Unit Rate with Complex Fractions
Lesson Objective
By the end of the lesson, we will be able to ____________________________________
_______________________________________________________________________
(AZ-7.RP.A.1) Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or
different units.
Do Now
Skills Preview
We will need to know each of these skills for today’s lesson. Reflect on what you know and
solve for each component.
Writing Proportions - (represent the word problem with a proportion; solve the proportion)
Monica is making a salad and uses two carrots for every 3 cucumber slices. How many carrots
will she need if there are 12 cucumber slices?
Unit Rates - (represent the word problem with a proportion; solve the proportion)
Julia is grading her students’ essays. She can grade 18 essays in 3 hours. How many essays can
she grade in 1 hour?
Reciprocals - (identify the reciprocal of each number)
What is the reciprocal of each of the values below?
3
4
2
8
1
7
[1]
5
Unit Rate with Complex Fractions
Essential Vocabulary
Vocabulary Term
Reciprocal
Teacher-Provided Definition
Visual
A fraction whose numbers are flipped;
The product of these two values is one.
A comparison of two numbers with different
Rate
units of measure; a special kind of ratio.
Unit Rate
Complex
Fraction
A rate that has a unit with a value of one;
(e.g. 1 hour, 1 mile, 1 cup, etc.)
The numerator or denominator of a fraction can
also be represented as a fraction.
Teacher Model
Let’s look at the example. Read the problem and notice that the answer can be found by writing a
proportion. (There are other ways as well). Be sure to label your answer as a rate.
1. Example 1
Unit Rates with Integers
A car is traveling on a freeway. After 2 hours, the driver estimates he has gone about 100 miles.
Based on this information, at what rate is the driver traveling?
1. Express the given information as a fraction
2. Set up a proportion with a second quantity as one
3. Simplify the proportion by identifying the “magic number”
Rate
Proportion
2. Example 2
[2]
Simplified
Unit Rate with Complex Fractions
Guided Practice
Steps: Computing Unit Rate
1. Express the given information as a fraction
2. Set up a proportion with a second quantity as one
Ask yourself!
How do I get to
one by multiplying
or dividing?
3. Simplify the proportion
a. When presented with integers, identify the mental math shortcut
.
b. When presented with fractions, multiply by the reciprocal
Example A
Clare is a baker. She can fry 15 donuts in 3 hours. How many donuts can she fry in one hour?
Example B
Erika works at the golf course giving putting lessons. She made $48 after giving 3 lessons.
How much does Erika make per lesson if each customer paid the same amount?
Practice
Diana is cooking dinner. A recipe calls for 4 cups of flour for every 2 cups of sugar. How much
flour is required for every cup of sugar?
[3]
Unit Rate with Complex Fractions
Guided Practice
Ask yourself!
How do I get to
one by multiplying
or dividing?
Computing Unit Rate with Complex Fractions
1. Express the given information as a fraction
2. Set up a proportion with a second quantity as one
3. Simplify the proportion
a. When presented with integers, identify the mental math shortcut
.
b. When presented with fractions, multiply by the reciprocal
Unit Rates (whole number + fraction)
4
After the sun goes down, the temperature of the ground decreases by 8 degrees Celsius every 5
hour. At what rate, as measured in degrees per hour, is the temperature decreasing?
Rate
Proportion
Simplified
Unit Rates (fraction + fraction)
1
2
In a science experiment, the temperature of an object decreases by 10 degree Celsius every 3 of
a minute. At what rate, as measured in degrees per minute, is the temperature decreasing?
Rate
Proportion
Simplified
[4]
Unit Rate with Complex Fractions
Guided Practice
Ask yourself!
How do I get to
one by multiplying
or dividing?
Computing Unit Rate with Complex Fractions
1. Express the given information as a fraction
2. Set up a proportion with a second quantity as one
3. Simplify the proportion
a. When presented with integers, identify the mental math shortcut
.
b. When presented with fractions, multiply by the reciprocal
Unit Rates (whole number + fraction)
1
A bicyclist travels at a constant rate of 11 miles in 4 hour. Find the unit rate of the hikers in
miles per hour.
Rate
Unit Rates (fraction + fraction)
Proportion
Simplified
4
1
A group of hikers travel at a constant rate of 7 mile in 3 hour. Find the unit rate of the hikers in
miles per hour.
Rate
Proportion
Simplified
[5]
Unit Rate with Complex Fractions
Guided Practice
Ask yourself!
How do I get to
one by multiplying
or dividing?
Steps: Computing Unit Rate
1. Express the given information as a fraction
2. Set up a proportion with a second quantity as one
3. Simplify the proportion
a. When presented with integers, identify the mental math shortcut
.
b. When presented with fractions, multiply by the reciprocal
Example A
1
A suggested planting rate for wildflower seeds is 2 pound per
pounds per acre?
Example B
2
1
8
acre. What is the unit rate in
1
It took 3 minute to fill a barrel 4 full of water. Find the unit rate in barrels per minute.
Practice
3
1
Diana is cooking a second dinner. This time the recipe calls for 5 cups of flour for every 3 cups
of sugar. How much flour is required for every cup of sugar?
[6]
Unit Rate with Complex Fractions
Partner Practice
Refer back to the examples given. Note: Part A uses whole numbers and Part B changes the
values to fractions. Be sure to write your proportion before solving and identify the magic
number!
1a. Marta runs 2 miles in the same time that it takes Kylie to run 3 miles. If Marta runs a total
distance of 6 miles, how far was Kylie able to run?
4
2
1b. Helen runs 5 of a mile in the same time that it takes Zoe to run 3 mile. If Helen runs one
mile, how far was Zoe able to run?
2a. Jane is planting a mixture of herbs in her kitchen garden. For every 6 ounces of rosemary
seeds she uses 3 ounces of dill seeds. How many ounces of rosemary seeds does Jane need if
she used one ounce of dill seeds?
2b. Janelle is planting a mixture of herbs in her kitchen garden. For every ounce of rosemary
seeds she uses two-fifths of an ounce of dill seeds. How many ounces of rosemary seeds does
Janelle need if she used one ounce of dill seeds?
[7]
Unit Rate with Complex Fractions
Group Practice
Refer back to the examples given. Be sure to write your proportion before solving and identify
the magic number!
7
6
1. A rotating object makes 8 of a revolution in10 second. Find the approximate speed in
revolutions per second. Challenge: Express your answer as a decimal and round to the nearest
hundredth.
1
2. A group of forest rangers work 5 days per week. It takes them 3 days to plant trees on 3
acre. What is the unit rate in acres per week? Challenge: Find how many acres they can plant
in 12 weeks.
[8]
Unit Rate with Complex Fractions
Extension of Complex Numbers
Two groups of hikers left camp at the same time. Each traveled at a constant rate.
3
1

Group A covered 4 mile in 2 hour

It took Group B 5 hour to travel 3 mile
3
1
Part A
Use complex fractions to compare the hiking speeds of the two groups. Show your work for
each group in the box provided.
Group A
Group B
Part B
Which group is traveling at a faster rate? How much faster?
[9]
Unit Rate with Complex Fractions
Extension of Complex Numbers
Compare each complex fraction to 1 (set each denominator to 1). Write the letter of the
complex fraction in the correct box. Use the space at the bottom of the page to show your work.
A
B
C
D
E
F
𝟏
𝟒
𝟑
𝟏𝟏
𝟐
𝟖
𝟑
𝟏𝟏
𝟔
𝟕
𝟏
𝟐
𝟔
𝟗
𝟕
𝟐
𝟏
𝟏𝟐
𝟒
𝟑
𝟐
𝟑
𝟏
𝟑
Less Than 1
Greater Than 1
A
[10]