Ratio and proportion
🎯 In this topic you will
- Use the language “in every” and “out of” to describe proportion
- Use the language “for every” to describe ratio
- Use the symbol “:” to represent ratios
🧠 Key Words
- proportion
- ratio
Show Definitions
- proportion: A way of comparing part of a whole or expressing how one quantity relates to an entire group.
- ratio: A comparison of two quantities showing how many times one value contains or is contained within the other.
What a Ratio Means
Ratio is used to compare two or more quantities. It tells us how much of one thing there is in relation to another.
Understanding Order in Ratios
The ratio of circles to squares is written as 3 : 1, while the ratio of squares to circles is written as 1 : 3. A ratio must always be written in the correct order so its meaning is clear.
❓ EXERCISES
1. Draw a bead pattern to match each of these descriptions.
a. For every $1$ black bead, $3$ beads are white.
b. $1$ in every $4$ beads is white.
👀 Show answer
1a. Any repeating pattern showing $1$ black bead followed by $3$ white beads.
1b. Any repeating pattern where exactly $1$ out of every $4$ beads is white.
2. $1$ in every $4$ squares in this pattern is black.
The pattern continues in the same way.
a. What is the ratio of white squares to black squares?
👀 Show answer
For every $1$ black square, there are $3$ white squares.
Ratio = $3 : 1$.
b. Copy and complete this table.
| White squares | Black squares |
|---|---|
| $3$ | |
| $6$ | |
| $3$ |
👀 Show answer
Using the ratio $3:1$:
White $3$ → Black $1$
White $6$ → Black $2$
Black $3$ → White $9$
3. Zara has $10$ blue pens and $5$ red pens.
Write whether each statement is true or false.
a. $\tfrac{1}{3}$ of the pens are red.
b. The ratio of red pens to blue pens is $10 : 5$.
c. $50\%$ of the pens are red.
d. $1$ in every $3$ pens is red.
👀 Show answer
Total pens = $15$. Red = $5$.
3a. $\tfrac{1}{3}$ of $15$ is $5$ → True.
3b. Red : Blue = $5 : 10$, not $10 : 5$ → False.
3c. $\tfrac{5}{15} = \tfrac{1}{3}$, which is $33.3\%$ → False.
3d. $1$ in every $3$ is $\tfrac{1}{3}$ → True because $\tfrac{5}{15} = \tfrac{1}{3}$.
4. A bag contains $2$ orange counters, $3$ blue counters and $5$ green counters.
a. What is the ratio of orange : blue : green?
b. What is the ratio of blue : green : orange?
c. What is the ratio of green : blue : orange?
d. What proportion of the counters are blue?
Write your answer as a fraction and as a percentage.
👀 Show answer
Total counters = $10$.
4a. $2 : 3 : 5$
4b. $3 : 5 : 2$
4c. $5 : 3 : 2$
4d. Blue = $3$. Proportion = $\tfrac{3}{10}$ = $30\%$.
5. Here is a recipe for pasta sauce.
a. What is the ratio of onions : tomatoes : mushrooms?
b. What is the ratio of tomatoes : mushrooms : onions?
c. What proportion of the recipe is tomatoes?
Write your answer as a fraction.
👀 Show answer
Recipe shown:
Mushrooms $1$ cup, Onions $2$ cups, Tomatoes $4$ cups.
5a. Onions : Tomatoes : Mushrooms = $2 : 4 : 1$.
5b. Tomatoes : Mushrooms : Onions = $4 : 1 : 2$.
5c. Total = $7$ cups. Tomatoes = $4$ → $\tfrac{4}{7}$.
6. Here is a string of grey and white beads.
What proportion of the beads are grey?
Give your answer as a fraction.
👀 Show answer
Count beads: Grey = $8$, Total = $16$ → $\tfrac{8}{16} = \tfrac{1}{2}$.
7. Sofia says her diagram shows black circles and white circles in the ratio $1 : 3$.
Sofia is not correct.
Explain how she can correct her answer.
👀 Show answer
Count circles: White = $3$, Black = $2$ → Correct ratio is $2 : 3$ (black : white).
She must recount the circles and write the ratio in the correct order.
🧠 Think like a Mathematician
a. How tall do you think the cactus is?
If the person is $160\ \text{cm}$ tall, how tall is the cactus?
b. Four girls describe a fruit smoothie made of kiwis and bananas. Three of the descriptions are right, but one is wrong. Which girl is wrong?
Alana: $\tfrac{3}{10}$ of the smoothie is banana.
Fatima: For every $2$ kiwis there is $1$ banana.
Haibo: For every $3$ bananas there are $7$ kiwis.
Orla: $70\%$ of the smoothie is kiwi.
If you explain your results, you will show you are convincing.
Show Answers
- a. Comparing the person’s height to the cactus in the picture, the cactus appears a little more than twice as tall as the person. If the person is $160\ \text{cm}$ tall, the cactus is approximately $\approx 2.2 \times 160 = 352\ \text{cm}$. A reasonable estimate is between $320$ and $360\ \text{cm}$.
- b. Convert all statements into the same form to compare them:
- Alana: bananas = $\tfrac{3}{10}$, so kiwi = $\tfrac{7}{10}$ → kiwi $70\%$.
- Fatima: ratio kiwi : banana = $2:1$ → kiwi fraction = $\tfrac{2}{3}$ (~$66.7\%$).
- Haibo: ratio kiwi : banana = $7:3$ → kiwi fraction = $\tfrac{7}{10}$ (= $70\%$).
- Orla: kiwi = $70\%$ = $\tfrac{7}{10}$.