Capacity
At the corner store they sell a bottle of water labeled "1.5 L," a juice box labeled "200 mL," and at the public pool the water is counted in thousands of liters. All of that is capacity: how much fits inside a container. How do we go from one unit to another without making mistakes?
The unit of capacity: the liter
To celebrate the end of the term, the 7th grade students have bought several bottles of orange juice. If each bottle has a capacity of $1.5\ L$, express this amount in…
To celebrate the end of the term, the 7th grade students have bought several bottles of orange juice. If each bottle has a capacity of $1.5\ L$, express this amount in centiliters ($cL$) and in milliliters ($mL$).
Choosing the right unit
To pick a unit, think of a container you already know and compare. The liter is the reference: a large bottle of water or a carton of milk.
| Container | Approximate capacity | Convenient unit |
|---|---|---|
| Vaccine syringe | $1\ \text{mL}$ to $5\ \text{mL}$ | milliliter ($\text{mL}$) |
| Soda can | $355\ \text{mL}$ ($35.5\ \text{cL}$) | milliliter or centiliter |
| Drinking glass | $2\ \text{dL}$ to $3\ \text{dL}$ | deciliter or milliliter |
| Milk carton | $1\ \text{L}$ | liter ($\text{L}$) |
| Mop bucket | $10\ \text{L}$ to $15\ \text{L}$ | liter |
| A house's water tank | $1{,}100\ \text{L}$ | liter or cubic meter |
| Olympic pool | about $2{,}500{,}000\ \text{L}$ | thousands of liters, or $\text{m}^3$ |
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…
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:
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An Olympic swimming pool.
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A can of cola.
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A syringe for giving a vaccine.
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A mop bucket.
Units smaller than the liter
The prefixes tell how much each unit is worth compared to the liter. You already know them from the meter:
| Unit | Meaning | Equals |
|---|---|---|
| liter ($\text{L}$) | main unit | $1\ \text{L}$ |
| deciliter ($\text{dL}$) | one tenth of a liter | $0.1\ \text{L}$ |
| centiliter ($\text{cL}$) | one hundredth of a liter | $0.01\ \text{L}$ |
| milliliter ($\text{mL}$) | one thousandth of a liter | $0.001\ \text{L}$ |
From this come the equivalences you will use most: $1\ \text{L} = 10\ \text{dL} = 100\ \text{cL} = 1{,}000\ \text{mL}$.
Why? The metric system is decimal: each unit is worth $10$ times the next smaller one. That is why $1\ \text{L} = 10\ \text{dL}$, $1\ \text{dL} = 10\ \text{cL}$, and $1\ \text{cL} = 10\ \text{mL}$.
Complete the following statements by converting to the required units: a) $25 L = ??? dL$ b) $4,500 mL = ??? L$ c) $0.75 kL = ??? L$ d) $32 cL = ??? mL$
Complete the following statements by converting to the required units:
- a) $25 L = ??? dL$
- b) $4,500 mL = ??? L$
- c) $0.75 kL = ??? L$
- d) $32 cL = ??? mL$
At a music festival for teenagers, the organizers have bought several types of water containers to distribute among attendees. They have a main tank of $2.5\text{ kL}$,…
At a music festival for teenagers, the organizers 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 liters, carry out the following conversions:
- a) Express the capacity of the tank in liters.
- b) How many liters are in a pack containing $12$ bottles of $50\text{ cl}$?
- c) If they have served $15$ full large bottles, how many deciliters have they distributed?
Changing units
Changing units means multiplying or dividing by $10$, $100$, $1{,}000$... according to the equivalence in the table:
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If you go to a smaller unit (from $\text{L}$ to $\text{mL}$), it fits more times: you multiply.
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If you go to a larger unit (from $\text{mL}$ to $\text{L}$), it fits fewer times: you divide.
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…
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 liters of water does she carry in total?
In the science laboratory, the students are using a graduated cylinder. If the graduated cylinder shows a capacity of $35\ cL$, express this measurement in compound form…
In the science laboratory, the students are using a graduated cylinder. If the graduated cylinder shows a capacity of $35\ cL$, express this measurement in compound form (that is, using liters and centiliters).
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:…
In the science laboratory, the students are preparing a mixture for an experiment on density. They have used the following amounts of different liquids:
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Distilled water: $1\text{ L}$ and $25\text{ cl}$.
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Alcohol: $4\text{ dl}$ and $5\text{ ml}$.
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Oil: $150\text{ ml}$.
Express the total amount of the mixture in simple form in centiliters ($cl$) and then in compound form using liters, deciliters, and milliliters.
Complex and incomplex forms
A measurement in complex form uses several units (for example, $2\ \text{L}$ and $35\ \text{cL}$). In incomplex form it is expressed with a single unit ($235\ \text{cL}$). To add, compare, or order, it is best to convert everything to a single unit.
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?
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?
A recipe for a sponge cake states that you need to add $2\ dL$ of milk. If we have an open one-liter carton in the fridge with $450\ mL$ left, do we have enough milk for…
A recipe for a sponge cake states that you need to add $2\ dL$ of milk. If we have an open one-liter 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.
Order the following measurements of capacity from smallest to largest. To compare them, first convert them all to liters ($L$): $0.007\text{ kL}$ $75\text{ dl}$…
Order the following measurements of capacity from smallest to largest. To compare them, first convert them all to liters ($L$):
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$0.007\text{ kL}$
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$75\text{ dl}$
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$725\text{ cl}$
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$7,100\text{ ml}$
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$0.8\text{ dal}$
Capacity problems
In long problems, always follow these steps: (1) convert all the data to the same unit, (2) calculate, (3) convert the result to the unit requested, and (4) check that it makes sense.
Price per liter
To find out which container is the better deal, calculate how much each liter costs: divide the price by the liters. If the capacity is given in $\text{mL}$ or $\text{cL}$, convert it to liters first.
A group of friends is going on a mountain hike. Each of them carries a backpack with a water bladder (camelbak). If the mass of one liter of water is approximately…
A group of friends is going on a mountain hike. Each of them carries a backpack with a water bladder (camelbak). If the mass of one liter of water is approximately $1\text{ kg}$:
- 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 backpack when full?
- 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?
- c) How many bottles like Lucía's ($75\text{ cl}$) could be filled with the water from a $1.5\text{ dal}$ drum?
For a birthday party, a teenager decides to make a homemade "fruit punch" by mixing the following ingredients: $2$ bottles of orange soda of $1.5\text{ L}$ each. $3$…
For a birthday party, a teenager decides to make a homemade "fruit punch" by mixing the following ingredients:
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$2$ bottles of orange soda of $1.5\text{ L}$ each.
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$3$ cartons of pineapple juice of $33\text{ cl}$ each.
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$1$ bottle of lemonade of $2\text{ L}$.
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$250\text{ ml}$ of strawberry syrup.
- a) How many liters of punch will they get in total?
- b) If they want to serve the punch in $200\text{ ml}$ glasses, how many full glasses can they fill?
- c) Will there be any punch left over? If so, express the leftover amount in centiliters.
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 $125\text{ ml}$…
A perfume factory wants to bottle a batch of $0.15\text{ m}^3$ of lavender essence. Three types of bottles are used:
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Half of the essence is bottled in $125\text{ ml}$ bottles.
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One third of the total is bottled in $0.5\text{ dl}$ bottles.
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The rest of the essence is bottled in $10\text{ cm}^3$ samples.
Calculate how many bottles of each type are needed.
Longer problems with several units
When the problem mixes many different units or asks about prices, apply the same four steps. If units larger than the liter appear, rely on the extension box in the section "Units smaller than the liter." Write down each conversion with its equivalence so you don't get lost, and to compare containers use the price per liter.
A farmer has an irrigation tank containing a compound volume of $2\text{ kl}$, $5\text{ hl}$ and $8\text{ dal}$ of water. To fertilize his trees, he must mix the water…
A farmer has an irrigation tank containing a compound volume of $2\text{ kl}$, $5\text{ hl}$ and $8\text{ dal}$ of water. To fertilize his trees, he must mix the water with a liquid fertilizer in a ratio of $15\text{ ml}$ of fertilizer for every liter of water. If he has $5\text{ liters}$ bottles of fertilizer, how many full bottles will he need to buy to use all the water in the tank? Will there be any fertilizer left over in the last bottle? Express the leftover amount in centiliters.
A tanker truck carries $12.5\text{ m}^3$ of milk. Upon arrival at the processing plant, the contents are unloaded into three different tanks: 1. $4,500\text{ liters}$ is…
A tanker truck carries $12.5\text{ m}^3$ of milk. Upon arrival at the processing plant, the contents are unloaded into three different tanks:
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$4,500\text{ liters}$ is unloaded into the first tank.
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$52\text{ hectoliters}$ is unloaded into the second tank.
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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 $\$90…
We want to compare the value for money of two containers of liquid detergent of the same brand:
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Container A: $1,800\text{ ml}$ at a price of $\$90$.
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Container B: $0.25\text{ dal}$ at a price of $\$130$.
Calculate the price per liter 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?
Liter and cubic decimeter
A cube of $1\ \text{dm}^3$ (with sides of $10\ \text{cm}$) has a capacity of exactly $1\ \text{L}$. So if you know the volume of a container in $\text{dm}^3$, you already know how many liters fit in it.
Remember that in the metric system there is a direct relationship between volume and capacity. If we have a cube-shaped container that has a volume of exactly $1\ dm^3$,…
Remember that in the metric system there is a direct relationship between volume and capacity. If we have a cube-shaped container that has a volume of exactly $1\ dm^3$, how many liters of water fit inside it? And how many milliliters fit in a volume of $1\ cm^3$?
An aquarium is in the shape of a rectangular prism with the dimensions shown in the image. Knowing that $1\text{ dm}^3$ is exactly equivalent to $1\text{ liter}$ of…
An aquarium is in the shape of a rectangular prism with the dimensions shown in the image. Knowing that $1\text{ dm}^3$ is exactly equivalent to $1\text{ liter}$ of capacity:
- a) Calculate the volume of the aquarium in $\text{dm}^3$.
- b) How many liters of water are needed to fill the aquarium completely?
- c) If the aquarium currently contains $45,000\text{ ml}$ of water, how many liters are needed to fill it?
An aquarium in the shape of a rectangular prism has interior dimensions of $80\text{ cm}$ long, $30\text{ cm}$ wide and $50\text{ cm}$ high. For safety reasons, it is…
An aquarium in the shape of a rectangular prism 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 $84\text{ liters}$ of water, how many $250\text{ cl}$ buckets are needed to reach the recommended maximum level?
Cubic meters and cubic millimeters
One cubic meter is $1{,}000\ \text{dm}^3$, that is, $1{,}000\ \text{L}$. And since $1\ \text{cm} = 10\ \text{mm}$, a $\text{cm}^3$ has $10 \times 10 \times 10 = 1{,}000\ \text{mm}^3$; therefore $1\ \text{mL} = 1{,}000\ \text{mm}^3$.
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 $25\text{ liters}$ per minute. a) How many…
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 $25\text{ liters}$ per minute.
- a) How many liters of water fit in the pool? (Remember that $1\text{ m}^3 = 1,000\text{ L}$).
- b) How many minutes will it take to fill the pool completely?
- c) Express the time above in hours and minutes.
A faulty faucet in a middle 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…
A faulty faucet in a middle 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 $2\text{ decaliters}$ drums would be filled if the faucet leaked water continuously throughout the month of April.
A municipal swimming pool $25\text{ m}$ long and $12\text{ m}$ wide is filled to an average depth of $1.8\text{ meters}$. Due to evaporation from the heat, the water…
A municipal swimming pool $25\text{ m}$ long and $12\text{ m}$ wide is filled to an average depth of $1.8\text{ meters}$. Due to evaporation from the heat, the water level drops by $5\text{ mm}$ each day.
- a) How many liters of water are lost through evaporation in a single day?
- b) If water costs $\$24$ per cubic meter ($m^3$), how much money does the water loss cost the municipality after a week of intense heat?
A laboratory experiment requires filling a graduated cylinder with three immiscible liquids: 1. $0.045\text{ dm}^3$ of mercury. 2. $12.5\text{ centiliters}$ of distilled…
A laboratory experiment requires filling a graduated cylinder with three immiscible liquids:
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$0.045\text{ dm}^3$ of mercury.
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$12.5\text{ centiliters}$ of distilled water.
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$850\text{ cubic millimeters}$ of oil.
What will be the total volume of the mixture in the graduated cylinder expressed in compound form (using liters, deciliters and milliliters)? Note: Remember that $1\text{ ml} = 1\text{ cm}^3$.
Back to the beginning: the $1.5\ \text{L}$ bottle is $150\ \text{cL}$ or $1{,}500\ \text{mL}$, and the liters in a pool come from its $\text{m}^3$ multiplied by $1{,}000$.