146. Understanding and Using Rates
Learning Intentions
- To understand that rates compare two quantities measured in different units
- Simplify rates
- Solve average rates
- To understand that a rate like $
(or $ per hour) means $ for hour
Pre-requisite Summary
- Know that a ratio compares quantities
- Understand that rates compare quantities with different units
- Be able to simplify ratios by dividing by a common factor
- Be able to divide whole numbers and decimals accurately
- Know that “per” means “for each” or “for one”
- Be able to read common units such as km, h, L, min, kg and dollars
- Understand that an average is found by dividing a total amount by the number of equal units
Worked Examples
Worked Example 1
State whether each comparison is a rate, and explain why:
a)
b)
c) $
Worked Example 2
Simplify each rate:
a)
b) $
c)
Worked Example 3
Write each rate in a “per 1” form:
a) $
b)
c)
Worked Example 4
Find the average rate:
a)
b)
c) $
Worked Example 5
Interpret each unit rate:
a) $
b)
c)
Worked Example 6
Solve the worded problem Use average rate:
a) A cyclist rides
b) A machine fills
Problems
Problem 1
State whether each comparison is a rate, and Explain why:
a)
b)
c) $
Problem 2
Simplify each rate:
a)
b) $
c)
Problem 3
Write each rate in a “per 1” form:
a) $
b)
c)
Problem 4
Find the average rate:
a)
b)
c) $
Problem 5
Interpret each unit rate:
a) $
b)
c)
Problem 6
Solve the worded problem using average rate:
a) A runner travels
b) A tap fills
Exercises
Understanding and Fluency
Exercise 1.
Complete each statement:
a) A rate compares two quantities measured in different ______
b) The word “per” means “for ______”
c) A rate of $
Exercise 2.
State whether each comparison is a rate:
a)
b)
c)
d) $
Exercise 3.
Simplify each rate:
a)
b) $
c)
d)
Exercise 4.
Write each rate in unit form:
a) $
b)
c)
d)
Exercise 5.
Find each average rate:
a)
b)
c)
d)
Exercise 6.
Interpret each rate in words:
a) $
b)
c)
d)
Exercise 7.
Solve each average rate problem:
a) A car travels
b) A worker earns $
c) A hose fills
Exercise 8.
Choose the most suitable rate and Calculate:
a) A typist types
b) A shopper pays $
c) A cyclist rides
Reasoning
Exercise 9.
Explain why
Exercise 10.
A student says that $
Exercise 11.
Noah says that to simplify a rate, you can divide only one quantity by a common factor. Is he correct? Explain.
Exercise 12.
Explain why finding an average rate involves dividing the total amount by the total number of units.
Exercise 13.
A student says that
Problem-solving
Exercise 14.
A bus travels
Exercise 15.
A plumber charges $
Exercise 16.
A juice machine fills
Exercise 17.
A reader finishes
Exercise 18.
A baker uses
Exercise 19.
A student earns $
Potential Misunderstandings
- Students may think a rate and a ratio are always the same thing
- Students may ignore the units when working with rates
- Students may forget that rates compare quantities in different units
- Students may simplify the numbers but not state the simplified rate with correct units
- Students may think “per hour” means for any number of hours instead of for one hour
- Students may confuse total amount with amount per unit
- Students may multiply instead of divide when finding an average rate
- Students may write a simplified rate without converting it to a “per 1” form