032. Ratios, Fractions and Proportions

Learning Intentions

• To understand the connection between ratios, fractions and proportions
• solve proportion problems using ratios and fractions
• divide a given quantity in a given ratio.

Pre-requisite Summary

• Understand that a ratio compares quantities in a given order
• Be able to simplify ratios using a common factor
• Understand that a fraction represents part of a whole
• Be able to find equivalent fractions
• Know that proportion describes two equal ratios or equal fractional relationships
• Be able to multiply and divide whole numbers accurately
• Understand how to find the total number of parts in a ratio
• Be able to interpret worded problems involving sharing quantities

Worked Examples

Worked Example 1

a) Write the ratio $3:5$ as a fraction in two ways.
b) Explain how the ratio $3:5$ is connected to the fraction ${\displaystyle \frac{3}{8}}$ of the total for the first part.
c) Explain the meaning of proportion in this context.

Worked Example 2

Use ratios and fractions to solve:
a) In a group, the ratio of boys to girls is $2:3$. What fraction of the group are boys?
b) What fraction of the group are girls?
c) Explain why the fractions add to $1$.

Worked Example 3

Solve a proportion problem:
a) A recipe uses flour and sugar in the ratio $4:1$. If there are $20$ cups in total, how many cups are flour and how many are sugar?
b) Show the fraction of the total for each ingredient.

Worked Example 4

Divide a quantity in a given ratio:
a) Divide $48$ in the ratio $1:3$.
b) Divide $90$ in the ratio $2:3$.

Worked Example 5

Divide a quantity in a given ratio:
a) Divide $84$ in the ratio $2:5$.
b) Divide $120$ in the ratio $3:2:1$.

Worked Example 6

Solve a worded proportion problem:
a) A class has red and blue counters in the ratio $3:7$. There are $40$ counters altogether. How many are red and how many are blue?
b) What fraction of the counters are red?
c) What fraction of the counters are blue?

Problems

Problem 1

a) Write the ratio $2:7$ as a fraction in two ways.
b) Explain how the ratio $2:7$ is connected to the fraction ${\displaystyle \frac{2}{9}}$ of the total for the first part.
c) Explain the meaning of proportion in this context.

Problem 2

Use ratios and fractions to solve:
a) In a group, the ratio of cats to dogs is $4:5$. What fraction of the group are cats?
b) What fraction of the group are dogs?
c) Explain why the fractions add to $1$.

Problem 3

Solve a proportion problem:
a) A drink uses cordial and water in the ratio $1:4$. If there are $25$ cups in total, how many cups are cordial and how many are water?
b) Show the fraction of the total for each ingredient.

Problem 4

Divide a quantity in a given ratio:
a) Divide $36$ in the ratio $1:2$.
b) Divide $75$ in the ratio $2:3$.

Problem 5

Divide a quantity in a given ratio:
a) Divide $98$ in the ratio $3:4$.
b) Divide $144$ in the ratio $2:1:3$.

Problem 6

Solve a worded proportion problem:
a) A bag has red and yellow beads in the ratio $2:5$. There are $42$ beads altogether. How many are red and how many are yellow?
b) What fraction of the beads are red?
c) What fraction of the beads are yellow?

Exercises

Understanding and Fluency

1. Write each ratio as fractions of the total:
   a) $1:3$
   b) $2:5$
   c) $4:7$

2. Write each ratio as fractions of the total:
   a) $3:2$
   b) $5:1$
   c) $2:3:5$

3. Find the fraction of the total for each part:
   a) boys:girls $=2:3$
   b) apples:oranges $=3:5$
   c) red:blue:green $=1:2:3$

4. Solve using ratios and fractions:
   a) The ratio of black pens to blue pens is $1:4$. What fraction are black pens?
   b) What fraction are blue pens?
   c) What is the total number of parts?

5. Divide each quantity in the given ratio:
   a) $24$ in the ratio $1:3$
   b) $35$ in the ratio $2:3$
   c) $54$ in the ratio $4:5$

6. Divide each quantity in the given ratio:
   a) $72$ in the ratio $1:2:3$
   b) $90$ in the ratio $2:3:4$
   c) $63$ in the ratio $4:2:1$

7. Solve the proportion problems:
   a) A recipe has oil and vinegar in the ratio $2:3$. If there are $25$ mL altogether, how much is oil?
   b) How much is vinegar?
   c) What fraction of the mixture is oil?

8. Solve the proportion problems:
   a) A collection has stamps and coins in the ratio $3:7$. If there are $50$ items, how many are stamps?
   b) How many are coins?
   c) What fraction of the collection is coins?

Reasoning

9. Explain why the ratio $2:3$ means the first quantity is ${\displaystyle \frac{2}{5}}$ of the total.

10. A student says the ratio $3:4$ means the first quantity is ${\displaystyle \frac{3}{4}}$ of the total. Explain the mistake.

11. Explain why dividing a quantity in the ratio $2:5$ requires finding $7$ equal parts first.

12. A student divides $48$ in the ratio $1:3$ and gets $1$ and $3$. Explain why this is incorrect.

Problem-solving

13. A class has boys and girls in the ratio $3:5$. There are $32$ students. How many are boys and how many are girls?

14. A farmer divides $56$ trees between two fields in the ratio $3:4$. How many trees go in each field?

15. A prize of $180 is shared in the ratio $2:3:4$. How much does each person receive?

16. A fruit drink is made from juice and water in the ratio $1:3$. If the total volume is $36$ L, how much is juice and how much is water?

17. A bag contains red, blue and green marbles in the ratio $2:5:3$. If there are $80$ marbles, how many of each colour are there?

18. A school divides $210$ books between three classrooms in the ratio $4:3:3$. How many books does each classroom receive?

Potential Misunderstandings

• Students may confuse a ratio such as $3:5$ with the fraction ${\displaystyle \frac{3}{5}}$ of the total instead of recognising that the total is $3+5=8$ parts
• Students may not recognise that fractions from a ratio must add to $1$ for the whole quantity
• Students may reverse the order of the ratio when interpreting the parts
• Students may divide a quantity by one part of the ratio instead of the total number of parts
• Students may simplify a ratio incorrectly before dividing a quantity
• Students may find one share correctly but not use the ratio to find the remaining share
• Students may confuse equal ratios with equal numerical totals
• Students may not understand that proportion compares equivalent relationships, not just any two fractions or ratios