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 using 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 find the value of one unit:
a) $5$ books cost $\mathrm{\$}20$
b) $8$ tickets cost $\mathrm{\$}56$
c) $3$ kg of apples cost $\mathrm{\$}12$

Worked Example 2

Solve each ratio or rate problem using the unitary method:
a) If $4$ pens cost $\mathrm{\$}10$, how much do $7$ pens cost?
b) If $6$ notebooks weigh $3$ kg, what is the weight of $1$ notebook?
c) If $1$ notebook weighs $0.5$ kg, what is the weight of $9$ notebooks?

Worked Example 3

Use the unitary method to solve a ratio problem:
a) If $3$ parts have value $18$, what is the value of $1$ part?
b) Divide $72$ in the ratio $2:4$ using the value of one part

Worked Example 4

Use the unitary method to solve a rate problem:
a) A car travels $180$ km in $3$ h. Find the distance travelled in $1$ h, then in $5$ h
b) A worker earns $\mathrm{\$}84$ in $7$ h. Find the hourly rate, then the pay for $10$ h

Worked Example 5

Convert rates between units using the unitary method:
a) Convert $72\text{ km/h}$ to $\text{km/min}$
b) Convert $300\text{ cm/min}$ to $\text{m/min}$
c) Convert $2.4\text{ L/h}$ to $\text{mL/min}$

Worked Example 6

Solve each problem using the unitary method and unit conversion where needed:
a) A tap fills $6$ L in $4$ min. How much does it fill in $1$ min and in $15$ min?
b) A cyclist rides $45$ km in $1.5$ h. Find the average speed in km/h

Problems

Problem 1

Use the unitary method to find the value of one unit:
a) $4$ books cost $\mathrm{\$}24$
b) $9$ tickets cost $\mathrm{\$}72$
c) $5$ kg of oranges cost $\mathrm{\$}20$

Problem 2

Solve each ratio or rate problem using the unitary method:
a) If $3$ pencils cost $\mathrm{\$}6$, how much do $8$ pencils cost?
b) If $4$ bags weigh $12$ kg, what is the weight of $1$ bag?
c) If $1$ bag weighs $3$ kg, what is the weight of $7$ bags?

Problem 3

Use the unitary method to solve a ratio problem:
a) If $5$ parts have value $25$, what is the value of $1$ part?
b) Divide $84$ in the ratio $3:4$ using the value of one part

Problem 4

Use the unitary method to solve a rate problem:
a) A bus travels $240$ km in $4$ h. Find the distance travelled in $1$ h, then in $6$ h
b) A cleaner earns $\mathrm{\$}96$ in $8$ h. Find the hourly rate, then the pay for $11$ h

Problem 5

Convert rates between units using the unitary method:
a) Convert $60\text{ km/h}$ to $\text{km/min}$
b) Convert $450\text{ cm/min}$ to $\text{m/min}$
c) Convert $3.6\text{ L/h}$ to $\text{mL/min}$

Problem 6

Solve each problem using the unitary method and unit conversion where needed:
a) A machine fills $9$ L in $6$ min. How much does it fill in $1$ min and in $20$ min?
b) A runner travels $18$ km in $1.5$ h. Find the average speed in km/h

Exercises

Understanding and Fluency

1. Complete each statement:
   a) The unitary method first finds the value of ______ unit
   b) In a rate problem, “per hour” means “for ______ hour”
   c) To divide a quantity in a ratio using the unitary method, first find the total number of ______

2. Use the unitary method to find the value of one unit:
   a) $6$ pens cost $\mathrm{\$}18$
   b) $5$ kg of rice cost $\mathrm{\$}25$
   c) $8$ tickets cost $\mathrm{\$}64$

3. Solve each using the unitary method:
   a) If $4$ apples cost $\mathrm{\$}6$, how much do $10$ apples cost?
   b) If $7$ books weigh $14$ kg, how much does $1$ book weigh?
   c) If $1$ book weighs $2$ kg, how much do $9$ books weigh?

4. Divide each quantity in the given ratio using the unitary method:
   a) Divide $42$ in the ratio $2:5$
   b) Divide $96$ in the ratio $3:5$
   c) Divide $\mathrm{\$}72$ in the ratio $1:2:3$

5. Solve each rate problem using the unitary method:
   a) A car travels $150$ km in $3$ h. How far does it travel in $1$ h?
   b) A worker earns $\mathrm{\$}70$ in $5$ h. What is the hourly rate?
   c) A tap fills $12$ L in $4$ min. How much does it fill in $1$ min?

6. Solve each rate problem using the unitary method:
   a) A cyclist rides $90$ km in $3$ h. How far will the cyclist ride in $7$ h?
   b) A baker uses $2$ kg of flour for $8$ trays. How much flour is needed for $15$ trays?
   c) A machine packs $48$ boxes in $6$ min. How many boxes does it pack in $1$ min and in $20$ min?

7. Convert each rate using the unitary method:
   a) $120\text{ km/h}$ to $\text{km/min}$
   b) $600\text{ mL/min}$ to $\text{L/min}$
   c) $1.8\text{ L/h}$ to $\text{mL/min}$
   d) $240\text{ cm/min}$ to $\text{m/min}$

8. Solve each problem using the unitary method and unit conversion if needed:
   a) A hose fills $15$ L in $5$ min. How much water does it fill in $12$ min?
   b) A train travels $270$ km in $3$ h. Find its average speed in km/h and km/min
   c) A map distance of $3$ cm represents $12$ km in real life. How many kilometres does $1$ cm represent?

Reasoning

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

10. A student says that if $5$ pens cost $\mathrm{\$}20$, then one pen costs $\mathrm{\$}20\times 5=\mathrm{\$}100$. Explain the mistake.

11. Noah says that to divide $56$ in the ratio $3:4$, you divide by $2$ because there are two numbers in the ratio. Is he correct? Explain.

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

13. A student says that $72\text{ km/h}$ means $72$ km in $72$ hours. Describe the error.

Problem-solving

14. A farmer buys $6$ bags of seed for $\mathrm{\$}54$. Find the cost of $1$ bag, then the cost of $11$ bags.

15. A class shares $\mathrm{\$}84$ in the ratio $2:5$. Find each share using the unitary method.

16. A car travels $210$ km in $3$ h. Find the average speed, then find how far it travels in $8$ h at the same rate.

17. A tap fills $7.2$ L in $6$ min. Find the amount filled in $1$ min, then in $25$ min.

18. Convert $90\text{ km/h}$ to $\text{km/min}$ using the unitary method.

19. A runner travels $24$ km in $2$ 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