Sample Algebra I Lesson: Ratios Proportions and Unit Rates

 

Objectives

At the end of this lesson you will be able to:

  1. Use ratios to compare two numbers
  2. Determine if a proportion is true
  3. Solve applications involving proportions
  4. Solve applications involving unit rates

Prior Knowledge

  • Multiplying Whole Numbers
  • Dividing Whole Numbers
  • Solving Equations

LESSON SUMMARY

When a fraction is used to compare two numbers, we call it a ratio. We can write ratios in several ways. If 2 out of every 5 people will vote in the next election, we can write the ratio of voters to the number of people. We can write this ratio one of two ways shown below.

The ratio of actual voters compared to the number of people is 2:5 or \(\frac25\)

A statement that two ratios are equal is called a proportion. If the statement is true, then when we cross-multiply, the cross-products will be equal.

In the proportion below, the cross products are found by multiplying the numerator of one fraction times the denominator of the other fraction. For the proportion below, the cross products are 2 x 10 and 4 x 5.

\(\frac25=\frac4{10}\)

The cross products are 2 x 10 and 4 x 5. Each cross-product is equal to 20, so this is a true proportion. If the cross products are not equal, the proportion is not true.

In problems involving proportions, we are always given three pieces of information and asked to find the fourth. We can use a letter, such as N, to represent the missing number. Then, we can set up a proportion, set the cross products equal to each other, and solve the equation.

Next, we can solve a proportion for an unknown quantity. In the school cafeteria, 1 out of 4 people order an ice cream cone. The cafeteria serves 300 people a day. How many ice cream cones will they sell in one day? Let’s call the answer n. We are saying 1 compares to 4 the same way that n compares to 300. We can write this as the following proportion.

\(\frac25=\frac n{300}\)

We have 2 ratios, 1 over 4 and n over 300, and we want to solve for n. The cross products are 1 times 300, which is 300, and 4 times n, which can be written as 4n.

300 = 4n

So, we have the equation 300 equals 4n. We want to solve for n, so we divide both sides by 4.

\(\frac{300}4=\frac{4n}4\)

Now 300 divided by 4 is 75 and 4n divided by 4 is n. So, we have 75 equals n. They sell 75 ice cream cones a day.

75 = n

Finally, a unit rate is a ratio with a denominator of 1. Examples of unit rates are miles per hour and words per minute. Let’s look at an application involving unit rates.

Jose can type 20 words in 40 seconds. How many words per minute can Jose type?

We can use proportions to solve this problem. We first need to recognize that there are 60 seconds in 1 minute. We will use this in our proportion since we know that Jose types 20 words in 40 seconds. We want to find how many words Jose can type in 60 seconds.

Next, we can set up a proportion to solve the problem. Jose can type 20 words in 40 seconds, and we can write this as 20 words / 40 seconds. This will be the fraction on the left side of the proportion.

Since we want to find his typing speed in words per minute, and there are 60 seconds in one minute, we can write the fraction x words / 1 minute as x words / 60 seconds. This will be the fraction on the right side of the proportion. We need to solve the equation 20 words /40 seconds = x words / 60 seconds, which is \(\frac{20}{40}=\frac x{60}\).

So, we have the following equation \(\frac{20}{40}=\frac x{60}\), and we want to solve for x.

Cross multiplying, we have 20(60) = 40x, or 1200=40𝑥 .

Dividing both sides by 40, we have \(\frac{1200}{40}=\frac{40x}{40}\). Simplifying both sides, we get 30 = x. So, we have x = 30, which means Jose can type 30 words in 60 seconds, which is 30 words per minute.

 

Examples

1. Are these ratios equal or not equal? \(\frac25\;and\;\frac3{10}\)

Solution: To see if these ratios are equal, we can write them as a proportion and find the cross products of \(\frac25\;and\;\frac3{10}\). The cross products are 2 x 10 and 3 x 5. 2 x 10 is 20 and 3 x 5 is 15. Since these are not equal, the proportion is false and the ratios are not equal.

2. Is this proportion true or false? \(\frac35\;=\;\frac6{10}\)

Solution: To see if this proportion is true, we find the cross products of \(\frac35\;=\;\frac6{10}\). The cross products are 3 x 10 and 6 x 5. 3 x 10 is 30 and 6 x 5 is also 30. Since these are equal, the proportion is true and the ratios are equal.

3. What is the value of N in this proportion? \(\frac25\;=\;\frac n{20}\)

Solution: To solve this proportion, we find the cross products for \(\frac25\;=\;\frac n{20}\). The cross products are 2 x 20 and N x 5. Setting the cross products equal to each other gives us 40 = 5N. To solve, we divide both sides by 5, which gives us 40/5 = 5N/5. So we have 8 = N, which is the answer.

4. If three t-shirts cost 15 dollars, how much would nine t-shirts cost?

Solution: To solve this application, we set up the proportion \(\frac3{15}\;=\;\frac9c\), where C is the cost of 9 t-shirts. To solve this, we set the cross products equal to each other, giving us 3C = 9 x 15. 9 x 15 is 135, so we have 3C = 135. Dividing both sides by 3, we get \(\frac{3c}3\;=\;\frac{135}3\), or C = 45. The answer is that 9 t-shirts will cost a total of $45.

5. Janet rides her bike 9 miles in 20 minutes. Find her speed in miles per hour.

Solution: We want to find Janet’s speed in miles per hour. Since there are 60 minutes in 1 hour, we want to find how far Janet can ride her bike in 60 minutes. To solve this application, we set up the proportion \(\frac9{20}\;=\;\frac D{60}\). Setting the cross products equal, we have 9 x 60 and D x 20. Setting the cross products equal to each other, we have 540 = 20D. Diving both sides by 20 give us
\(\frac{540}{20}\;=\;\frac{20D}{20}\). Simplifying both sides, we get 27 = D, so Janet can ride her bike 27 miles in 60 minutes, which is 27 miles per hour.

Practice Problems

1. Are these ratios equal or not equal? \(\frac4{16}\;and\;\frac5{20}\)<\p>

2. Is this proportion true or false? \(\frac45\;=\;\frac{15}{20}\)

3. Solve the following proportion. \(\frac45\;=\;\frac n{30}\)

4. If four t-shirts cost 12 dollars, how much would twenty t-shirts cost?

5. Olivia rides her bike 6 miles in 24 minutes. Find her speed in miles per hour.

Answers to Practice Problems

1. The ratios are equal.

2. The proportion is false.

3. N = 24

4. Twenty t-shirts would cost $60.

5. Olivia rides her bike 15 miles per hour.

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