147. Modelling and Solving Rate Problems

Learning Intentions

• To understand that rates can be used to model many situations
• solve problems involving rates

Pre-requisite Summary

• Know that a rate compares two quantities measured in different units
• Understand that the word “per” means “for each” or “for one”
• Be able to simplify rates to unit rates where useful
• Be able to multiply and divide whole numbers and decimals accurately
• Know common rates such as km/h, $\mathrm{\$}/\text{h}$, $\mathrm{\$}/\text{kg}$ and mL/min
• Understand that a rate can be used as a model for a real situation
• Be able to identify the two quantities and their units in a worded problem

Worked Examples

Worked Example 1

State the rate that models each situation:
a) A car travels $180$ km in $3$ h
b) Apples cost $\mathrm{\$}12$ for $3$ kg
c) A tap fills $2.4$ L in $6$ min

Worked Example 2

Find the unit rate for each situation:
a) $\mathrm{\$}45$ for $5$ h
b) $300$ km in $4$ h
c) $900$ mL in $3$ min

Worked Example 3

Use a rate to solve the problem:
a) At $\mathrm{\$}8/\text{h}$, how much is earned in $6$ h?
b) At $60\text{ km/h}$, how far is travelled in $3$ h?

Worked Example 4

Use a rate to solve the problem:
a) At $\mathrm{\$}4/\text{kg}$, what is the cost of $7$ kg?
b) At $250\text{ mL/min}$, how much water flows in $8$ min?

Worked Example 5

Solve the problem by first finding the rate:
a) A cyclist rides $120$ km in $4$ h. How far will the cyclist ride in $7$ h at the same average speed?
b) A worker earns $\mathrm{\$}84$ in $7$ h. How much will the worker earn in $10$ h at the same rate?

Worked Example 6

Choose an appropriate rate model and solve:
a) A machine packs $48$ boxes in $6$ min. How many boxes does it pack in $15$ min?
b) A recipe uses $300$ g of flour for $4$ loaves. How much flour is needed for $10$ loaves?

Problems

Problem 1

State the rate that models each situation:
a) A train travels $240$ km in $4$ h
b) Rice costs $\mathrm{\$}15$ for $5$ kg
c) A hose fills $3$ L in $5$ min

Problem 2

Find the unit rate for each situation:
a) $\mathrm{\$}63$ for $7$ h
b) $360$ km in $6$ h
c) $1200$ mL in $4$ min

Problem 3

Use a rate to solve the problem:
a) At $\mathrm{\$}9/\text{h}$, how much is earned in $5$ h?
b) At $70\text{ km/h}$, how far is travelled in $4$ h?

Problem 4

Use a rate to solve the problem:
a) At $\mathrm{\$}6/\text{kg}$, what is the cost of $8$ kg?
b) At $200\text{ mL/min}$, how much water flows in $9$ min?

Problem 5

Solve the problem by first finding the rate:
a) A runner travels $30$ km in $2$ h. How far will the runner travel in $5$ h at the same average speed?
b) A worker earns $\mathrm{\$}96$ in $8$ h. How much will the worker earn in $12$ h at the same rate?

Problem 6

Choose an appropriate rate model and solve:
a) A printer prints $72$ pages in $6$ min. How many pages does it print in $20$ min?
b) A juice recipe uses $500$ mL for $4$ serves. How much juice is needed for $9$ serves?

Exercises

Understanding and Fluency

1. State the rate that models each situation:
   a) $150$ km in $3$ h
   b) $\mathrm{\$}20$ for $4$ kg
   c) $600$ mL in $2$ min
   d) $48$ pages in $6$ days

2. Find the unit rate for each situation:
   a) $\mathrm{\$}40$ for $5$ h
   b) $280$ km in $4$ h
   c) $1.5$ L in $3$ min
   d) $36$ questions in $6$ min

3. Solve each problem using the given rate:
   a) At $\mathrm{\$}11/\text{h}$, how much is earned in $4$ h?
   b) At $80\text{ km/h}$, how far is travelled in $2$ h?
   c) At $\mathrm{\$}5/\text{kg}$, what is the cost of $9$ kg?

4. Solve each problem using the given rate:
   a) At $300\text{ mL/min}$, how much flows in $7$ min?
   b) At $12$ pages/day, how many pages are read in $5$ days?
   c) At $8$ boxes/min, how many boxes are packed in $9$ min?

5. First find the rate, then solve:
   a) A car travels $210$ km in $3$ h. How far will it travel in $6$ h at the same average speed?
   b) A worker earns $\mathrm{\$}54$ in $6$ h. How much will the worker earn in $9$ h?
   c) A tap fills $2.4$ L in $4$ min. How much water flows in $10$ min?

6. First find the rate, then solve:
   a) A typist types $360$ words in $6$ min. How many words are typed in $15$ min?
   b) A baker uses $2$ kg of flour for $8$ trays. How much flour is needed for $14$ trays?
   c) A bus travels $320$ km in $5$ h. How far will it travel in $8$ h?

7. Choose an appropriate rate and solve:
   a) Apples cost $\mathrm{\$}18$ for $6$ kg. What is the cost of $11$ kg?
   b) A machine fills $4.5$ L in $9$ min. How much does it fill in $12$ min?
   c) A cyclist rides $90$ km in $3$ h. How far does the cyclist ride in $1$ h?

8. Solve each multi-step rate problem:
   a) A runner travels $24$ km in $2$ h. How far will the runner travel in $7$ h at the same rate?
   b) A teacher prints $90$ worksheets in $15$ min. How many worksheets can be printed in $40$ min?
   c) A cleaner charges $\mathrm{\$}25/\text{h}$. How much will $3.5$ h of work cost?

Reasoning

9. Explain why a rate must compare quantities with different units.

10. A student says that $\mathrm{\$}24$ for $3$ h means the rate is $\mathrm{\$}24/\text{h}$. Explain the mistake.

11. Noah says that if a car travels $180$ km in $3$ h, then in $6$ h it will travel $183$ km. Is he correct? Explain.

12. Explain why finding a unit rate is often useful when solving a rate problem.

13. A student says that $400$ mL in $2$ min means $200$ min/mL. Describe the error.

Problem-solving

14. A plumber charges $\mathrm{\$}18$ per hour. How much does $7$ hours of work cost?

15. A van travels $420$ km in $6$ h. How far will it travel in $9$ h at the same average speed?

16. A recipe uses $250$ g of butter for $5$ batches. How much butter is needed for $12$ batches?

17. A tap fills $3.6$ L in $8$ min. How much water will it fill in $25$ min at the same rate?

18. A reader finishes $84$ pages in $7$ days. How many pages will the reader finish in $12$ days at the same average rate?

19. A shop sells rice at $\mathrm{\$}3.50/\text{kg}$. What is the cost of $8$ kg of rice?

Potential Misunderstandings

• Students may confuse a rate with a ratio between quantities in the same units
• Students may ignore the units when identifying or using a rate
• Students may multiply when they should divide to find the unit rate
• Students may divide when they should multiply to use a known unit rate
• Students may think a total amount is the same as an amount per unit
• Students may reverse the units in a rate, such as writing min/L instead of L/min
• Students may assume a rate model applies without checking that the situation is constant
• Students may not recognise that solving many rate problems becomes easier after finding the value for one unit