Capacity & volume
🎯 In this topic you will
- Look at the difference between capacity and volume.
🧠 Key Words
- capacity
- volume
Show Definitions
- capacity: The amount of liquid that a container can hold.
- volume: The amount of space that an object or substance occupies.
Measuring Ingredients
When you are cooking or baking you need to measure out ingredients. Solid ingredients such as rice, pasta or vegetables can be weighed on kitchen scales. When you measure liquid ingredients such as milk, oil or water you will need to use a measuring jug.
Getting the Right Amount
If you want the perfect pancakes, you need to measure the correct amount of milk!
❓ EXERCISES
$1$. For each of these jugs write down:
a. the capacity of the jug
b. the volume of water in the jug
👀 Show answer
a. Capacity is the maximum value shown on each jug scale.
b. Volume is the level where the water reaches on the scale.
$2$. Read what Sofia says.
a. Explain why Sofia is correct.
b. Use Sofia’s example to help you copy and complete this table.
| millilitres | litres and millilitres | litres |
|---|---|---|
| $2500\ \text{ml}$ | $2\ 500\ \text{ml}$ | $2.5\ \text{l}$ |
| $3200\ \text{ml}$ | ||
| $4\ 300\ \text{ml}$ | ||
| $3.7\ \text{l}$ | ||
| $0\ 800\ \text{ml}$ | ||
| $12\ 100\ \text{ml}$ |
👀 Show answer
a. Sofia is correct because $1\ \text{litre} = 1000\ \text{ml}$. Therefore $2500\ \text{ml} = 2.5\ \text{litres}$.
b.
- $3200\ \text{ml} = 3\ 200\ \text{ml} = 3.2\ \text{l}$
- $4\ 300\ \text{ml} = 4.3\ \text{l}$
- $3.7\ \text{l} = 3700\ \text{ml}$
- $0\ 800\ \text{ml} = 800\ \text{ml} = 0.8\ \text{l}$
- $12\ 100\ \text{ml} = 12.1\ \text{l}$
🧠 Think like a Mathematician
Question: Marcus and Arun are looking at this question. What is the volume of water in this jug?
What they say:
Follow-up Questions:
Show Answers
- a: Arun is correct. The water level is at the second small division above $6$, so the volume is $6.2$ litres.
- b: Marcus has counted the scale incorrectly. He has read the first small division above $6$ as the water level instead of the second one.
- c: The jug’s capacity is $10$ litres, so the amount needed is $10 - 6.2 = 3.8$ litres.
- d: To read a scale correctly, first check the labelled numbers, then count how many equal small divisions are between them, and work out the value of each division before reading the level. For part $c$, you can subtract the current volume from the capacity, count up from $6.2$ to $10$, or use number facts with decimals.
❓ EXERCISES
$3$. For each of these jugs write down:
a. the capacity of the jug
b. the volume of water in the jug
👀 Show answer
a. Capacity is the largest value shown on each jug scale.
b. Volume is the level where the water reaches on the scale.
$4$. What volume of water must be added to these jugs to fill them to capacity?
👀 Show answer
Subtract the volume shown from the capacity of each jug to find the amount of water needed.
$5$. Chipo needs to measure out $2.3$ litres of milk.
She only has the measuring jug shown.Explain how she can use this measuring jug to measure out $2.3$ litres of milk.
👀 Show answer
$2.3$ litres equals $2300$ ml. She can fill the $500$ ml jug four times to get $2000$ ml and then add another $300$ ml.
$6$. Vishan buys a fish tank with a capacity of $120$ litres.
He pours water into the tank until it is $\dfrac{3}{4}$ full.
What is the volume of the water in the tank?
👀 Show answer
$\dfrac{3}{4} \times 120 = 90$. The volume of water in the tank is $90$ litres.
🧠 Think like a Mathematician
Mair has four measuring cups A, B, C and D. The capacity of each cup, in millilitres, is shown.
Question: How can the cups be used to measure the following volumes?
Show Answers
- a.i:$240 + 160 = 400$ ml (cup A and cup B).
- a.ii:$240 + 120 = 360$ ml (cup A and cup C).
- a.iii:$240 + 120 + 60 = 420$ ml (cups A, C and D).
- a.iv:$160 + 160 = 320$ ml (cup B twice).
- a.v:$120 + 60 = 180$ ml (cups C and D).
- a.vi:$240 + 240 + 120 = 600$ ml (cups A, A and C).
- b: Accurate volumes can be measured by adding together the capacities of different cups. By combining cups with different sizes, many different totals can be made.
❓ EXERCISES
$7$. Each of the containers below is marked with its capacity. Estimate the volume of liquid in each container.
Order the volumes of liquid from the least to the greatest.
👀 Show answer
Estimate the fraction of each container that is full and multiply by its capacity.
An example ordering from least to greatest may be:
$B < D < A < E < C < F$
$8$. Copy and complete the Carroll diagram to sort the containers by their capacity and the volume of liquid they contain.
| Volume of $500\ \text{ml}$ or less | Volume of more than $500\ \text{ml}$ | |
|---|---|---|
| Capacity of $1$ litre or less | $C,\ E$ | $A$ |
| Capacity of more than $1$ litre | $B$ | $D,\ F$ |
👀 Show answer
First determine the capacity of each container. Then read the scale to estimate the amount of liquid inside. Finally place each container in the correct part of the Carroll diagram based on both properties.
🧠 Think like a Mathematician
Investigation: Investigate what amounts you can make with jug A that has a capacity of $3$ litres and jug B that has a capacity of $4$ litres, when neither jug has a measurement scale.
It is possible to measure a volume of any whole number of litres from $1$ to $7$ with the two jugs by filling a whole jug, pouring water from one jug to the other or emptying a jug.
Tasks:
- Investigate and record how each volume from $1$ litre to $7$ litres can be made using the $3$ litre and $4$ litre jugs.
- Investigate which amounts of whole litres can be made with a $3$ litre jug and a $5$ litre jug.
- Predict what amounts you will be able to make with $3$ litre and $6$ litre jugs.
- Check your prediction and record which numbers of whole litres can be made and how they are made.
- Suggest an explanation for what you have found.
- Choose two jug sizes of your own to investigate. Predict what volumes can be made. Test your prediction and record your results.
Show Answers
- Using 3 L and 4 L jugs: All whole numbers from $1$ to $7$ litres can be made by repeatedly filling, pouring and emptying the jugs.
- Using 3 L and 5 L jugs: All whole numbers from $1$ to $8$ litres can be made.
- Using 3 L and 6 L jugs: Only multiples of $3$ litres can be made ($3$, $6$, $9$…), because both jugs share the same factor.
- Explanation: The volumes that can be made depend on the common factors of the jug sizes. When the capacities are relatively prime (like $3$ and $4$), every whole number up to their total can be created.