Ratio
🎯 In this topic you will
- Find and use equivalent ratios to represent the same relationship between quantities.
- Simplify ratios by writing them in their simplest form.
🧠 Key Words
- ratio
- equivalent ratio
- simplest form
Show Definitions
- ratio: A comparison between two or more quantities showing how many times one value contains or relates to another.
- equivalent ratio: Ratios that represent the same relationship between quantities even though the numbers used are different.
- simplest form: A ratio written using the smallest whole numbers that still keep the same relationship between the quantities.
🍎 What Is a Ratio?
A ratio is a relationship between two or more numbers. It tells us how one quantity compares with another quantity.
🍊 Comparing Apples and Oranges
T his bowl of fruit contains six apples and three oranges. The ratio of apples to oranges is six to three. We can write this ratio as $6 : 3$. This ratio is equivalent to $2 : 1$. In this unit, you will learn how to find and use equivalent ratios.
❓ EXERCISES
1. Look at these shapes.
Write in its simplest form:
a. the ratio of circles to squares
b. the ratio of squares to pentagons
c. the ratio of pentagons to squares.
👀 Show answer
b. There are $4$ squares and $6$ pentagons. Ratio $4:6 = 2:3$.
c. There are $6$ pentagons and $4$ squares. Ratio $6:4 = 3:2$.
2. Write these ratios in their simplest form.
a. $3 : 12$
b. $21 : 14$
c. $12 : 9$
d. $24 : 8$
e. $150 : 25$
f. $15 : 300$
👀 Show answer
b. $21:14 = 3:2$
c. $12:9 = 4:3$
d. $24:8 = 3:1$
e. $150:25 = 6:1$
f. $15:300 = 1:20$
3. Write the missing numbers.
a. $2 : 3 = 12 : \square$
b. $5 : 7 = \square : 21$
c. $3 : 5 = 15 : \square$
Check your answers to question $3$ with your partner.
👀 Show answer
b. $5:7 = 15:21$ → missing number $=15$
c. $3:5 = 15:25$ → missing number $=25$
4. Place the ratios in the correct position in the table. One ratio cannot be placed. Which one is it?
$18:27,\ 12:16,\ 28:35,\ 16:20,\ 4:6,\ 14:21,\ 24:32,\ 18:24,\ 36:27,\ 36:45,\ 21:28,\ 8:12,\ 32:40$
| Equivalent to $2:3$ | Equivalent to $3:4$ | Equivalent to $4:5$ |
|---|---|---|
👀 Show answer
Equivalent to $3:4$: $12:16,\ 18:24,\ 21:28,\ 24:32$
Equivalent to $4:5$: $16:20,\ 28:35,\ 32:40,\ 36:45$
Ratio that cannot be placed: $36:27$
5. Pierre plants $4$ carrots for every $3$ onions. He plants $48$ carrots.
a. How many onions does he plant?
b. How many carrots and onions does he plant altogether?
👀 Show answer
a. Onions $=3 \times 12 = 36$.
b. Total plants $=48 + 36 = 84$.
6. The ratio of girls to boys in a dance competition is $3 : 2$.
$42$ girls take part in the competition.
How many boys and girls take part in the competition altogether?
👀 Show answer
If $3$ parts $=42$, one part $=14$.
Boys $=2 \times 14 = 28$.
Total $=42 + 28 = 70$.
7. Ollie and Igor share $24$ counters in different ways.
a. How many counters does Igor get when he has twice as many as Ollie?
b. How many counters does Igor get when he has one counter for every three counters Ollie has?
c. How many counters does Igor get when they share the counters in the ratio Igor to Ollie is $3 : 5$?
👀 Show answer
Igor $= \frac{2}{3} \times 24 = 16$.
b. Ratio Igor:Ollie $=1:3$. Total parts $=4$.
Igor $= \frac{1}{4} \times 24 = 6$.
c. Ratio Igor:Ollie $=3:5$. Total parts $=8$.
Igor $= \frac{3}{8} \times 24 = 9$.
🧠 Think like a Mathematician
Problem:
Suzy spent $4 on beads to make necklaces.
White beads cost $10$ cents each and coloured beads cost $20$ cents each.
Suzy bought beads in the ratio of white to coloured $3:1$.
How many white and coloured beads did she buy?
Tip
Try tabulating your information.
| Number of white beads | Cost of white beads | Number of coloured beads | Cost of coloured beads | Total cost |
|---|---|---|---|---|
| 24 | $24 \times 10 = 240$ cents | 8 | $8 \times 20 = 160$ cents | $240 + 160 = 400$ cents |
Follow-up Questions:
👀 show answer
- 1: A ratio of $3:1$ means each group contains $3+1=4$ beads.
- 2: Let the number of coloured-bead groups be $x$. White beads = $3x$, coloured beads = $x$. Total cost equation: $3x \times 10 + x \times 20 = 400$
- 3:$30x + 20x = 400$$50x = 400$$x = 8$ Coloured beads = $8$ White beads = $3 \times 8 = 24$Suzy bought 24 white beads and 8 coloured beads.