Exercises: Capacity

To celebrate the end of term, the Year 7 students have bought several bottles of orange juice. If each bottle has a capacity of $1,5\ L$, express this amount in centilitres ($cL$) and in millilitres ($mL$).

Match each of the following objects with the unit of capacity of the metric system that you think is most suitable for measuring what fits inside it:

• An Olympic swimming pool.

• A can of cola.

• A syringe for giving a vaccine.

• A mop bucket.

Complete the following statements by converting to the required units:

1. a) $25 L = ??? dL$
2. b) $4.500 mL = ??? L$
3. c) $0,75 kL = ??? L$
4. d) $32 cL = ??? mL$

An athlete carries a hydration pack for her training. If she has filled the reservoir with $1.200\ mL$ of water and then added an extra bottle of $0,5\ L$, how many litres of water does she carry in total?

In the Science laboratory, the students are using a measuring cylinder. If the measuring cylinder shows a capacity of $35\ cL$, express this measurement in compound form (that is, using litres and centilitres).

A farmer has a small tank to water his pots, with a capacity of $2\ dal$ and $5\ L$. How many $1\ L$ bottles can he fill when the tank is completely full?

Remember that in the metric system there is a direct relationship between volume and capacity. If we have a cube-shaped container with a volume of exactly $1\ dm^3$, how many liters of water fit inside it? And how many milliliters fit in a volume of $1\ cm^3$?

(Figure not to scale.)

A recipe for a sponge cake states that you need to add $2\ dL$ of milk. If we have an open one-litre carton in the fridge with $450\ mL$ left, do we have enough milk for the recipe? Justify your answer by converting both quantities to the same unit.

Lucía is preparing a homemade soft drink for her school's end-of-year party. Into a pitcher, she pours $1,5\text{ L}$ of sparkling water, $35\text{ cL}$ of freshly squeezed lemon juice, and $150\text{ mL}$ of strawberry syrup. To calculate how much drink she has made in total, she writes in her notebook:

1. a) Identify the conceptual error Lucía made. Explain why her result is absurd by thinking about everyday containers.
2. b) Correct the calculation: first convert all quantities to litres ($L$) and calculate the exact total capacity of the soft drink.
3. c) Express that same total quantity in decilitres ($dL$) and in centilitres ($cL$), stating in each case how many steps you move on the metric scale.

You have a $5\text{ L}$ water jug and want to distribute it completely into individual hiking bottles of $33\text{ cL}$ capacity each.

1. a) Before doing any mathematical calculation, make a mental estimate: round the capacity of the bottle to a simple fraction of a litre or to whole decilitres and reason whether more or fewer than $15$ bottles can be filled.
2. b) Now perform the exact calculation by expressing both capacities in centilitres ($cL$). How many full bottles can be filled?
3. c) How many centilitres of water are left over in the jug? Express that leftover amount in millilitres ($mL$) and in litres ($L$).
4. d) Compare your initial estimate from part a) with the actual result obtained and explain whether your previous approximation was reasonable.

At a music festival for teenagers, the organisers have bought several types of water containers to distribute among attendees. They have a main tank of $2,5\text{ kL}$, packs of bottles of $50\text{ cl}$ and large bottles of $8\text{ L}$.

If they want to know the total water available in litres, carry out the following conversions:

1. a) Express the capacity of the tank in litres.
2. b) How many litres are in a pack containing $12$ bottles of $50\text{ cl}$?
3. c) If they have served $15$ full large bottles, how many decilitres have they distributed?

In the Science laboratory, the students are preparing a mixture for an experiment on density. They have used the following amounts of different liquids:

• Distilled water: $1\text{ L}$ and $25\text{ cl}$.

• Alcohol: $4\text{ dl}$ and $5\text{ ml}$.

• Oil: $150\text{ ml}$.

Express the total amount of the mixture in simple form in centilitres ($cl$) and then in compound form using litres, decilitres and millilitres.

An aquarium is in the shape of a cuboid (rectangular prism) with the dimensions shown in the image. Knowing that $1\text{ dm}^3$ is exactly equivalent to $1\text{ litre}$ of capacity:

1. a) Calculate the volume of the aquarium in $\text{dm}^3$.
2. b) How many litres of water are needed to fill the aquarium completely?
3. c) If the aquarium currently contains $45.000\text{ ml}$ of water, how many litres are needed to fill it?

A group of friends is going on a mountain hike. Each of them carries a rucksack with a water bladder (camelbak). If the mass of one litre of water is approximately $1\text{ kg}$:

1. a) If Marcos's water bladder has a capacity of $2.500\text{ ml}$, how many kilograms does the water add to the weight of his rucksack when full?
2. b) Lucía carries a $75\text{ cl}$ bottle. If she has already drunk one third of the contents, how many grams of water does she have left?
3. c) How many bottles like Lucía's ($75\text{ cl}$) could be filled with the water from a $1,5\text{ dal}$ drum?

Order the following measurements of capacity from smallest to largest. To compare them, first convert them all to litres ($L$):

• $0,007\text{ kL}$

• $75\text{ dl}$

• $725\text{ cl}$

• $7.100\text{ ml}$

• $0,8\text{ dal}$

For a birthday party, a teenager decides to make a homemade "fruit punch" by mixing the following ingredients:

• $2$ bottles of orange fizzy drink of $1,5\text{ L}$ each.

• $3$ cartons of pineapple juice of $33\text{ cl}$ each.

• $1$ bottle of lemonade of $2\text{ L}$.

• $250\text{ ml}$ of strawberry syrup.

1. a) How many litres of punch will they get in total?
2. b) If they want to serve the punch in $200\text{ ml}$ glasses, how many full glasses can they fill?
3. c) Will there be any punch left over? If so, express the leftover amount in centilitres.

A garden swimming pool has a total capacity of $12\text{ m}^3$. We want to fill it using a hose delivering a flow rate of $25\text{ litres}$ per minute.

1. a) How many litres of water fit in the pool? (Remember that $1\text{ m}^3 = 1.000\text{ L}$).
2. b) How many minutes will it take to fill the pool completely?
3. c) Express the time above in hours and minutes.

According to records from the State Meteorological Agency (AEMET), during an autumn storm in the Valencian Community, $65\text{ L/m}^2$ (litres per square metre) fell. The flat roof of a school sports hall measures $40\text{ m}$ long and $25\text{ m}$ wide, and all the water that falls onto it is channelled entirely into a municipal cistern.

1. a) Model the situation by calculating how many litres of rainwater were collected on the roof during that storm.
2. b) Express the total volume collected in cubic metres ($m^3$) and in hectolitres ($hL$), taking into account that $1\text{ m}^3 = 1.000\text{ L}$.
3. c) If the municipal cistern has a maximum capacity of $80\text{ kL}$ and already contained $28.000\text{ dm}^3$ of water before the storm, determine with reasoning whether the cistern will overflow or if there will still be free space left. If it does not overflow, indicate how many litres would be needed to fill it completely.

An olive oil cooperative in Jaén has a tank with $1,36\text{ kL}$ of freshly pressed extra virgin olive oil. It is decided to bottle the entire contents of the tank without any oil left over, combining only two commercial formats: $5\text{ L}$ jugs and $75\text{ cL}$ gourmet bottles. Due to market demand, it is agreed to package exactly four gourmet bottles for every $5\text{ L}$ jug.

1. a) Express the total capacity of the tank and that of a gourmet bottle in litres ($L$).
2. b) Consider an 'elementary pack' consisting of $1\text{ jug}$ of $5\text{ L}$ and $4\text{ bottles}$ of $75\text{ cL}$. Exactly how many litres of oil does one of these packs contain?
3. c) Set up the necessary operation and determine how many family jugs and how many gourmet bottles will be packaged in total. Check that the combined volume matches the initial $1,36\text{ kL}$.

An aquarium in the shape of a cuboid has interior dimensions of $80\text{ cm}$ long, $30\text{ cm}$ wide and $50\text{ cm}$ high. For safety reasons, it is recommended not to fill it to more than $\frac{4}{5}$ of its total capacity. If it currently contains $84\text{ litres}$ of water, how many $250\text{ cl}$ buckets are needed to reach the recommended maximum level?

A perfume factory wants to bottle a batch of $0,15\text{ m}^3$ of lavender essence. Three types of bottles are used:

• Half of the essence is bottled in $125\text{ ml}$ bottles.

• One third of the total is bottled in $0,5\text{ dl}$ bottles.

• The rest of the essence is bottled in $10\text{ cm}^3$ samples.

Calculate how many bottles of each type are needed.

A faulty tap in a secondary school drips one drop of water every $2\text{ seconds}$. If each drop has an average volume of $50\text{ mm}^3$, calculate how many $2\text{ decalitres}$ drums would be filled if the tap leaked water continuously throughout the month of April.

A farmer has an irrigation tank containing a compound volume of $2\text{ kl}$, $5\text{ hl}$ and $8\text{ dal}$ of water. To fertilise his trees, he must mix the water with a liquid fertiliser in a ratio of $15\text{ ml}$ of fertiliser for every litre of water. If he has $5\text{ litres}$ bottles of fertiliser, how many full bottles will he need to buy to use all the water in the tank? Will there be any fertiliser left over in the last bottle? Express the leftover amount in centilitres.

A tanker lorry carries $12,5\text{ m}^3$ of milk. Upon arrival at the processing plant, the contents are unloaded into three different tanks:

1. $4.500\text{ litres}$ is unloaded into the first tank.

2. $52\text{ hectolitres}$ is unloaded into the second tank.

3. The rest is divided equally into $200$ industrial containers.

What is the capacity of each of those $200$ containers expressed in $dm^3$?

We want to compare the value for money of two containers of liquid detergent of the same brand:

• Container A: $1.800\text{ ml}$ at a price of $4,50\text{ €}$.

• Container B: $0,25\text{ dal}$ at a price of $6,50\text{ €}$.

Calculate the price per litre for each container and determine which is cheaper. If a family uses $15\text{ cl}$ of detergent per wash, how many washes can they do with the larger container?

A laboratory experiment requires filling a measuring cylinder with three immiscible liquids:

1. $0,045\text{ dm}^3$ of mercury.

2. $12,5\text{ centilitres}$ of distilled water.

3. $850\text{ cubic millimetres}$ of oil.

What will be the total volume of the mixture in the measuring cylinder expressed in compound form (using litres, decilitres and millilitres)? Note: Remember that $1\text{ ml} = 1\text{ cm}^3$.

A local council swimming pool $25\text{ m}$ long and $12\text{ m}$ wide is filled to an average depth of $1,8\text{ metres}$. Due to evaporation from the heat, the water level drops by $5\text{ mm}$ each day.

1. a) How many litres of water are lost through evaporation in a single day?
2. b) If water costs $1,20\text{ €}$ per cubic metre ($m^3$), how much money does the water loss cost the council after a week of intense heat?

In the Science lab, a glass container shaped like a rectangular cuboid (rectangular prism) whose interior measurements are $20\text{ cm}$ wide, $35\text{ cm}$ long, and $40\text{ cm}$ high needs to be filled to the brim. A $250\text{ mL}$ graduated cylinder is used to pour the water. Two students propose different strategies to calculate how many full graduated cylinders will be needed:

• Carla's strategy: She calculates the volume of the container by multiplying the measurements in centimetres to get $\text{cm}^3$. Then she applies the equivalence $1\text{ cm}^3 = 1\text{ mL}$ and divides the result directly by the capacity of the graduated cylinder in millilitres.

• Dani's strategy: He first converts each of the three dimensions to decimetres ($dm$), calculates the volume in cubic decimetres ($\text{dm}^3$), converts it to litres knowing that $1\text{ dm}^3 = 1\text{ L}$, converts the cylinder's capacity to litres ($L$), and divides both measurements.

1. a) Solve the problem rigorously applying Carla's strategy step by step.
2. b) Solve the problem rigorously applying Dani's strategy step by step.
3. c) Check whether both arrive at the same number of cylinders and compare the two methods: which of the two paths do you consider more advisable to minimise calculation errors involving many zeros or decimal commas? Justify your choice by analysing the intermediate steps.

According to data from the National Statistics Institute (INE), the average water consumption in Spanish households is approximately $130\text{ L}$ per person per day. A family of 4 in Seville installs a greywater recycling system from showers and washing machines.

The manufacturer's technical study establishes that:

• In that household, $40\,\%$ of the total water consumed daily is used for showers and washing machines.

• The filtration equipment effectively recovers $75\,\%$ of that greywater for reuse.

The treated water is stored in a prismatic tank whose rectangular base measures $1,2\text{ m}$ long by $0,65\text{ m}$ wide and has a maximum usable height of $1,5\text{ m}$.

1. a) Model the amount of recovered water: calculate how many litres the family recovers in a day and how many hectolitres ($hL$) that accumulated amount represents after one week ($7\text{ days}$).
2. b) Calculate the maximum capacity in litres of the storage tank ($1\text{ m}^3 = 1.000\text{ L}$).
3. c) If at the start of a period the tank already contained $210\text{ dm}^3$ of water and none of the incoming water is removed, how many full days can pass before the tank begins to overflow? How many centimetres does the water level in the tank rise each day?