144. Dividing Quantities in a Ratio

Learning Intentions

• To understand that a quantity can be divided into a ratio
• find an unknown value for items in a given ratio
• divide a quantity in a particular ratio

Pre-requisite Summary

• Know that a ratio compares quantities
• Be able to write ratios using the $:$ symbol
• Understand that equivalent ratios represent the same comparison
• Be able to simplify ratios where appropriate
• Know that the total number of parts in a ratio is found by adding the ratio numbers
• Be able to multiply and divide whole numbers accurately
• Understand that one part can be found by dividing the total quantity by the total number of parts

Worked Examples

Worked Example 1

A quantity is divided in the ratio shown. Find the total number of parts:

a) $2:3$
b) $4:1$
c) $3:5:2$

Worked Example 2

Find the unknown value:

a) If $x:y=3:4$ and $x=12$, find $y$
b) If $a:b=2:5$ and $b=20$, find $a$

Worked Example 3

Find the unknown value:

a) If boys:girls $=4:7$ and there are $28$ girls, how many boys are there?
b) If red:blue $=5:3$ and there are $15$ red counters, how many blue counters are there?

Worked Example 4

Divide the quantity in the given ratio:

a) Divide $35$ in the ratio $2:3$
b) Divide $48$ in the ratio $5:3$

Worked Example 5

Divide the quantity in the given ratio:

a) Divide $90$ in the ratio $2:3:4$
b) Divide $\mathrm{\$}72$ in the ratio $1:2:3$

Worked Example 6

Solve the worded problem:

a) A prize of $\mathrm{\$}84$ is shared between Amy and Ben in the ratio $3:4$. How much does each person receive?
b) A rope of length $60$ m is cut into two pieces in the ratio $1:4$. Find the length of each piece.

Problems

Problem 1

A quantity is divided in the ratio shown. Find the total number of parts:

a) $3:2$
b) $5:1$
c) $2:4:3$

Problem 2

Find the unknown value:

a) If $x:y=2:7$ and $x=10$, find $y$
b) If $a:b=3:8$ and $b=32$, find $a$

Problem 3

Find the unknown value:

a) If boys:girls $=5:6$ and there are $30$ girls, how many boys are there?
b) If red:blue $=4:9$ and there are $16$ red counters, how many blue counters are there?

Problem 4

Divide the quantity in the given ratio:

a) Divide $42$ in the ratio $1:5$
b) Divide $56$ in the ratio $3:4$

Problem 5

Divide the quantity in the given ratio:

a) Divide $72$ in the ratio $2:3:4$
b) Divide $\mathrm{\$}96$ in the ratio $1:3:4$

Problem 6

Solve the worded problem:

a) A prize of $\mathrm{\$}63$ is shared between Mia and Noah in the ratio $2:5$. How much does each person receive?
b) A ribbon of length $45$ m is cut into two pieces in the ratio $2:3$. Find the length of each piece.

Exercises

Understanding and Fluency

1. Find the total number of parts in each ratio:
   a) $2:5$
   b) $3:7$
   c) $4:1$
   d) $1:2:3$

2. Find the unknown value:
   a) If $x:y=1:4$ and $x=6$, find $y$
   b) If $a:b=3:5$ and $a=12$, find $b$
   c) If $m:n=2:9$ and $n=27$, find $m$

3. Find the unknown value:
   a) If cats:dogs $=4:3$ and there are $12$ dogs, how many cats are there?
   b) If apples:oranges $=5:2$ and there are $20$ apples, how many oranges are there?
   c) If red:green $=3:8$ and there are $24$ green counters, how many red counters are there?

4. Divide each quantity in the given ratio:
   a) Divide $20$ in the ratio $1:3$
   b) Divide $45$ in the ratio $2:7$
   c) Divide $64$ in the ratio $3:5$

5. Divide each quantity in the given ratio:
   a) Divide $54$ in the ratio $2:3:4$
   b) Divide $80$ in the ratio $1:1:3$
   c) Divide $\mathrm{\$}70$ in the ratio $2:5:7$

6. Solve each ratio problem:
   a) A total of $36$ stickers is shared in the ratio $1:2$. How many stickers does each part get?
   b) A class has boys:girls in the ratio $3:4$. If there are $12$ boys, how many girls are there?
   c) A rope is cut in the ratio $3:5$ and has total length $48$ m. Find the two lengths.

7. Write the steps needed to divide a quantity in a ratio:
   a) find the total number of parts
   b) find the value of one part
   c) multiply by each part of the ratio

8. Solve each:
   a) Divide $96$ in the ratio $5:7$
   b) Divide $120$ in the ratio $2:3:5$
   c) A sum of $\mathrm{\$}90$ is shared in the ratio $4:5$. Find each share.

Reasoning

9. Explain why dividing a quantity in the ratio $2:3$ does not mean dividing it into two equal parts.

10. A student says that to divide $40$ in the ratio $1:3$, you should do $40÷2=20$ and $20$. Explain the mistake.

11. Noah says that if the ratio is $2:5$ and one part is worth $6$, then the total is $2+5+6=13$. Is he correct? Explain.

12. Explain why the first step in dividing a quantity in a ratio is to find the total number of parts.

13. A student divides $\mathrm{\$}72$ in the ratio $1:2:3$ and writes $\mathrm{\$}72,\mathrm{\$}144,\mathrm{\$}216$. Describe the error.

Problem-solving

14. A prize of $\mathrm{\$}56$ is shared between two students in the ratio $3:4$. Find how much each student receives.

15. A class has boys:girls in the ratio $2:3$. If there are $18$ girls, how many boys are there?

16. A $72$ m ribbon is cut into three pieces in the ratio $2:3:4$. Find the length of each piece.

17. A farmer divides $84$ kg of feed between two animals in the ratio $5:7$. How much feed does each animal get?

18. A total of $\mathrm{\$}108$ is shared among three friends in the ratio $1:2:6$. Find each share.

19. In a bag, red and blue marbles are in the ratio $4:9$. If there are $36$ blue marbles, how many red marbles are there?

Potential Misunderstandings

• Students may think a ratio such as $2:3$ means the total is always $5$
• Students may divide the quantity by the number of terms in the ratio instead of by the total number of parts
• Students may forget to add all parts of the ratio first
• Students may find the value of one part correctly but then multiply incorrectly
• Students may reverse the order of the ratio when giving answers
• Students may assume the shares are equal because the quantity is being divided
• Students may confuse finding an unknown value from a ratio with simply adding or subtracting the ratio numbers