# Lesson 2

Ratios and Rates With Fractions

Let’s calculate some rates with fractions.

### 2.1: Number Talk: Division

Find each quotient mentally.

$$5\div\frac13$$

$$2\div\frac13$$

$$\frac12\div\frac13$$

$$2\frac12\div\frac13$$

### 2.2: A Train is Traveling at . . .

A train is traveling at a constant speed and goes 7.5 kilometers in 6 minutes. At that rate:

1. How far does the train go in 1 minute?
2. How far does the train go in 100 minutes?

### 2.3: Comparing Running Speeds

Lin ran $$2 \frac34$$ miles in $$\frac25$$ of an hour. Noah ran $$8 \frac23$$ miles in $$\frac43$$ of an hour.

1. Pick one of the questions that was displayed, but don’t tell anyone which question you picked. Find the answer to the question.
2. When you and your partner are both done, share the answer you got (do not share the question) and ask your partner to guess which question you answered. If your partner can’t guess, explain the process you used to answer the question.
3. Switch with your partner and take a turn guessing the question that your partner answered.

Nothing can go faster than the speed of light, which is 299,792,458 meters per second. Which of these are possible?

1. Traveling a billion meters in 5 seconds.

2. Traveling a meter in 2.5 nanoseconds. (A nanosecond is a billionth of a second.)

3. Traveling a parsec in a year. (A parsec is about 3.26 light years and a light year is the distance light can travel in a year.)

### 2.4: Scaling the Mona Lisa

In real life, the Mona Lisa measures $$2 \frac12$$ feet by $$1 \frac34$$ feet. A company that makes office supplies wants to print a scaled copy of the Mona Lisa on the cover of a notebook that measures 11 inches by 9 inches.

The applet is here to help you experiment with the situation. (It won't solve the problems for you.) Use the sliders to scale the image and drag the red circle to place it on the book. Measure the side lengths with the Distance or Length tool.

1. What size should they use for the scaled copy of the Mona Lisa on the notebook cover?

2. What is the scale factor from the real painting to its copy on the notebook cover?

3. Discuss your thinking with your partner. Did you use the same scale factor? If not, is one more reasonable than the other?

### Summary

There are 12 inches in a foot, so we can say that for every 1 foot, there are 12 inches, or the ratio of feet to inches is $$1:12$$. We can find the unit rates by dividing the numbers in the ratio:

$$1\div 12 = \frac{1}{12}$$
so there is $$\frac{1}{12}$$ foot per inch.

$$12 \div 1 = 12$$
so there are 12 inches per foot.

The numbers in a ratio can be fractions, and we calculate the unit rates the same way: by dividing the numbers in the ratio. For example, if someone runs $$\frac34$$ mile in $$\frac{11}{2}$$ minutes, the ratio of minutes to miles is $$\frac{11}{2}:\frac34$$.

$$\frac{11}{2} \div \frac34 = \frac{22}{3}$$, so the person’s
pace is $$\frac{22}{3}$$ minutes per mile.

$$\frac34 \div \frac{11}{2} = \frac{3}{22}$$, so the person’s
speed is $$\frac{3}{22}$$ miles per minute.

### Glossary Entries

• percentage

A percentage is a rate per 100.

For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.

• unit rate

A unit rate is a rate per 1.

For example, 12 people share 2 pies equally. One unit rate is 6 people per pie, because $$12 \div 2 = 6$$. The other unit rate is $$\frac16$$ of a pie per person, because $$2 \div 12 = \frac16$$.