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, $ , $ 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 km in h

b) Apples cost $ for kg

c) A tap fills L in min

Worked Example 2

Find the unit rate for each situation:

a) $ for h

b) km in h

c) mL in min

Worked Example 3

Use a rate to solve the problem:

a) At $ , how much is earned in h?

b) At , how far is travelled in h?

Worked Example 4

Use a rate to solve the problem:

a) At $ , what is the cost of kg?

b) At , how much water flows in min?

Worked Example 5

Solve the problem by first finding the rate:

a) A cyclist rides km in h. How far will the cyclist ride in h at the same average speed?

b) A worker earns $ in h. How much will the worker earn in h at the same rate?

Worked Example 6

Choose an appropriate rate model and solve:

a) A machine packs boxes in min. How many boxes does it pack in min?

b) A recipe uses g of flour for loaves. How much flour is needed for loaves?

Problems

Problem 1

State the rate that models each situation:

a) A train travels km in h

b) Rice costs $ for kg

c) A hose fills L in min

Problem 2

Solve the unit rate for each situation:

a) $ for h

b) km in h

c) mL in min

Problem 3

Use a rate to solve the problem:

a) At $ , how much is earned in h?

b) At , how far is travelled in h?

Problem 4

Use a rate to solve the problem:

a) At $ , what is the cost of kg?

b) At , how much water flows in min?

Problem 5

Solve the problem by first finding the rate:

a) A runner travels km in h. How far will the runner travel in h at the same average speed?

b) A worker earns $ in h. How much will the worker earn in h at the same rate?

Problem 6

Choose an appropriate rate model and solve:

a) A printer prints pages in min. How many pages does it print in min?

b) A juice recipe uses mL for serves. How much juice is needed for serves?

Exercises

Understanding and Fluency

Exercise 1.

State the rate that models each situation:

a) km in h

b) $ for kg

c) mL in min

d) pages in days

Exercise 2.

Find the unit rate for each situation:

a) $ for h

b) km in h

c) L in min

d) questions in min

Exercise 3.

Solve each problem Use the given rate:

a) At $ , how much is earned in h?

b) At , how far is travelled in h?

c) At $ , what is the cost of kg?

Exercise 4.

Solve each problem using the given rate:

a) At , how much flows in min?

b) At pages/day, how many pages are read in days?

c) At boxes/min, how many boxes are packed in min?

Exercise 5.

First find the rate, then solve:

a) A car travels km in h. How far will it travel in h at the same average speed?

b) A worker earns $ in h. How much will the worker earn in h?

c) A tap fills L in min. How much water flows in min?

Exercise 6.

First find the rate, then solve:

a) A typist types words in min. How many words are typed in min?

b) A baker uses kg of flour for trays. How much flour is needed for trays?

c) A bus travels km in h. How far will it travel in h?

Exercise 7.

Choose an appropriate rate and solve:

a) Apples cost $ for kg. What is the cost of kg?

b) A machine fills L in min. How much does it fill in min?

c) A cyclist rides km in h. How far does the cyclist ride in h?

Exercise 8.

Solve each multi-step rate problem:

a) A runner travels km in h. How far will the runner travel in h at the same rate?

b) A teacher prints worksheets in min. How many worksheets can be printed in min?

c) A cleaner charges $ . How much will h of work cost?

Reasoning

Exercise 9.

Explain why a rate must compare quantities with different units.

Exercise 10.

A student says that $ for h means the rate is $ . Explain the mistake.

Exercise 11.

Noah says that if a car travels km in h, then in h it will travel km. Is he correct? Explain.

Exercise 12.

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

Exercise 13.

A student says that mL in min means min/mL. Describe the error.

Problem-solving

Exercise 14.

A plumber charges $ per hour. How much does hours of work cost?

Exercise 15.

A van travels km in h. How far will it travel in h at the same average speed?

Exercise 16.

A recipe uses g of butter for batches. How much butter is needed for batches?

Exercise 17.

A tap fills L in min. How much water will it fill in min at the same rate?

Exercise 18.

A reader finishes pages in days. How many pages will the reader finish in days at the same average rate?

Exercise 19.

A shop sells rice at $ . What is the cost of 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

Next: 148. Speed, Distance and Time