149e. Unitary Method for Ratios and Rates

Learning Intentions

• To understand that the unitary method involves finding the value of ‘one unit’ first
• Solve ratio and rates problems Use the unitary method
• convert rates between different units using the unitary method

Pre-requisite Summary

• Know that a ratio compares quantities
• Know that a rate compares quantities measured in different units
• Be able to divide a quantity by the total number of parts in a ratio
• Understand that “per” means “for one”
• Be able to convert between basic units of length, mass, volume and time
• Be able to multiply and divide whole numbers and decimals accurately
• Understand that the unitary method finds the value of one unit first

Worked Examples

Worked Example 1

Use the unitary method to Solve the value of one unit:

a) books cost $

b) tickets cost $

c) kg of apples cost $

Worked Example 2

Solve each ratio or rate problem using the unitary method:

a) If pens cost $ , how much do pens cost?

b) If notebooks weigh kg, what is the weight of notebook?

c) If notebook weighs kg, what is the weight of notebooks?

Worked Example 3

Use the unitary method to solve a ratio problem:

a) If parts have value , what is the value of part?

b) Divide in the ratio using the value of one part

Worked Example 4

Use the unitary method to solve a rate problem:

a) A car travels km in h. Find the distance travelled in h, then in h

b) A worker earns $ in h. Find the hourly rate, then the pay for h

Worked Example 5

Convert rates between units using the unitary method:

a) Convert to

b) Convert to

c) Convert to

Worked Example 6

Solve each problem using the unitary method and unit conversion where needed:

a) A tap fills L in min. How much does it fill in min and in min?

b) A cyclist rides km in h. Find the average speed in km/h

Problems

Problem 1

Use the unitary method to find the value of one unit:

a) books cost $

b) tickets cost $

c) kg of oranges cost $

Problem 2

Solve each ratio or rate problem using the unitary method:

a) If pencils cost $ , how much do pencils cost?

b) If bags weigh kg, what is the weight of bag?

c) If bag weighs kg, what is the weight of bags?

Problem 3

Use the unitary method to solve a ratio problem:

a) If parts have value , what is the value of part?

b) Divide in the ratio using the value of one part

Problem 4

Use the unitary method to solve a rate problem:

a) A bus travels km in h. Find the distance travelled in h, then in h

b) A cleaner earns $ in h. Find the hourly rate, then the pay for h

Problem 5

Convert rates between units using the unitary method:

a) Convert to

b) Convert to

c) Convert to

Problem 6

Solve each problem using the unitary method and unit conversion where needed:

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

b) A runner travels km in h. Find the average speed in km/h

Exercises

Understanding and Fluency

Exercise 1.

Complete each statement:

a) The unitary method first finds the value of what unit?

b) In a rate problem, “per hour” means “for what hour”

c) To divide a quantity in a ratio using the unitary method, first find the total number of what?

Exercise 2.

Use the unitary method to find the value of one unit:

a) pens cost $

b) kg of rice cost $

c) tickets cost $

Exercise 3.

Solve each using the unitary method:

a) If apples cost $ , how much do apples cost?

b) If books weigh kg, how much does book weigh?

c) If book weighs kg, how much do books weigh?

Exercise 4.

Divide each quantity in the given ratio using the unitary method:

a) Divide in the ratio

b) Divide in the ratio

c) Divide $ in the ratio

Exercise 5.

Solve each rate problem using the unitary method:

a) A car travels km in h. How far does it travel in h?

b) A worker earns $ in h. What is the hourly rate?

c) A tap fills L in min. How much does it fill in min?

Exercise 6.

Solve each rate problem using the unitary method:

a) A cyclist rides km in h. How far will the cyclist ride in h?

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

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

Exercise 7.

Convert each rate using the unitary method:

a) to

b) to

c) to

d) to

Exercise 8.

Solve each problem using the unitary method and unit conversion if needed:

a) A hose fills L in min. How much water does it fill in min?

b) A train travels km in h. Find its average speed in km/h and km/min

c) A map distance of cm represents km in real life. How many kilometres does cm represent?

Reasoning

Exercise 9.

Explain why the unitary method always starts by finding the value of one unit.

Exercise 10.

A student says that if pens cost $ , then one pen costs $ 100$. Explain the mistake.

Exercise 11.

Noah says that to divide in the ratio , you divide by because there are two numbers in the ratio. Is he correct? Explain.

Exercise 12.

Explain why converting rates often becomes easier after rewriting the rate as a “per 1” value.

Exercise 13.

A student says that means km in hours. Describe the error.

Problem-solving

Exercise 14.

A farmer buys bags of seed for $ . Find the cost of bag, then the cost of bags.

Exercise 15.

A class shares $ in the ratio . Find each share using the unitary method.

Exercise 16.

A car travels km in h. Find the average speed, then find how far it travels in h at the same rate.

Exercise 17.

A tap fills L in min. Find the amount filled in min, then in min.

Exercise 18.

Convert to using the unitary method.

Exercise 19.

A runner travels km in h. Find the average speed in km/h, then convert it to km/min using the unitary method.

Potential Misunderstandings

• Students may think the unitary method means multiplying first instead of finding one unit first
• Students may divide by the wrong number when finding one unit
• Students may forget to find the total number of parts in a ratio before finding one part
• Students may confuse a rate with a ratio
• Students may find the value of one unit correctly but then multiply by the wrong number of units
• Students may forget to convert units before or after working with a rate
• Students may not recognise that “per” already describes a one-unit comparison
• Students may divide a quantity in a ratio by the number of terms instead of by the total number of parts