1. Ratios and Rates
Lesson 4.2.1

2. Ratios and Rates 4.2.1 California Standard: What it means for you:
Lesson
4.2.1
Ratios and Rates
California Standard:
Measurement and Geometry 3.2
Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.
What it means for you:
You’ll learn what rates are and how you can use them to compare things — such as which size product is better value.
Key words:
rate
ratio
fraction
denominator
unit rate

3. Lesson
4.2.1
Ratios and Rates
Rates are used a lot in daily life. You often hear people talk about speed in miles per hour, or the cost of groceries in dollars per pound.
A rate tells you how much one thing changes when something else changes by a certain amount.
Imagine buying apples for $2 per pound — the cost will increase by $2 for every pound you buy.
Best apples $2 per pound

4. Ratios and Rates 4.2.1 Ratios are Used to Compare Two Numbers
Lesson
4.2.1
Ratios and Rates
Ratios are Used to Compare Two Numbers
You might remember ratios from grade 6. Ratios compare two numbers, and don’t have any units.
For example, the ratio of boys to girls in a class might be 5 : 6.
There are three ways of expressing a ratio.
The ratio 5 : 6 could also be expressed as “5 to 6” or as the fraction . 5 6

5. Lesson
4.2.1
Ratios and Rates
Example 1
There are four nuts between three squirrels. What is the ratio of nuts to squirrels?
Solution
There are 4 nuts to 3 squirrels so the ratio of nuts to squirrels is 4 : 3.
This could also be written “4 to 3” or . 4 3
Solution follows…

6. Ratios and Rates 4.2.1 Ratios Compare Quantities With Different Units
Lesson
4.2.1
Ratios and Rates
Ratios Compare Quantities With Different Units
A rate is a special kind of ratio, because it compares two quantities that have different units.
For example, if you travel 60 miles in 3 hours you would be traveling at a rate of
60 miles
3 hours
You’d normally write this as a unit rate. That’s one with a denominator of 1.
60 miles
3 hours
So = , or 20 miles per hour.
20 miles
1 hour

7. Lesson
4.2.1
Ratios and Rates
Example 2
John takes 110 steps in 2 minutes. What is his unit rate in steps per minute?
Solution
110 steps in 2 minutes means a rate of:
55 steps
1 minute
110 steps
2 minutes
= = 55 steps per minute
Solution follows…

8. Ratios and Rates 4.2.1 Numerator ÷ Denominator Gives a Unit Rate
Lesson
4.2.1
Ratios and Rates
Numerator ÷ Denominator Gives a Unit Rate
Dividing the numerator by the denominator of a rate gives the unit rate.
So, if it costs 2 dollars for 3 apples, the unit rate is the price per apple, which is = 2 dollars ÷ 3 apples = 0.67 dollars per apple
2 dollars
3 apples

9. Lesson
4.2.1
Ratios and Rates
Example 3
A car goes 54 miles in 3 hours. Write this as a unit rate in miles per hour.
Solution
Divide the top by the bottom of the rate.
= (54 ÷ 3) miles per hour = 18 miles per hour.
54 miles
3 hours
This is a unit rate because the denominator is now 1 (it’s equivalent to mi/h). 18 1
Solution follows…

10. Lesson
4.2.1
Ratios and Rates
Example 4
If a wheel spins 420 times in 7 minutes, what is its unit rate in revolutions per minute?
Solution
The rate is revolutions per minute.
420 7
Divide the top by the bottom of the rate.
(420 ÷ 7) revolutions per minute = 60 revolutions per minute.
This is a unit rate because 60 revolutions per minute has a denominator of 1 (60 = ). 60 1
Solution follows…

11. Ratios and Rates 4.2.1 Guided Practice
Lesson
4.2.1
Ratios and Rates
Guided Practice
In Exercises 1–3, find the unit rates.
1. $3.60 for 3 pounds of tomatoes.
2. $25 for 500 cell phone minutes.
words typed in 8 minutes.
4. Joaquin buys 2 meters of fabric, which costs him $ What was the price per meter?
5. Mischa buys a $19.98 ticket for unlimited rides at a fairground. She goes on six rides. How much did she pay per ride?
$1.20 per pound of tomatoes
$0.05 per minute
81 words per minute
$4.75 per meter
$3.33 per ride
Solution follows…

12. Ratios and Rates 4.2.1 Use “Unit Rates” to Find the Better Buy
Lesson
4.2.1
Ratios and Rates
Use “Unit Rates” to Find the Better Buy
Stores often sell different sizes of the same thing, such as clothes detergent or fruit juice.
A bigger size is often a better buy — meaning that you get more product for the same amount of money.
But this isn’t always the case, so it’s useful to be able to work out which is the better buy.
You can do this by finding the price for a single unit of each product. The units can be ounces, liters, meters, or whatever is most sensible.

13. Lesson
4.2.1
Ratios and Rates
Example 5
A store sells two sizes of cereal. Which is the better buy?
CEREAL
$3.20 for 16 ounce box
$4.32 for 24 ounce box
Solution
3.20 dollars
16 ounces
Rate is
16 ounce box:
Unit rate = (3.20 ÷ 16) dollars per ounce = $0.20 per ounce
4.32 dollars
24 ounces
Rate is
24 ounce box:
Unit rate = (4.32 ÷ 24) dollars per ounce = $0.18 per ounce
The 24 ounce box is the better buy — the price per ounce is lower.
Solution follows…

14. Ratios and Rates 4.2.1 Guided Practice
Lesson
4.2.1
Ratios and Rates
Guided Practice
6. Determine which phone company offers the better deal: Phone Company A: $40 for 800 minutes. Phone Company B: $26 for 650 minutes.
7. Determine which is the better deal on carrots: $1.20 for 2 lb or $2.30 for 5 lb.
Unit rate from Company A = $40 ÷ 800 minutes = 5 ¢ per minute Unit rate from Company B = $26 ÷ 650 minutes = 4 ¢ per minute So, phone Company B offers the best deal.
Unit rate deal 1 = $1.20 ÷ 2 lb = 60 ¢ per lb Unit rate deal 2 = $2.30 ÷ 5 lb = 46 ¢ per lb So, deal 2 — $2.30 for 5 lb is best.
Solution follows…

15. Ratios and Rates 4.2.1 Independent Practice
Lesson
4.2.1
Ratios and Rates
Independent Practice
In Exercises 1–6, write each as a unit rate.
1. $4.50 for 6 pens miles in 8 h
pages in 5 days miles in 2 h
5. $400 for 10 items 6. $36 in 6 hours
$0.75 per pen
12.5 miles per hour
40 pages per day
60 miles per hour
$40 per item
$6 per hour
Solution follows…

16. Ratios and Rates 4.2.1 Independent Practice
Lesson
4.2.1
Ratios and Rates
Independent Practice
7. Peanuts are either $1.70 per pound or $8 for 5 pounds. Which is the better buy?
8. Lemons sell for $4.50 for 6, or $10.50 for Which is the better buy?
9. “$40 for 500 pins or $60 for 800 pins.” Which is the better buy?
$8 for 5 pounds
$10.50 for 15
$60 for 800 pins
Solution follows…

17. Ratios and Rates 4.2.1 Round Up
Lesson
4.2.1
Ratios and Rates
Round Up
Rates compare one thing to another and always have units.
A unit rate is a rate that has a denominator of one.
In the next Lesson you’ll see how rate is related to the slope of a graph.