146. Understanding and Using Rates

Learning Intentions

• To understand that rates compare two quantities measured in different units
• simplify rates
• find average rates
• To understand that a rate like $\mathrm{\$}12/\text{h}$ (or $\mathrm{\$}12$ per hour) means $\mathrm{\$}12$ for $1$ 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) $60$ km in $2$ h
b) $3:5$
c) $\mathrm{\$}18$ for $3$ kg

Worked Example 2

Simplify each rate:
a) $120$ km in $2$ h
b) $\mathrm{\$}24$ for $4$ h
c) $900$ mL in $3$ min

Worked Example 3

Write each rate in a “per 1” form:
a) $\mathrm{\$}36$ for $3$ h
b) $150$ km in $5$ h
c) $48$ pages in $6$ days

Worked Example 4

Find the average rate:
a) $240$ km travelled in $4$ h
b) $360$ words typed in $6$ min
c) $\mathrm{\$}84$ earned in $7$ h

Worked Example 5

Interpret each unit rate:
a) $\mathrm{\$}12/\text{h}$
b) $80\text{ km/h}$
c) $250\text{ mL/min}$

Worked Example 6

Solve the worded problem using average rate:
a) A cyclist rides $90$ km in $3$ h. Find the average speed.
b) A machine fills $2.4$ L in $6$ min. Find the average filling rate.

Problems

Problem 1

State whether each comparison is a rate, and explain why:
a) $48$ L in $4$ min
b) $2:7$
c) $\mathrm{\$}30$ for $5$ tickets

Problem 2

Simplify each rate:
a) $180$ km in $3$ h
b) $\mathrm{\$}40$ for $5$ h
c) $600$ mL in $2$ min

Problem 3

Write each rate in a “per 1” form:
a) $\mathrm{\$}45$ for $5$ h
b) $210$ km in $7$ h
c) $35$ questions in $5$ min

Problem 4

Find the average rate:
a) $300$ km travelled in $5$ h
b) $420$ words typed in $7$ min
c) $\mathrm{\$}96$ earned in $8$ h

Problem 5

Interpret each unit rate:
a) $\mathrm{\$}15/\text{h}$
b) $60\text{ km/h}$
c) $120\text{ mL/min}$

Problem 6

Solve the worded problem using average rate:
a) A runner travels $24$ km in $2$ h. Find the average speed.
b) A tap fills $3$ L in $5$ min. Find the average filling rate.

Exercises

Understanding and Fluency

1. Complete each statement:
   a) A rate compares two quantities measured in different ______
   b) The word “per” means “for ______”
   c) A rate of $\mathrm{\$}10/\text{h}$ means $\mathrm{\$}10$ for ______ hour

2. State whether each comparison is a rate:
   a) $100$ km in $2$ h
   b) $4:9$
   c) $12$ books in $3$ boxes
   d) $\mathrm{\$}20$ for $4$ kg

3. Simplify each rate:
   a) $150$ km in $3$ h
   b) $\mathrm{\$}28$ for $4$ h
   c) $800$ mL in $4$ min
   d) $45$ questions in $5$ min

4. Write each rate in unit form:
   a) $\mathrm{\$}54$ for $6$ h
   b) $360$ km in $6$ h
   c) $72$ pages in $8$ days
   d) $2.5$ L in $5$ min

5. Find each average rate:
   a) $280$ km in $4$ h
   b) $96$ dollars in $8$ h
   c) $540$ words in $9$ min
   d) $1.2$ L in $3$ min

6. Interpret each rate in words:
   a) $\mathrm{\$}18/\text{h}$
   b) $90\text{ km/h}$
   c) $300\text{ mL/min}$
   d) $25$ pages/day

7. Solve each average rate problem:
   a) A car travels $420$ km in $6$ h. Find the average speed.
   b) A worker earns $\mathrm{\$}110$ in $5$ h. Find the average pay rate.
   c) A hose fills $4.8$ L in $8$ min. Find the average filling rate.

8. Choose the most suitable rate and calculate:
   a) A typist types $250$ words in $5$ min. Find the average rate in words per minute.
   b) A shopper pays $\mathrm{\$}36$ for $3$ kg of apples. Find the price per kilogram.
   c) A cyclist rides $56$ km in $4$ h. Find the average speed.

Reasoning

9. Explain why $80$ km in $2$ h is a rate, but $2:5$ is only a ratio.

10. A student says that $\mathrm{\$}12/\text{h}$ means $\mathrm{\$}12$ for any number of hours. Explain the mistake.

11. Noah says that to simplify a rate, you can divide only one quantity by a common factor. Is he correct? Explain.

12. Explain why finding an average rate involves dividing the total amount by the total number of units.

13. A student says that $240$ km in $4$ h simplifies to $240:4$ instead of $60\text{ km/h}$. Describe the error.

Problem-solving

14. A bus travels $180$ km in $3$ h. Find its average speed.

15. A plumber charges $\mathrm{\$}96$ for $8$ h of work. Find the hourly rate.

16. A juice machine fills $1.8$ L in $6$ min. Find the average rate in L/min.

17. A reader finishes $84$ pages in $7$ days. Find the average number of pages read per day.

18. A baker uses $6$ kg of flour for $12$ trays of bread. Find the average amount of flour per tray.

19. A student earns $\mathrm{\$}72$ by working $6$ hours. Explain what the rate $\mathrm{\$}12/\text{h}$ means in this situation.

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