Unit 3 Unit Rates And Percentages — Unit Plan

Suppose a farm lets us pick 2 pounds of blueberries for 5 dollars. We can say:

A unit rate tells us how much of one quantity for 1 of the other quantity. Each of these numbers is useful in the right situation.

Which unit rate is most useful depends on what question we want to answer, so be ready to find either one!

The table shows different amounts of apples selling at the same rate. This means that all of the ratios of weight (in pounds) to price (in dollars) are equivalent. 

We can find the unit price in dollars per pound by dividing the price (in dollars) by the weight of apples (in pounds).

In each case, the unit price is always the same. Whether we buy 4 pounds of apples for 10 dollars or 8 pounds of apples for 20 dollars, the apples cost 2.50 dollars per pound.

This is true in all situations: When two ratios are equivalent, their unit rates will be equal.

Sometimes we can find and use more than one unit rate to solve a problem.

Suppose a small bag of powder detergent holds 16 ounces and is sold for $2. A large bag that holds 2 kilograms is sold for $8. Which is a better deal? 

Because the bags are in different units of weight, it helps to make comparisons using the same unit. Here are two different ways:

Another way to solve the problem would be to compare the unit prices of each bag in dollars per ounce. Try it!

There are many real-world situations in which something keeps happening at the same rate. In these situations, we can use equivalent ratios or unit rates to make predictions or to answer questions about the quantities.

For example, the school cafeteria serves 600 students in 40 minutes. At this rate, how long will it take the cafeteria to serve 750 students?

We can use a table or a double number line diagram to find ratios that are equivalent to the given ratio.

Both the double number line diagram and table show that it will take the cafeteria 50 minutes to serve 750 students.

Double number line, 4 evenly spaced tick marks. Top line, number of students. Beginning at first tick mark, labels: 0, 150, 300, 450, 600, 750, 900. Bottom line, time in minutes. Beginning at first tick mark, labels: 0, 10, 20, 30, 40, 50, 60. 

How many students can the cafeteria serve in 27 minutes?

In this case, it is helpful to find a unit rate—the number of students the cafeteria can serve per minute. Dividing the number of students, 600, by the number of minutes, 40, gives us this unit rate.  $600 \div 40 = 15$, so the cafeteria can serve 15 students per minute. This means that in 27 minutes it can serve $27 \boldcdot 15$, or 405 students.

Percent means “per 100.” A percentage is a rate per 100.

We can use percentages to describe the relationship between two quantities. For instance:

• The value of a quarter is 25 percent of a dollar because there are 25 cents in a quarter per 100 cents in a dollar. The ratio of their values is 25 to 100.
• The value of a half-dollar coin is 50 percent of a dollar because there are 50 cents in a half-dollar coin per 100 cents in a dollar. The ratio of their values is 50 to 100.
• The value of five quarters is 125 percent of a dollar because there are 125 cents in five quarters per 100 cents in a dollar. The ratio of their values is 125 to 100.

Here is a double number line diagram that can represent these coin values. Because we’re representing percentages of a dollar, 100% is aligned to $1 on the number line.

Sometimes we are interested in percentages of an amount other than 100 or 1. For example, what is 30% of 50 pounds? We can use a double number line diagram to solve problems about percentages.

Because we are looking for a percentage of 50 pounds, 100% is aligned to 50 pounds on the double number line diagram, like this:

We divide the distance between 0% and 100% and that between 0 and 50 pounds into ten equal parts. The tick marks on the top line can be labeled by counting by 5s ($50\div10=5$). Those on the bottom line can be labeled by counting by 10% ($100\div10=10$). We can see that 30% of 50 pounds is 15 pounds.

Double number line diagrams can also help us find the value of 100% when we know the value of another percentage.

Suppose Mai read for 90 minutes on Monday and this was 125% as much time as she spent reading on Sunday. How long did she read on Sunday?

In this case, the value we’re looking for is 100% of the number of reading minutes on Sunday. On a double number line diagram, we can align 90 minutes and 125%. Then, we can divide the interval between 0 and 90 and between 0 and 125% into five equal parts.

Each part on the top line represents 18 minutes ($90\div 5=18$) and each part on the bottom line represents 25% ($125 \div 5 = 25$).

From the diagram, we can see that 72 minutes corresponds to 100%, so Mai read for 72 minutes on Sunday.

Tables and tape diagrams can also help us make sense of percentages.

Consider two problems that we solved earlier using a double number line diagram: 

• What is 30% of 50 pounds? Here is a tape diagram that shows that 30% of 50 pounds is 15 pounds.

  

• Mai spent 90 minutes reading on Monday. This is 125% as much time she spent reading on Sunday. How long did she read on Sunday? 

  In other words: If 90 is 125% of a number, what is 100% of that number? A table can help us reason about problems like this.

  Here is one that shows that 100% of that number must be 72.

  

In a situation that involves percentages, there are often three questions we are interested in answering.

Suppose a tank is filled with some water.

1. If we know that the tank is 25% filled and can hold 36 liters, we can ask: What is 25% of 36 liters?
2. If we know that the tank has 9 liters and is 25% filled, we can ask: How many liters are in a full tank?
3. If we know that the tank has 9 liters but can hold 36 liters when full, we can ask: What percentage of 36 liters is 9 liters?

We can use a double number line diagram, a table, or a tape diagram to help us reason about each question.

We can also use our knowledge of fractions or relationships between numbers. For instance, we know that 9 is $\frac{1}{4}$ of 36, or $36 \div 4$, so it is $\frac{1}{4}$ of 100% or 25%.

In general, in a situation where $A$ of $B$ is $C$, we can find the value of $A$, $B$, or $C$ if we know the other two values.

Suppose a business donates 1% of its profits to charity each year. How much would it donate if it made $7,500 in profits?

To find 1% of 7,500, we can multiply 7,500 by $\frac{1}{100}$ or 0.01.

$\frac{1}{100} \boldcdot 7,500 = 75$, so the business would donate $75.

What if the business donates 6% of its profits to charity? Because 6% of 7,500 is 6 times as much as 1% of 7,500, we can calculate $6 \boldcdot \frac{1}{100} \boldcdot 7,500$ or $\frac{6}{100} \boldcdot 7,500$.

$\frac{6}{100} \boldcdot 7,500 = 450$, so the business would donate $450.

The same reasoning can help us find 1%, 6%, and other percentages of another number:

• To find 1% of a number, we can multiply that number by $\frac{1}{100}$ or 0.01.
• To find 6% of a number, we can multiply the number by $\frac{6}{100}$ or 0.06.
• To find 49% of a number, we can multiply the number by $\frac{49}{100}$ or 0.49.
• To find 135% of a number, we can multiply the number by $\frac{135}{100}$ or 1.35.

In general, to find $P\%$ of any number, $x$, we can calculate: $\frac{P}{100}\boldcdot x$.