We will look at these last two problems below.

**Problem 2: **18 is 40% of what number?

__Identify:__The phrase**18 is**means that 18 is the part.**40%**means that 40 will replace percent in our proportion.- The phrase
**of what number**represents the whole and is the unknown quantity. We will let variable*x*represent this unknown quantity in our proportion. __Substitute:__Now we can substitute these values into our proportion.**PART / WHOLE = PERCENT / 100**becomes 18/*x*= 40/100__Solve:__Cross multiply and we get: 40*x*= 18(100) or 40*x*= 1800- Divide both sides by 40 to solve for
*x*and we get:*x*= 45 __Solution:__18 is 40% of 45

**Problem 3:**What is 20% of 45?

__Identify:__The phrase**what is**means represents the part and is the unknown quantity. We will let variable*x*represent this unknown quantity in our proportion.**20%**means that 20 will replace percent in our proportion.- The phrase
**of 45**means that 45 is the whole. __Substitute:__Now we can substitute these values into our proportion.**PART / WHOLE = PERCENT / 100**becomes*x*/45 = 20/100__Solve:__Cross multiply and we get: 100*x*= 45(20) or 100*x*= 900- Divide both sides by 100 to solve for
*x*and we get:*x*= 9 __Solution:__9 is 20% of 45

In Problems 1, 2 and 3 we are given two numbers and asked to find the third by using a proportion. However, the unknown quantity was different for each problem. Let's compare these problems.

In Problem 1 we let *x* represent the unknown quantity "**what percent**"; in Problem 2 we let *x* represent the unknown quantity "**of what number**"; and in Problem 3 we let *x* represent the unknown quantity "**What is."** Thus, we solved three different percent problems, where in each problem, two numbers were given and we were asked to find the third. We did this by letting a variable represent the unknown quantity and then substituting the given values into a proportion to solve for the unknown quantity.

Note that in all three percent statements, __the whole always follows the word "of"__ and __the part always precedes the word "is."__ This is not surprising since our original statement is, "*One number is some percent of another number." *Thus, we can revise our proportion as follows:

**PART / WHOLE = PERCENT / 100**becomes**IS (part) / OF (whole) = PERCENT / 100**

**Problem 4:** What is 25% of 52?

__Identify:__25% means that 25 will replace**PERCENT**in our proportion.- 52 is the whole and will replace
**OF**in our proportion. - The part is the unknown quantity and will be represented by
*p*in our proportion. __Substitute:__Now we can substitute these values into our proportion.

**IS (part) / OF (whole) = PERCENT / 100**becomes*p*/52 = 25/100__Solve:__Cross multiply and we get: 100*p*= 52(25) or 100*p*= 1300- Divide both sides by 100 to solve for
*p*and we get:*p*= 13 __Solution:__13 is 25% of 52

Note that we could restate this problem as, "**Find 25% of 52", **and get the same answer. However, in the interest of consistency, we will use proportions to solve percent problems throughout this lesson. In Problems 5 through 7, we will use *n *to represent the unknown quantity.

**Problem 5:** What percent of 56 is 14?

__Identify:__56 is the whole and will replace*OF*in our proportion.- 14 is the part and will replace
*IS*in our proportion. *PERCENT*is the unknown quantity in our proportion, to be represented by*n*.__Substitute:__**IS (part) / OF (whole) = PERCENT / 100**becomes 14/56 =*n*/100__Solve:__Cross multiply and we get: 56*n**n*= 1400- Divide both sides by 56 and we get:
*n*= 25 __Solution:__25% of 56 is 14

**Problem 6:**18 is 75% of what number?

__Identify:__18 is the part and will replace*IS*in our proportion.- 75% means that 75 will replace
*PERCENT*in our proportion. - The
*whole*is the unknown quantity in our proportion, to be represented by*n*. __Substitute:__**IS (part) / OF (whole) = PERCENT / 100**becomes 18/*n*= 75/100__Solve:__Cross multiply and we get: 75*n*= 18(100) or 75*n*= 1800- Divide both sides by 75 and we get:
*n*= 24 __Solution:__18 is 75% of 24

**Problem 7:** What is 15% of 200?

__Identify:__15% means that 25 will replace*PERCENT*in our proportion.- 200 is the
*whole*and will replace*OF*in our proportion. - The
*part*is the unknown quantity in our proportion, to be represented by*n* __Substitute:__**IS (part) / OF (whole) = PERCENT / 100**becomes*n*/200 = 15/100__Solve:__Cross multiply and we get: 100*n*= 200(15) or 100*n*= 3000- Divide both sides by 100 and we get:
*n*= 30 __Solution:__30 is 15% of 200

Now that we have solved a number of percent problems using proportions, we can go back to the type of problem presented at the beginning of this lesson. In Problems 8 through 10 we will solve real world problems, using different variables to represent the unknown quantity in each problem.

**Problem 8:** At Little Rock School, 476 students ride their bike to school. If this number is 85% of the school enrollment, then how many students are enrolled?

__Identify:__This problem can be rewritten as**476 is 85% of what number?**- 476 is the part and will replace
*IS*in our proportion. - The
*percent*given is 85%. - The
*whole*is the unknown quantity, so*y*will*OF*in our proportion. __Substitute:__**IS / OF = PERCENT / 100**becomes 476 /*y*= 85 / 100 Solve:Cross multiply and we get: 85*y*= 47600- Divide both sides by 85 and we get:
*y*= 560 __Solution:__There are 560 students enrolled at Little Rock School.

**Problem 9:** A football team won 75% of 120 games in a season. How many games is that?

__Identify:__This problem can be rewritten as**What is 75% of 120?**- 120 is the whole and will replace the
*OF*in our proportion. - The
*percent*given is 75%. - The
*part*is the unknown quantity, so*p*will*IS*in our proportion. __Substitute:__**IS / OF = PERCENT / 100**becomes*p*/120 = 75/100__Solve:__Cross multiply and we get: 100*p*= 9000- Divide both sides by 85 and we get:
*p*= 90 __Solution:__The team won 90 games.

**Problem 10:** Jennie has $300 and she spends $15. What percent of her money is spent?

__Identify:__This problem can be rewritten as**$15 is what percent of $300?**- 15 is the
*part*and will replace the*IS*in our proportion. - 300 is the
*whole*and will replace the*OF*in our proportion. *Percent*is the unknown quantity, so*x*will represent the*PERCENT*in our proportion.__Substitute:__IS / OF = PERCENT / 100 becomes 15/300 =*x*/100__Solve:__Cross multiply and we get: 300*x*= 1500- Divide both sides by 300 and we get:
*x*= 5 __Solution:__Jennie spent 5% of her money.

**Summary:** Every statement of percent can be expressed verbally as: "*One number is some percent of another number.*" Percent statements will always involve three numbers. Given two of these numbers, we can find the third by substituting into one of the proportions below.

__PART / WHOLE = PERCENT / 100__**OR**__IS (part) / OF (whole) = PERCENT / 100__

In this lesson, we solved percent problems using proportions by following this procedure:

- Read the percent problem.
- Identify what information is given.
- Identify what information is unknown.
- Use a variable to represent the unknown quantity.
- Set up a proportion for the problem by substituting the given information and the variable into one of the proportions listed above.
- Evaluate and solve the proportion in Step 5 to find the unknown quantity.

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