Measuring volume

Volume is a measurement of how much space an object takes up (e.g. 600 ml of water). Capacity is a measure of how much liquid a container can hold when its full. (e.g. a 2 litre bottle). For example, if you have a 500 ml bottle of cola, with 200 ml of cola left inside it, the capacity of the bottle is 500 ml, while the volume of cola inside it is 200 ml.

As with length and weight, we use different containers or instruments to measure the volume of different quantities of liquid or dry ingredients. Some examples are given below:

Table 1

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Measuring spoons come in different standard sizes or capacities, including a teaspoon (5 ml) and a tablespoon (15 ml). Some sets of spoons also include \(\frac{\text{1}}{\text{2}}\) and \(\frac{\text{1}}{\text{4}}\) teaspoons.

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Measuring cups also come in standard capacities, including 1 cup (250 ml), \(\frac{\text{1}}{\text{2}}\) a cup (125 ml) and \(\frac{\text{1}}{\text{4}}\) cup (63 ml).

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Measuring jugs come in many different sizes, but the most common capacity is 1 litre. The measuring jug on the left gives measurements in litres and millilitres. It has a capacity of 1 \(\ell\).

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Flasks, like measuring jugs, come in different capacities. They usually don't come with any calibrated measurements, (just a capacity measurement) so the only way to know what the volume of liquid inside a flask is, is to pour it out into a measuring jug.

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The capacity of an average household bucket is 10 litres. Some buckets have litre markings on the inside that enable you to measure off a volume of liquid less than 10 litres. (This is only accurate to the nearest litre though!)

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The capacity of a wheelbarrow is usually about 170 litres.

As we learned in Chapter 3, it is possible to estimate the quantities of a substance that we need - for example heaped teaspoons. Another common way of estimating is using a fraction of a standard quantity, for example a quarter teaspoon of salt, or half a brick of butter.

Example 1: Measuring volume

Question

  1. An urn of boiling water in an office has a capacity of 20 litres.

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    1. If it is filled to maximum capacity, calculate the number of 250 ml cups that can be shared from it.
    2. After everyone has their morning tea, there are only 6 litres of water left in the urn.

      1. How much water is this in ml?
      2. How many 250 ml cups of water are left in the urn now?
      3. What percentage of the urn still has water in it?
  2. Jabu is building a new flower bed and is using a bucket to carry soil from another part of the garden to the new bed. He knows his bucket has a capacity of 10 \(\ell\).

    1. If he has 300 \(\ell\) of soil that needs to be moved, and for each trip he fills the bucket to the top with soil, how many trips will Jabu have to make with the bucket to move all the soil?
    2. Jabu decides that 10 litres of soil is too heavy to carry. How many trips will he have to make to move all the soil if he only fills the bucket with 7 litres of soil at a time?
    3. Jabu's friend Matthew arrives with his wheelbarrow and a spade. He suggests that Jabu should rather move the soil using the wheelbarrow. If the wheelbarrow has a capacity of 150 litres and they fill it to the brim, how many trips will Jabu have to make to move all the soil?
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  3. Dorothy goes hiking with her friends every Sunday morning. She always takes a flask of tea. She knows that the lid of the flask (which doubles as a cup) can hold 200 ml of water. If she can get five and a half cups of tea out of the flask, calculate the capacity of the flask, in litres.

Answer

  1. 20 litres = 20 000 ml
    20 000 ml \(\div\) 250 ml = 80
    80 250 ml cups can be poured from the urn.
    1. 6 \(\ell\) = 6000 ml
    2. 6000 ml \(\div\) 250 ml = 24
      There are 24 cups of water left in the urn.
    3. \(\frac{\text{6}\text{ ℓ}}{\text{20}\text{ ℓ}} \times \text{100} = \text{30}\%\)
      The urn is 30% full.
    1. \(\text{300}\text{ ℓ} \div \text{10}\text{ ℓ}\) = 30 trips.
    2. \(\text{300}\text{ ℓ} \div \text{7}\text{ ℓ} = \text{42,8}\). Jabu can't make 0,8 of a trip so we round this up to 43 trips (even though the bucket won't have 7 litres of soil in it for the last trip).
    3. \(\text{300}\text{ ℓ} \div \text{150}\text{ ℓ}\) = 2 trips.
  2. 200 ml \(\times\) 5,5 cups = 1100 ml = 1,1 \(\ell\).
    The capacity of her flask is 1,1 litres.

Exercise 1: Measuring and comparing volume

A six pack of soft drinks contains 6 cans of 330 ml each. What is the total volume of soft drink in a six pack? Give your answer in litres.

6 \(\times\) 330 ml = 1980 ml = 1,98 litres

A large juice container has a capacity of 30 litres.

  1. If the container is 75% full, calculate the amount of juice in the container in litres.

  2. How many 300 ml cups of juice can you fill (to the top)?

  1. 30 litres \(\times\) 0,75 = 22,5 litres

  2. 22,5 litres \(\div\) 300 ml = 2250 ml \(\div\) 300 ml = 6,8 cups = 6 full cups.

Jonathan uses the following recipe to make chocolate muffins:
\(\frac{\text{2}}{\text{3}}\) cup of baking cocoa
2 large eggs
2 cups of flour
\(\frac{\text{1}}{\text{2}}\) cup of sugar
2 teaspoons of baking soda
\(\text{1}\frac{\text{1}}{\text{3}}\) cups of milk
\(\frac{\text{1}}{\text{3}}\) cup of sunflower oil
1 teaspoon of vanilla essence
\(\frac{\text{1}}{\text{2}}\) teaspoon of salt

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  1. If 1 teaspoon = 5 ml, calculate how much baking soda Jonathan will use. Give your answer in ml.

  2. Calculate the amount of vanilla essence Jonathan will use in this recipe. Give your answer in ml.

  3. Jonathan does not own measuring cups but he does own a measuring jug calibrated in ml. How many ml of flour does he need? (1 cup = 250 ml).

  4. If Jonathan buys a 100 ml bottle of vanilla essence, how many times will he be able to use the same bottle, if he bakes the same amount of muffins each time?

  5. The recipe above is used to make 30 muffins. Calculate how many cups of flour Jonathan will need to make 45 muffins.

  1. 2 \(\times\) 5 ml = 10 ml.

  2. 1 tsp = 5 ml.

  3. 2 cups flour \(\times\) 250 ml = 500 ml.

  4. 100 ml \(\div\) 5 ml = 20 times.

  5. \(\dfrac{\text{2} \text{ cups flour}}{\text{30} \text{ cupcakes}} = \dfrac{\text{3} \text{ cups flour}}{\text{45} \text{ cupcakes}}\) so he will need 3 cups of flour.

Example 2: Calculating costs

Question

  1. Suppose paraffin is sold at R 7,80 per litre at the local service station.

    1. How much will you pay for 5 litres of paraffin?
    2. How many litres of paraffin will you be able to buy for R 20? Round off your answer to two decimal places.
    3. If you have a paraffin lamp at home that can hold 500 ml of paraffin, how many times will you be able to refill the lamp if you buy 3 litres of paraffin?
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  2. If petrol costs R 11,72 a litre:

    1. Calculate how much it costs to fill up a car that has a tank of 50 litres.
    2. Calculate how many litres you could buy with R 200. Round off your answers to two decimal places.
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Answer

    1. Number of litres \(\times\) Cost per litre
      = 5 litres \(\times\) R 7,80 = R 39
    2. Amount of money \(\div\) Cost per litre
      = R 20 \(\div\) R 7,80 = 2,564 102 56\(\ldots\)
      \(\approx\) 2,56 litres (to two decimal places)
    3. 3 litres = 3000 ml
      3000 ml \(\div\) 500 ml = 6
      You would be able to refill the lamp 6 times.
    1. Number of litres \(\times\) Cost per litre
      = 50 litres \(\times\) R 10,72 = R 536
    2. Amount of money \(\div\) Cost per litre
      = R 200 \(\div\) R 10,72 = 18,656 716 4\(\ldots\)
      \(\approx\) 18,66 litres (to two decimal places)

Exercise 2: Measuring volume and calculating costs

Thandi is baking cupcakes and the recipe she has requires \(\text{1}\frac{\text{1}}{\text{3}}\) cups of milk.

  1. Calculate how many ml of milk she will need if 1 cup = 250 ml.

  2. If the recipe is designed to produce 20 cupcakes, calculate the amount of milk required to bake 30 cupcakes. Give your answer in litres.

  3. Milk is sold in bottles of 1 litre each for R 8,50 at the local store. Calculate the amount of money Thandi will need to spend on milk to make the 30 cupcakes.

  1. \(\text{1}\frac{\text{1}}{\text{3}}\) cups = \(\frac{\text{4}}{\text{3}}\). \(\frac{\text{4}}{\text{3}} \times \text{250}\) ml = 333 ml of milk.

  2. \(\dfrac{\text{330}\text{ ml}}{\text{20} \text{ cupcakes}} = \dfrac{\text{500}\text{ ml}}{\text{30} \text{ cupcakes}}\). She will need 500 ml of milk.

  3. 1 bottle = R 8,50 (she will only use half).

Thabiso decides to sell homemade lemonade. He has made 5 litres of lemonade to sell at the local schools' rugby tournament.

  1. Thabiso will be selling his lemonade in 250 ml plastic cups. Calculate the number of cups of lemonade he will be able to sell.

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  2. If he sells the lemonade at R 5 per cup, how much money will he make from the lemonade? (Assume that he sold all of his lemonade).

  3. If it cost Thabiso R 120 to make the lemonade, how many cups would he need to sell (at R 5 each) before he's made back the money he spent?

  1. 5 litres = 5000 ml. 5000 ml \(\div\) 250 ml = 20 cups.

  2. 20 cups \(\times\) R 5 = R 100

  3. R 120 \(\div\) R 5 = 24. He would need to sell 24 cups just to recoup his costs.