Liquid volume: liters and milliliters
How much does a jug, a glass or a spoon hold, and how do you read the amount off the marks on a measuring cup when most of the marks carry no number?
Where you are 9 in Volume and capacity
Why this exists
Seydou’s family keeps its drinking water in a big jug by the kitchen door. Awa works the counter at a juice bar, where every smoothie starts with a pour into the blender. Sundiata’s class measures water in a tall, thin measuring cylinder. All three need to say how much liquid there is, in a way someone else can check.
“A jugful” or “a cupful” does not do it, because jugs and cups come in every size. A unit fixes one size that everybody shares. For liquids, the two everyday metric units are the liter and the milliliter. This lesson gives each one a size you can picture, then reads the marks on the side of a container, where the real work is that most of the marks carry no number.
The metric system and SI units made the liter the base unit for liquid volume, and Metric prefixes showed that milli- cuts a base unit into 1{,}000 equal parts. Working out what one space on a marked scale stands for is the idea of Measuring tools and reading their scales; here it is used on cylinders, jugs and a blender.
The lesson runs over three days. Day 1 is the two units and their sizes (Examples 1 and 2). Day 2 reads measuring cylinders and jugs marked in milliliters (Examples 3 to 5). Day 3 puts liters and milliliters on the same container (Examples 6 and 7).
Definition
Day 1: the liter and the milliliter
Liquid volume is how much space a liquid takes up. The liquid volume a container holds when it is full is its capacity. Liquid volume is about space filled, not about how heavy the liquid is or how tall it stands in its container.
- The liter, written \text{L}, is the metric base unit of liquid volume.
- The milliliter, written \text{mL}, is one of 1{,}000 equal parts of a liter. So 1{,}000 milliliters fill exactly one liter:
A liquid volume is a number together with a unit: 4 L, 70 mL. The number alone does not say how much, since 4 L and 4 mL are very different amounts.
Day 2: a measuring scale
A measuring cylinder is a tall, narrow, see-through tube with marks up its side. A measuring jug is a wider container with marks. Each mark stands for an amount of liquid counted up from the bottom, which is 0. Every space between two neighboring marks stands for the same amount; call it the space size of the scale. Only some marks carry a number. The reading is the amount at the mark where the top of the liquid sits.
Day 3: liters and milliliters on one container
A container may be numbered in milliliters, in liters, or in both. A mark labeled 1 L and a mark labeled 1{,}000 mL are the same mark, because they name the same amount.
Intuition
Day 1: sizes you can picture
A unit is easiest to use when you can see its size in your mind. These benchmarks are close, not exact.
| Unit | About the size of |
|---|---|
| 1 mL | the water that fills one centimeter cube, the small cube from a counting kit |
| 15 mL | one soup spoon, filled to the brim |
| 250 mL | an ordinary drinking glass |
| 1 L | a milk carton labeled 1 L (a quart carton holds a little less) |
| 4 L | a big water jug, about the size of a gallon jug |
The cube is the one The metric system and SI units stacked a thousand times into a box that holds a liter. A milliliter is tiny: a soup spoon takes about fifteen of them. That is why even a glass of water is a few hundred milliliters.
So the choice of unit follows the size of the thing. A spoonful, a sip, a dose of medicine, a glass or a mug is measured in milliliters, because a liter would be far too big and the amount would be a small part of one. A jug, a bucket, a kitchen sink or a fish tank is measured in liters, because in milliliters the number would run into the thousands.
To estimate an amount, hold it up against the nearest benchmark. “A bit more than a glass” is a bit more than 250 mL. “About four cartons” is about 4 L.
Day 2: what one space stands for
Look at the cylinder in Figure 1. Most marks are bare lines. Only 0, 50 and 100 are numbered. To read the water, first find what one space stands for.
From the 0 mark to the 50 mark there are four bare marks, and they cut the gap into five spaces. Count the spaces, the gaps between marks, not the marks themselves: four marks inside a gap make five spaces. The five spaces are equal, and together they make 50 mL, so each one is 50 \div 5 = 10 mL, because 5 \times 10 = 50. This scale counts by tens.
Now read the water. Start at the nearest number below the top of the liquid, here 50, and count up one space at a time: 60, 70. The water reads 70 mL.
The same picture works on any scale. Two numbered marks tell you how much the gap between them is worth. The number of spaces in that gap tells you how many equal parts it is cut into. Each space is one of those equal parts.
In a narrow tube the water climbs the glass a little at the edges, so its top is not quite flat. Crouch until your eye is level with the water and read the flat middle.
Day 3: liters and milliliters on one container
A bigger container is often numbered in liters with bare marks in between. A blender jar might show 1 L and 2 L and nothing else. Since 1 L is 1{,}000 mL, the bare marks between 0 and 1 L split those 1{,}000 milliliters into equal spaces, and the reading can be given in milliliters.
Results
Days 1 to 3
> A liter is a thousand milliliters. 1 \text{ L} = 1{,}000 \text{ mL}.
> The space size. When two numbered marks differ by some amount and the gap between them is cut > into equal spaces, each space stands for that amount divided by the number of spaces.
> A reading. Start at the nearest numbered mark below the top of the liquid and count up one > space size for each space. A top that falls between two marks gives an estimate, not an exact > reading.
> Comparing. To compare an amount in liters with one in milliliters, name both in milliliters: > 1 L is more than any amount under 1{,}000 mL.
Why it is true
Why a liter is a thousand milliliters. That is what the prefix says. Milli- means one of 1{,}000 equal parts of the base unit (Metric prefixes), and here the base unit is the liter. Cut a liter into 1{,}000 equal parts and each part is a milliliter, so 1{,}000 of them put back together make the liter again.
Why every space holds the same amount. A measuring cylinder is the same width from bottom to top. Fill it a little at a time with equal amounts and the water rises by the same height each time, since every slice of the tube the same height holds the same amount. The maker puts a mark at each of those heights, so the marks are evenly spaced and each space holds the same amount.
Why the space size is a division. The spaces in a gap are equal groups that together make the gap’s amount. If 5 equal spaces make 50 mL, then 5 \times (\text{space size}) = 50, and the space size is 50 \div 5 = 10 mL. This is the same equal-groups reasoning as any division: the whole is known, the number of groups is known, and the size of each group is the unknown.
Why counting up from a number below gives the reading. The reading at any mark is the amount from the bottom up to that mark. That amount is the amount up to the numbered mark below, plus one space size for each space above it. So reading 50, 60, 70 is adding 10 twice to 50.
Worked examples
Day 1 is Examples 1 and 2, Day 2 is Examples 3 to 5, and Day 3 is Examples 6 and 7.
A jug that holds four
Seydou’s family buys a water jug for the kitchen. The label says it holds 4 of something, but the unit has rubbed off. Does the jug hold 4 L or 4 mL?
SolutionHold each amount up against a benchmark.
4 mL is less than one soup spoon, since a soup spoon holds about 15 mL and 4 < 15. That is a sip. A jug that held only 4 mL would hold less than one spoonful.
4 L is about four milk cartons of water, the size of a big jug a family drinks from over a day or two. That fits a water jug by the kitchen door.
So the jug holds 4 L. The milliliter is the wrong size for a jug; it suits a spoonful.
The size of a smoothie cup
At the juice bar, Awa serves smoothies in a cup that holds a bit more than an ordinary drinking glass. Which is the best estimate for the cup: 40 mL, 400 mL or 4 L?
SolutionCompare each choice with a benchmark.
- 40 mL is less than three soup spoons, since 15 + 15 + 15 = 45 and 40 < 45. That is a few sips, far less than a glass.
- 4 L is about four milk cartons, as much as a whole family water jug. Nobody drinks that from one cup.
- 400 mL is more than a glass, since 400 > 250, and less than a milk carton, since 400 < 1{,}000.
The cup holds a bit more than a glass, so the best estimate is 400 mL. Only the middle choice sits between the glass and the carton, which is exactly where “a bit more than a glass” belongs.
Reading a measuring cylinder
Sundiata pours water into the measuring cylinder in Figure 1. Only 0, 50 and 100 are numbered. How much water is in the cylinder?
SolutionFirst find the space size. From the 0 mark to the 50 mark there are four bare marks, which cut the gap into 5 equal spaces. Together the 5 spaces make 50 mL. Since 5 \times 10 = 50, each space is 10 mL.
Now find the top of the water. It sits on the second mark above 50. Count up from 50 by the space size: one space is 60, two spaces is 70.
The cylinder holds 70 mL of water.
Check. Count up from 0 instead. The top of the water is 7 spaces above the bottom, and 7 \times 10 = 70. The same reading.
A jug that counts by fifties
Awa measures orange juice for a recipe in the measuring jug of Figure 2. Its numbers go 200, 400, 600, 800, 1{,}000 mL, with bare marks between them. How much juice is in the jug?
SolutionFind the space size. Between the 200 mark and the 400 mark there are three bare marks, cutting the gap into 4 equal spaces. The gap is worth 400 - 200 = 200 mL. Since 4 \times 50 = 200, each space is 50 mL.
The top of the juice sits on the third mark above 400. Count up by fifties: 450, 500, 550.
The jug holds 550 mL of juice.
Check. Count down from the number above instead. The top of the juice is one mark below 600, and 600 - 50 = 550.
A space on Sundiata’s cylinder was 10 mL; a space on Awa’s jug is 50 mL. The spaces look alike on paper, but they stand for different amounts. The space size belongs to the scale, so it is found again for every new container.
Between two marks
Sundiata uses the cylinder from Example 3 again, with a mark every 10 mL. This time he measures lemon juice for a class experiment, and its top sits about halfway between the 30 mark and the 40 mark. How much lemon juice is there?
SolutionThe top is on no mark, so the scale cannot give an exact reading. What it does say for sure is that the amount is more than 30 mL and less than 40 mL.
Halfway between 30 and 40 is 35, because 35 is 5 more than 30 and 5 less than 40. So there is about 35 mL of lemon juice.
The word “about” matters. A scale reads exactly only on its marks. Between marks, the best answer names the two marks it lies between, or makes a careful estimate such as “about 35 mL”.
A milk carton in a measuring jug
Seydou pours a whole 1 L carton of milk into a measuring jug numbered 200, 400, 600, 800 and 1{,}000 mL, with one bare mark between each pair of numbers.
(a) Which mark does the milk reach?
(b) He pours a glass of milk for his sister, and the top of the milk falls to the bare mark just above 600. How much milk went into the glass?
Solution(a) The carton holds 1 L, and 1 L is 1{,}000 mL. So the milk reaches the 1{,}000 mark, the top number on the jug. The carton and the jug name the same amount with different units.
(b) First the space size. One bare mark cuts each gap of 200 mL into 2 equal spaces, and 2 \times 100 = 200, so each space is 100 mL. The bare mark just above 600 is one space higher, so it stands for 700 mL.
The jug had 1{,}000 mL and now has 700 mL. The glass took the difference:
So 300 mL of milk went into the glass. Check: 700 + 300 = 1{,}000, the full carton.
The blender at the juice bar
The blender jar at the juice bar holds 2 L. Its side is numbered only 1 L and 2 L, with bare marks between (Figure 3). Awa pours smoothie mix into the jar up to the level shown.
(a) What does one space stand for?
(b) How much smoothie mix is in the jar?
(c) Is that more or less than 1 L? How much more mix would bring it up to the 1 L mark?
Solution(a) From 0 to the 1 L mark there are nine bare marks, which cut the gap into 10 equal spaces. The gap is 1 L, which is 1{,}000 mL. Ten equal spaces make 1{,}000 mL, and 10 \times 100 = 1{,}000 (ten hundreds make a thousand), so each space is 100 mL.
(b) The top of the mix sits on the ninth mark above 0. Counting up by hundreds, nine spaces make 9 \times 100 = 900, so the jar holds 900 mL. Counting down gives the same: the top is one mark below 1 L, and 1{,}000 - 100 = 900.
(c) 900 mL is less than 1{,}000 mL, so it is less than 1 L, even though 900 is the bigger number. The two numbers count in different units, so they are compared only after both are in milliliters. To reach the 1 L mark takes one more space:
So 100 mL more mix brings the jar up to 1 L. Above that mark the spaces go on, 100 mL each, up to the 2 L mark at the top.
Boundaries
- A reading is only as fine as the marks. On a scale marked every 10 mL, an amount between marks can only be estimated. A finer amount needs a scale with smaller spaces.
- Equal spaces need equal amounts, not equal distances. On a straight-sided cylinder, equal amounts sit at equal heights, so the marks are evenly spaced. On a jug that widens toward the top, each centimeter of height holds more liquid higher up, so the maker draws the upper marks closer together. Each space still holds the same amount, and counting spaces still works; measuring the distance with a ruler would not.
- No marks, no reading. A plain glass or bottle has no scale. Its contents can only be estimated against benchmarks, or poured into a marked container and read there.
- Liquid volume is not mass. A liter of water and a liter of honey take up the same space, but the honey is heavier. How heavy something is (its mass) is measured in grams and kilograms: Metric mass units.
- Pouring does not change the amount. The same 250 mL stands tall in a thin glass and low in a wide bowl. The height of the liquid depends on the container; the liquid volume does not (Volume does not change when it is poured).
- A number needs its unit. 4 alone is not a liquid volume, and 900 mL is less than 1 L. Two amounts are compared only in the same unit.
Where this goes next
Story problems that add, subtract, multiply or divide liquid volumes, all in the same unit, are Story problems with grams, kilograms and liters. The United States also uses cups, pints, quarts and gallons for liquids, and Customary capacity units builds that family and sets it beside the liter. Amounts like 1 L 250 mL, and changing between liters and milliliters with decimals, come in Converting metric units by powers of ten. The centimeter cube that holds 1 mL returns in Volume as filling, where the space inside a solid is counted in cubes. And how fast a liquid pours, in liters each minute, is Flow rate.
Exercises
Day 1: liters, milliliters and sizes
- Would you measure each one in liters or in milliliters? (a) the water in a kitchen sink; (b) one dose of cough syrup; (c) the water in a fish tank; (d) the milk poured on one bowl of cereal.
- Choose the best estimate for how much a watering can holds: 5 mL, 50 mL or 5 L.
- Choose the best estimate for a mug of hot cocoa: 3 mL, 300 mL or 30 L.
- Seydou says his soup spoon holds about 15. Fifteen what? How do you know it is not the other unit?
- A carton holds 1 L and a bottle holds 800 mL. Which holds more, and how much more?
- Put these in order from least to most: 600 mL, 1 L, 60 mL, 900 mL.
Day 2: reading cylinders and jugs
- A cylinder is numbered 0 and 20, and the gap between them is cut into 4 equal spaces. What does one space stand for?
- A jug is numbered every 100 mL, and each gap is cut into 2 equal spaces. What does one space stand for?
- A jug is numbered every 100 mL, and each gap is cut into 5 equal spaces. What does one space stand for?
- Read container A in Figure 4.
- Read container B in Figure 4.
- Read container C in Figure 4.
- On a cylinder with a mark every 10 mL, the top of the water sits about halfway between the 80 mark and the 90 mark. About how much water is there?
- On a jug with a mark every 50 mL, the top of the juice sits about halfway between the 250 mark and the 300 mark. About how much juice is there?
- Sundiata’s cylinder has a mark every 10 mL. The water reads 60 mL. He adds more water and it now reads 90 mL. How much water did he add?
Day 3: liters and milliliters together
- Fill in the blanks: 1 L = ___ mL, and 1{,}000 mL = ___ L.
- Which is more, 1 L or 990 mL? How much more?
- A jug’s top mark says 1 L, and the gap from 0 to 1 L is cut into 5 equal spaces. What does one space stand for? The juice reaches the third mark above 0; how much juice is there?
- A 1 L bottle of mango juice stands at the juice bar. After Awa pours some out, 350 mL is left. How much did she pour?
- A camping water jug holds 10 L and has a mark at every liter. The water reaches the 6 L mark. How many more liters fill the jug?
- Which is more, 2 L or 1{,}000 mL? Explain.
Reasoning
- Why must you work out what one space stands for before you read a measuring jug? Describe two containers whose marks look alike but whose spaces stand for different amounts.
- Is the amount with the bigger number always the bigger amount? Give an example that shows your answer.
- Why does it make sense to measure a glass of water in milliliters but a fish tank in liters?
- A cylinder has only one number, 100 mL, printed at its top mark, and the gap from the bottom up to it is cut into 10 equal spaces. Can you still read it? Explain how.
Answers
- (a) Liters; (b) milliliters; (c) liters; (d) milliliters. A sink and a fish tank hold many cartons, so liters keep the number small. A dose of syrup is about a spoonful and the milk on cereal is less than a glass, both small parts of a liter, so milliliters fit.
- 5 L. 5 mL is a third of a soup spoon and 50 mL is a few spoonfuls, far too little to water plants. A watering can holds several cartons of water.
- 300 mL. A mug holds a little more than a glass (250 mL). 3 mL is less than a soup spoon and 30 L is thirty milk cartons.
- Milliliters. 15 mL is the soup spoon benchmark. 15 L would be fifteen milk cartons, more than three big 4 L water jugs, which no spoon holds.
- The carton, by 200 mL. The carton holds 1{,}000 mL, and 1{,}000 - 800 = 200.
- 60 mL, 600 mL, 900 mL, 1 L. Name all four in milliliters: 1 L is 1{,}000 mL, the most, and 60 < 600 < 900 < 1{,}000.
- 5 mL. Four equal spaces make 20 mL, and 4 \times 5 = 20.
- 50 mL. Two equal spaces make 100 mL, and 2 \times 50 = 100.
- 20 mL. Five equal spaces make 100 mL, and 5 \times 20 = 100.
- 50 mL. The gap from 40 to 60 is 20 mL in 2 spaces, so each space is 10 mL. The top is one space above 40: 40 + 10 = 50.
- 350 mL. Each gap of 100 mL holds 2 spaces, so each space is 50 mL. The top is one space above 300: 300 + 50 = 350.
- 800 mL. The gap from 500 to 1{,}000 is 500 mL in 5 spaces, so each space is 100 mL (5 \times 100 = 500). The top is three spaces above 500: 600, 700, 800.
- About 85 mL. Halfway between 80 and 90 is 85, which is 5 more than 80 and 5 less than 90.
- About 275 mL. The gap from 250 to 300 is 50 mL, and halfway along is 25 mL in: 250 + 25 = 275.
- 30 mL. The water rose from 60 mL to 90 mL, and 90 - 60 = 30.
- 1 L = 1{,}000 mL, and 1{,}000 mL = 1 L. Milli- cuts the liter into 1{,}000 equal parts.
- 1 L, by 10 mL. 1 L is 1{,}000 mL, and 1{,}000 - 990 = 10.
- 200 mL a space; 600 mL of juice. Five equal spaces make 1{,}000 mL, and 5 \times 200 = 1{,}000. Three spaces up: 200, 400, 600.
- 650 mL. The bottle held 1{,}000 mL, and 1{,}000 - 350 = 650.
- 4 L. The jug holds 10 L and has 6 L in it, and 10 - 6 = 4.
- 2 L. One liter alone is 1{,}000 mL, so 2 L is 1{,}000 mL and another whole liter on top: 2 L is 2{,}000 mL, more than 1{,}000 mL.
- The marks are only lines; what a space is worth depends on the numbers printed on that container. A cylinder numbered 0, 50, 100 with 5 spaces in each gap counts by 10 mL, while a jug numbered every 200 mL with 4 spaces in each gap counts by 50 mL. Their spaces can look the same size, yet one is five times the other. Reading without the space size would give the wrong amount on one of them.
- No, not when the units differ. 900 mL is less than 1 L, since 1 L is 1{,}000 mL. Only amounts named in the same unit can be compared by their numbers.
- A glass holds a few hundred milliliters, a small part of a liter, so milliliters give a whole number of a sensible size. A fish tank holds many liters; in milliliters its number would run into the thousands. Each unit is chosen near the size of the thing measured.
- Yes. The bottom of a cylinder is always 0, so the gap from 0 to 100 mL is known. Ten equal spaces make 100 mL, and 10 \times 10 = 100, so each space is 10 mL. Count up from the bottom by tens to the top of the liquid.