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CHAPTER 4: RATIOS AND RATES
Chapter Objectives By the end of this chapter, students should be able to: Represent ratios in multiple ways Find rates and unit rates Solve proportions involving decimals or fractions
Contents
CHAPTER 4: RATIOS AND RATES ............................................................................... 1
SECTION 4.1 RATIOS ................................................................................................ 2
A. WORKING WITH RATIOS OF FRACTIONS AND DECIMALS....................... 4
I. Ratios with Decimals....................................................................................... 5 II. Ratio with Fractions ..................................................................................... 5 III. Ratios with Mixed Numbers: ........................................................................ 5 EXERCISES............................................................................................................. 7 SECTION 4.2 RATES AND UNIT RATES ................................................................. 10 A. WRITING RATES AND UNIT RATES........................................................... 10 I. Rates ............................................................................................................ 10 I. Unit rates ...................................................................................................... 11 B. DETERMINE BETTER BUY USING UNIT RATES ....................................... 13 EXERCISES........................................................................................................... 14 SECTION 4.3 PROPORTIONS ................................................................................. 19 A. USE RATES TO SOLVE PROPORTIONAL PROBLEMS ............................ 19 B. USE UNIT RATES TO SOLVE PROPORTIONAL PROBLEMS ................... 21 C. USE CROSS PRODUCT TO SOLVE PROPORTIONAL PROBLEMS ......... 22 EXERCISES........................................................................................................... 24
SECTION 4.1 RATIOS
A ratio is a comparison of two quantities that are measured in the same unit. If we
compare 𝒂 and 𝒃, the ratio can be written as
, 𝒂: 𝒃 , or 𝒂 𝒕𝒐 𝒃.
Example: The ratio of 6 miles to 3 miles can be written in the following forms.
Fraction:
Colon: 6 miles : 3 miles
“ 𝒂 𝒕𝒐 𝒃 ” language : 6 miles to 3 miles
Example: Kate is traveling 100 miles to visit Rick. So far she has traveled 40 miles.
The ratio of miles Kate has traveled to the total number of miles is
We can also write this ratio as 40 ∶ 100 or as 40 𝑡𝑜 100.
We usually represent ratios as fractions. The first number listed in the ratio is used as the numerator and the second number in the ratio is used as the denominator. You can simplify a ratio just as you simplify a fraction.
Media Lesson Ratios (Duration 4:15)
View the video lesson, take notes and complete the problems below.
A ratio is a comparison of two quantities in the form of a quotient.
The ratio of A and B can be written ______ ways:
____________________ ____________________ ____________________
Ratios can be _____________ just like fractions.
A class has 15 female students and 12 male students.
What is the ratio of males to females?
What is the ratio of females to males?
A. WORKING WITH RATIOS OF FRACTIONS AND DECIMALS
Media Lesson Simplifying Ratios Involving Decimals and Fractions (Duration 6:08) View the video lesson, take notes and complete the problems below.
Examples: Write the ratios as simplified fractions.
- 2 ∶ 0. 6 3. 5 ∶ 4. 25
I. Ratios with Decimals We eliminate decimals in a ratio by following the following steps.
1. Set up the ratio in the colon form 2. Get rid of the decimal point by multiplying by 10, 100 or 1000, … depends on the number with the highest decimal places. 3. Reduce your new ratio to the lowest term.
Another way to set your ratio with decimal is to rewrite it in fraction form and get rid of the decimal point just like in the example below.
Example: Consider the ratio 0.8 𝑡𝑜 0.05.
We write this ratio as the following fraction.
Note 0.8 has one decimal place and 0.05 has two decimal places. We will move the decimal point to the right by two decimal places.
We end up with the ratio
which reduces to 161.
II. Ratio with Fractions To solve ratios with fractions we use the following steps.
- Setup the ratio as in the colon form.
- Convert both fractions to have the lowest common denominator (LCD).
- Get rid of the LCD in both fractions.
Media Lesson Simplifying Ratios Involving Decimals and Fractions (Duration 6:08) View the video lesson, take notes and complete the problems below.
III. Ratios with Mixed Numbers: To solve ratios with fractions we use the following steps.
- Setup the ratio in the colon form.
- Change any mixed number to an improper fraction.
- Follow the same step as above by converting the fractions to the same LCD and get rid of the LCD.
EXERCISES
In the following exercises write each ratio in fraction and colon notation.
12 hours to 16 hours 2) 30 miles to 9 miles
$26 to $2 4) 5 minutes to 45 minutes
In the following exercises, write the ratio in fraction form. Simplify if possible.
- $3 to $11 6) 5 days to 7 days 7) 2 hours to 9 hours
$18 to $
$16 to $72 24) $1.21 to $0.44 25) $1.38 to $0.
28 oz. to 84 oz. 27) 32 oz. to 128 oz. 28) 12 feet to 46 feet
15 feet to 57 feet 30) 246 mg to 45 mg 31) 304 mg to 48 mg
Consider the rectangle with width 10 cm and length 15 cm, write a ratio of the length to the width.
Using the rectangle in number 16, write the ratio of the width to the length.
If you spend 4 hours a week studying for English and 5.5 hours studying for math what is the ratio of time spent studying in math to studying in English?
An employee pays $125 towards health insurance, while the employer pays $550. What is the ratio of the employer’s contribution to the employee’s contribution?
SECTION 4.2 RATES AND UNIT RATES
A. WRITING RATES AND UNIT RATES
Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. A rate is a comparison with different units , such as miles per gallon and money per hours.
Like ratios we usually write rates as fractions. We put the first given in the numerator and the second amount in the denominator. When rates are simplified, the units remain in the numerator and denominator.
A special type of rate is called a unit rate. A unit rate is a rate where the denominator is a 1. Unit rates allow us to see relationships better.
For example, you are offered a job and your new employer says that you will be paid at a rate of $805 per 25 hours. We can express this rate as the following fraction.
$ 25 ℎ𝑜𝑢𝑟𝑠 This is your rate of pay, but it may be more useful to know how much you will be paid per 1 hour instead of 25 hours. The unit rate of pay can be founds as shown below.
$ 25 ℎ𝑜𝑢𝑟𝑠
$805 ÷ 25
25 ℎ𝑜𝑢𝑟𝑠 ÷ 25
I. Rates
Media Lesson Rates (Stop at 1”35) View the video lesson, take notes and complete the problems below.
If the quantities you are comparing have different units, then your ratio is known as a rate. Units are especially important here and should absolutely be included.
Example: Write “12 miles in 10 hours” as a ratio in simplest form.
Example: In a small bag of mixed nuts, 15 were peanuts, 20 were almonds, and 5 were Brazil nuts. Write the ratio of peanuts to almonds in simplest form.
Note: With ratios, the units will cancel out. With rates, the units will not cancel out.
YOU TRY:
Use the information to write a ratio in simplest form. Indicate if the ratio is also a rate.
a) 5 feet:10 feet b) 12 geese to 15 ducks
I. Unit rates
Media Lesson Unit Rates (Duration 1:58) View the video lesson, take notes and complete the problems below.
A unit rate is a special kind of rate in which the denominator of the ratio is__________.
This kind of rate allows for easier comparison of different rates as seen in the example below. As with rates, units ______________________________________________.
Example 1: Which is faster, “12 miles in 10 hours” or “10 miles in 8 hours”? Use unit rates to compare.
Media Lesson Unit Rates – example 2. (Duration 1:16) View the video lesson, take notes and complete the problems below.
Example 2: Write each of the following as a unit rate:
a. There are 5280 feet in a mile
b. There are 60 seconds per each minute
c. Gasoline costs $3.49 a gallon
B. DETERMINE BETTER BUY USING UNIT RATES
By comparing the unit rates of two different products it is easy to identify which is the best buy. The better buy is the item that cost less per unit.
Media Lesson Unit Rates – better buy. (Duration 2:17)
View the video lesson, take notes and complete the problems below.
Example 2: Determine which bag of Cheetos is the better buy.
Bag A: $4.99 for 20.50 oz. Bag B: $4.29 for 12.50 oz.
YOU TRY:
Round any answers to the hundredths place.
e) Callie is buying cereal at the grocery store. A 12.2-ounce box costs $4.39. A 27.5-ounce box costs $10.19. Which is the better buy?
f) Hector is buying cookies for a party. A regular sized bag has 34 cookies and costs $2.46. The family size bag has 48 cookies and costs $3.39 a bag. Which is the better buy?
EXERCISES
Write the following rates as a fraction in lowest terms.
1) 140 calories per 12 ounces 2) 180 calories per 16 ounces
3) 9.5 pounds per 4 square inches 4) 8.2 pounds per 3 square inches
5) 488 miles in 7 hours 6) 527 miles in 9 hours
7) $595 for 40 hours 8) $798 for 40 hours
In the following exercises, find the unit rate. Round to two decimal places, if necessary.
9) $45 dollars for 5 pounds 10) $24 dollars for 2 pounds
11) $27 dollars on 3 ounces 12) $44 dollars on 4 ounces
13) $252 per 12 people 14) $231 for 21 tees
27) 46 beats in 0.5 minutes 28) 54 beats in 0.5 minutes
In the following exercises, find each unit price and then identify the better buy.
Round to three decimal places.
29) Toothpaste, 6 ounce size for $3. or 7 ounce size for $5.
30) Breakfast cereal, 18 ounces for $3.99 or 14 ounces for $3.
31) Ketchup, 40 ounce regular bottle for $2.99 or 64 ounce squeeze bottle for $4.
32) Mayonnaise, 15 ounce regular bottle for $3.49 or 22 ounce squeeze bottle for $4.
33) Black beans, 16 ounce for $1.28 or 32 ounce for $2.
34) The grocery store has a special on macaroni and cheese. The price is $3.87 for 3 boxes. How much does each box cost?
35) A store sells two different kinds of toothpaste. The first toothpaste comes in a 6- ounce tube and costs $3.19. The second toothpaste comes in a 7.8-ounce tube and costs $5.19. Find the unit rate and identify the better buy.
36) Sven drives his car 455 miles using 14 gallons of gasoline. How many miles per gallon does his car get?
37) You are renting a house in Cancun for a week at $3,600, what is the cost per day?
38) The bindery at a printing plant assembles 96,000 magazines in 12 hours. How many magazines are assembled in one hour?
39) One elementary school in Ohio has 684 students and 45 teachers. Write the student-to-teacher ratio as a unit rate.
40) The average American produces about 1,600 pounds of paper trash per year (365 days). How many pounds of paper trash does the average American produce each day? (Round to the nearest tenth of a pound.)
SECTION 4.3 PROPORTIONS
When two ratios or rates are equal the equation relating them is called a proportion.
A proportion for the ratios
and
is written as an equation of the form 𝒂 𝒃
=
𝒄 𝒅
, where 𝑏 ≠ 0 and 𝑑 ≠ 0.
When we say things are proportional we are saying they have the same rate or ratio.
We will explore different ways of solving proportions.
A. USE RATES TO SOLVE PROPORTIONAL PROBLEMS
Proportions and rates allow us to solve many applications.
Example:
You are making cookies. A recipe calls for 29 grams of sugar and makes 2 dozen cookies. You want to make 6 dozen cookies. We can use a proportion to figure out how much grams of sugar you will need.
Our first step is to write the rate 29 grams of sugar for 2 dozen cookies in fraction form.
29 𝑔𝑟𝑎𝑚𝑠 2 𝑑𝑜𝑧𝑒𝑛
You want to make 6 dozen cookies but do not know how much sugar we need. We will use this information to write a proportion.
29 𝑔𝑟𝑎𝑚𝑠 2 𝑑𝑜𝑧𝑒𝑛
Notice the units are lined up. Both rates have grams in the numerator and dozens in the denominator. If we think of these rates as equivalent fractions we notice 2 × 3 = 6.
Your total earnings for the week are 29 grams × 3 or 87 grams.
Media Lesson Use Proportions to Solve Applications (No Cross Product) (Duration 5: 42 ) View the video lesson, take notes and complete the problems below.
a) The ratio of the lengths of corresponding sides of two similar decagons is 1:2. If the perimeter of the smaller decagon is 76 cm, what is the perimeter of the larger decagon? (^1) : 2 76cm (^)? cm
b) A cookie recipe requires 4 cups of flour to make 5 dozen cookies. If Amy needs to make 15 dozen cookies, how many cups of flour will she need?
c) The president of the student body estimated that 2 out of every 3 students at school would attend the Spring Festival. If there are 1,140 students at this school, according to the estimate, how many students will not attend the Spring Festival?
d) If one bus holds 60 students, how many buses are needed to take 780 students to the Valley Fair?
