Grade 4

Customary capacity units

A kiddie pool, a milk jug and a mug of cocoa all hold liquid. Which unit fits each one, and how can you tell whether 3 quarts is more or less than a gallon?

You need

Liquid volume: liters and millilitersGrade 3 The customary systemGrade 4

Where you are 9 in Volume and capacity

Measurement12

Volume and capacity9

Volume does not change when it is poured Liquid volume: liters and milliliters

Customary capacity units

Volume as filling Volume with unit cubes ▾ Show all 9 (4 more)

Why this exists

Yash, Yarema and Nayra are filling a kiddie pool in the backyard with a garden hose. Yarema wonders how much water the pool holds. A ruler is no help: water has no length of its own, it takes the shape of whatever it is in. Water in a tall jug and the same water in a flat pool look nothing alike.

So liquid is measured a different way. Pick one container as the unit, and count how many times it would take to fill the pool. The answer to “how much does it hold?” is that count, together with the name of the container. How much a container holds is its capacity, also called liquid volume.

In the United States the everyday capacity units are the cup, the pint, the quart and the gallon, with the fluid ounce for small amounts. They belong to the customary system, met in The customary system. This lesson looks closely at these five units: how big each one is, how they fit inside each other, which one suits which job, and how to compare amounts given in different units.

This lesson takes two days. Day 1 (Examples 1 to 3) is the units, their sizes, and how each one splits into halves. Day 2 (Examples 4 to 7) is comparing capacities given in different units, and how capacity is related to volume.

Definition

Day 1: five units for liquid

The capacity of a container is how much it holds. A capacity is a number together with a unit: “the jug holds 1 gallon”.

unit	short form	equals
fluid ounce	\text{fl oz}	the smallest unit here
cup	\text{c}	8 fluid ounces
pint	\text{pt}	2 cups
quart	\text{qt}	2 pints, which is 4 cups
gallon	\text{gal}	4 quarts, which is 8 pints or 16 cups

Every fact in the table is exact. A gallon is not “about” 4 quarts; it is exactly 4 quarts.

Milk and juice are also sold by the half gallon, which is just what it says: half of a gallon, or 2 quarts. It is a handy container size more than a separate unit.

The fluid ounce measures room for liquid. It is not the same as the ounce, which measures weight (see Customary weight units). The two share a name and nothing else.

Most of the world measures capacity in liters and milliliters instead, the units of Liquid volume: liters and milliliters. A liter is a little more than a quart.

Intuition

Day 1: a container for each unit

To feel how big each unit is, keep a familiar container in mind, called a benchmark. A benchmark is “about”, never exact.

unit	about how much
fluid ounce	two spoonfuls from a big kitchen spoon; a small medicine cup
cup	a measuring cup; a school lunch carton of milk
pint	a small carton of ice cream
quart	a carton of milk from the dairy aisle
gallon	a big plastic jug of milk

A kiddie pool holds many milk jugs, so its capacity is told in gallons. A mug of cocoa holds about a cup. The best unit is the one closest in size to the container, so the number is easy to picture.

The halving chain. Start with a gallon of milk and pour exactly half into another jug. Each half is a half gallon. Halve one of those, and each part is a quart. Halve a quart, and each part is a pint. Halve a pint, and each part is a cup. Four halvings take a gallon down to a cup.

The picture shows the same chain inside one gallon. The gallon is drawn as a square of 16 cups. Its left half is a half gallon. Half of what is left is a quart, half of what is left after that is a pint, and then a cup. One more cup fills the square.

Figure 1A gallon is 16 cups. Half of it is a half gallon, half of what is left is a quart, half again is a pint, then a cup, and one last cup fills the gallon.

Counting squares reads the facts off the picture. The quart is 4 squares, so a quart is 4 cups and \displaystyle \frac{1}{4} of the gallon. The pint is 2 squares, \displaystyle \frac{1}{8} of the gallon. The only step that is not a halving is fluid ounces to cups: a cup is 8 fluid ounces.

Bigger unit, smaller number. The same pool can be described in gallons or in quarts. The water does not change. A quart is smaller, so it takes more quarts than gallons to fill the pool. The number in the smaller unit is always the bigger number.

Results

Day 2: comparing amounts, and capacity as volume

The ladder in cups and fluid ounces. Each unit holds a fixed number of each smaller one:

	\text{fl oz}	\text{c}	\text{pt}	\text{qt}
1 \text{ c}	8	1
1 \text{ pt}	16	2	1
1 \text{ qt}	32	4	2	1
\tfrac{1}{2} \text{ gal}	64	8	4	2
1 \text{ gal}	128	16	8	4

Comparing capacities. Two amounts in different units are compared by first putting them in the same unit, usually the smaller one, and then comparing the numbers. The numbers alone can mislead: 3 is more than 1, but 3 quarts is less than 1 gallon, because a gallon is 4 quarts. Changing a whole amount from one unit to another is the work of Customary conversions; here it takes only one fact from the ladder.

Capacity is a volume. The capacity of a container is the volume of the empty space inside it, the idea of Volume as filling. Solid things are often measured in cubic units such as cubic inches; liquids are measured in cups, quarts and gallons. They measure the same kind of thing, and the two are tied together exactly: a gallon is 231 cubic inches.

Why it is true

Why a gallon is 16 cups. A gallon is 4 quarts. Each quart is 2 pints and each pint is 2 cups, so each quart is 2 \times 2 = 4 cups. Four groups of 4 cups is 4 \times 4 = 16 cups. The halving chain gives the same number another way: four halvings split a gallon into 2 \times 2 \times 2 \times 2 = 16 equal parts, and each part is a cup.

Why comparing needs one unit. A number of units counts how many times the unit fits. Two counts can be compared only if they count the same thing. “3 quarts” and “1 gallon” count different containers, so their numbers cannot be lined up until both count quarts: 3 quarts against 4 quarts.

Why capacity can measure a solid. Water fills every gap in a container. A solid pushed under the water takes up space the water was using, so the water rises by exactly the volume of the part underwater. Reading that rise in gallons or cups measures the solid’s volume in a capacity unit.

Worked examples

Examples 1 to 3 go with Day 1, and Examples 4 to 7 with Day 2.

Example 1

A unit for each container

Yarema lists five amounts of liquid at the backyard party: the water in the kiddie pool, the milk poured on one bowl of cereal, one dose of cough syrup, the lemonade in a pitcher, and the water in a mop bucket. Which unit suits each?

Solution

Compare each amount with the benchmarks, and pick the unit closest in size.

The kiddie pool holds many milk jugs of water, so gallons suit it.
The milk on a bowl of cereal is about a measuring cup, so cups suit it.
A dose of cough syrup is a spoonful or two, far less than a cup, so fluid ounces suit it.
A pitcher of lemonade holds a few cartons from the dairy aisle, so quarts suit it.
A mop bucket holds a few milk jugs, so gallons suit it.

Each choice keeps the number small and easy to picture: “2 quarts of lemonade” rather than “64 fluid ounces of lemonade”, although both are true.

Example 2

Filling a gallon with smaller cartons

Yash has a full gallon jug of milk and five empty containers: a half gallon carton, a quart carton, a pint carton and two cups. Is there room for all the milk? Will any be left over?

Solution

Put every container in cups, using the ladder or the gallon square:

\begin{aligned} \tfrac{1}{2}\text{ gal} &= 8 \text{ c}, \quad 1 \text{ qt} = 4 \text{ c}, \\ 1 \text{ pt} &= 2 \text{ c} \end{aligned}

Together the containers hold

8 + 4 + 2 + 1 + 1 = 16 \text{ cups}

A gallon is 16 cups, so the milk fills every container exactly, with none left over. This is the gallon square in the figure: each container is one of its five pieces.

Example 3

One bucket, two counts

Nayra finds out what a bucket holds by filling it from a quart carton. It takes 9 cartons. Yash measures the same bucket with a pint carton. How many times does he pour? Who has the bigger number, and does that mean the bucket got bigger?

Solution

Each quart is 2 pints, so every quart carton Nayra poured is two of Yash’s pints. Nine quarts is 9 groups of 2 pints:

9 \times 2 = 18 \text{ pints}

Yash pours 18 times. His number is bigger, but the bucket is the same: 9 quarts and 18 pints are the same amount. A pint is the smaller unit, so it takes more of them.

Example 4

Three quarts against a gallon

Yash has 3 quarts of milk in cartons. Nayra has a full gallon jug. Yash says his milk is more, because 3 is more than 1. Who has more, and by how much?

Solution

The numbers count different units, so put both in quarts. A gallon is 4 quarts. Now compare: 3 \text{ qt} < 4 \text{ qt}, so Nayra has more, by 4 - 3 = 1 quart.

The picture says the same thing. Pour Yash’s milk into an empty gallon jug, and it reaches the third quart mark. One quart of room is left.

Figure 2Three quarts poured into a gallon jug reach the third mark. One quart of room is left, so the gallon is more.

The difference can be told in a smaller unit too: 1 quart is 2 pints, or 4 cups.

Example 5

Ordering three drinks

Yarema has a 40 fluid ounce bottle of juice, a quart carton of milk and a pitcher with 5 cups of lemonade. Put them in order from least to greatest.

Solution

The fluid ounce is the smallest unit in the problem, so change every amount to fluid ounces. A cup is 8 fluid ounces, and a quart is 4 cups, or 4 \times 8 = 32 fluid ounces.

The juice is 40 fluid ounces.
The milk is 1 quart, which is 32 fluid ounces.
The lemonade is 5 cups, which is 5 \times 8 = 40 fluid ounces.

So the juice and the lemonade are exactly the same amount, and the milk is the least:

1 \text{ qt} < 40 \text{ fl oz} = 5 \text{ c}

Example 6

Room left in the cooler

Yash’s cooler holds 2 gallons. Nayra pours in 3 quarts of lemonade, and Yarema adds 5 cups of juice. Is there room for it all? How much room is left?

Solution

Three different units are in one problem, so put every amount in cups, the smallest of them. A gallon is 16 cups, so the cooler holds 2 \times 16 = 32 cups. A quart is 4 cups, so the lemonade is 3 \times 4 = 12 cups. The juice is already in cups. Together the drinks are

12 + 5 = 17 \text{ cups}

Since 17 < 32, it all fits, with 32 - 17 = 15 cups of room left. That is 1 cup less than a gallon, because a gallon is 16 cups.

Check: the drinks and the room left fill the cooler, 17 + 15 = 32 cups, which is 2 gallons.

Example 7

Measuring any amount with four containers

Nayra has four empty containers: a half gallon, a quart, a pint and a cup. She may fill each one at most once and pour them all into a punch bowl. How can she measure exactly 11 cups? Can she measure every whole number of cups from 1 to 15 this way?

Solution

In cups the containers hold 8, 4, 2 and 1. Start with the biggest that fits: 8 cups leaves 3 to go. The quart is too big, the pint gives 2, and the cup gives the last 1:

11 = 8 + 2 + 1

So she fills the half gallon, the pint and the cup.

Now every amount from 1 to 15. The cup and the pint alone make 1, 2 and 3 cups. Adding the quart’s 4 cups to those makes 4, 5, 6 and 7. Adding the half gallon’s 8 cups to everything so far makes 8 up to 15. So every whole number of cups from 1 to 15 can be measured, and 15 is all four containers together. It works because each container is exactly twice the next one down, the halving chain again.

Boundaries

United States units only. The units here are the United States liquid units. The British imperial pint, quart and gallon have the same names but are larger.
Liquid units, not dry ones. Berries are sometimes sold by the dry pint and dry quart, which are slightly larger than the liquid pint and quart. This lesson uses the liquid units throughout.
A fluid ounce is not an ounce. One measures room, the other weight. A cup of water and a cup of honey both fill 8 fluid ounces of space, but they do not weigh the same.
Compare only capacity with capacity. A gallon can be compared with a quart, never with a pound or a foot.
Benchmarks are approximate; unit facts are exact. Milk cartons vary a little in shape. A quart is exactly 2 pints, always.

Where this goes next

Customary conversions changes whole amounts between these units, such as 5 gallons into quarts, and back again with remainders. Multi-step problems with unit conversions chains several facts, gallons all the way down to fluid ounces, inside a longer problem, and Word problems with distance, time, volume, mass and money puts capacity to work in story problems. The weight family, with its own ounce, is Customary weight units. Volume as filling measures the space inside a container and the space a solid takes up as one idea, and Conversion factors and dimensional analysis later moves amounts between gallons and liters.

Exercises

Day 1: the units and how big they are

Which of these are units of capacity: pint, pound, gallon, foot, fluid ounce, ounce?
Put these in order from smallest to largest: quart, cup, gallon, fluid ounce, pint.
Choose the best unit for each: (a) the water in a bathtub; (b) the juice in a juice box; (c) the soup in a big soup pot; (d) the vanilla in one spoonful; (e) the water in a goldfish bowl.
Choose the sensible measurement: (a) a milk jug: 1 cup, 1 quart or 1 gallon; (b) a mug of cocoa: 1 fluid ounce, 1 cup or 1 gallon; (c) a kiddie pool: 40 cups, 40 quarts or 40 gallons; (d) a water bottle: 16 fluid ounces, 16 quarts or 16 gallons.
Each label has the wrong unit. Fix it by changing only the unit: (a) a watering can holds 2 cups; (b) a juice box holds 6 gallons; (c) a bathtub holds 40 pints.
Fill in the blanks: (a) 1 half gallon = ___ quarts = ___ pints; (b) 1 pint = ___ fluid ounces.
In the gallon square, how many small squares make the half gallon, the quart and the pint? What fraction of the gallon is each?
Yarema fills a cooler using a pint carton, and it takes 12 cartons. How many times would she pour with a cup? With a quart carton?
Nayra pours a full gallon jug of milk into pint jars. How many jars does she fill? How many half gallon cartons would the same milk fill?
Yash fills a half gallon carton, a quart carton and a cup with water and pours all three into a bucket. How many cups of water are in the bucket?
A school lunch carton holds 1 cup of milk. How many cartons hold as much as a gallon jug?
A cup of water and a cup of honey each take up 8 fluid ounces. Does that mean they weigh the same? Explain.

Day 2: comparing capacities

Which is more? (a) 3 pints or 1 quart; (b) 7 cups or 2 quarts; (c) 1 gallon or 9 pints; (d) 20 fluid ounces or 2 cups.
Yarema has 6 pints of lemonade. Yash has a full gallon jug of lemonade. Who has more, and by how much?
Put these in order from least to greatest: 1 gallon, 3 quarts, 7 pints, 13 cups.
Put these in order from least to greatest: 30 fluid ounces, 1 quart, 3 cups, 1 pint.
A bigger pool holds 75 gallons. How many quarts does it hold?
Nayra has 6 cups of lemonade. Will it all fit in a quart carton? In a half gallon carton? If it fits, how much room is left?
Say whether each sentence is about a capacity or about the volume of a solid: (a) a juice box holds 6 fluid ounces; (b) a brick takes up 80 cubic inches; (c) a rock dropped in a measuring cup pushes the water up by a quarter cup.
A small pool holds 40 gallons, and the hose has put in 34 gallons so far. How many more gallons will fill it? Yarema finishes the job with a quart pitcher. How many pitchers does she pour?
Nayra has a half gallon, a quart, a pint and a cup, and may fill each one at most once. Which ones should she fill to measure exactly 14 cups?

Reasoning

Use the halving chain to explain why 16 cups make a gallon.
Yash says 3 quarts is more than a gallon because 3 is more than 1. Explain what is wrong.
One bucket is measured in cups and in quarts. Explain why the number of cups is always 4 times the number of quarts.
Yarema says capacity and volume are the same thing. Nayra says a pool’s capacity is not the room it takes up on the lawn. Explain what is right in each view.

Answers

Pint, gallon and fluid ounce. The pound and the ounce measure weight, and the foot measures length.
Fluid ounce, cup, pint, quart, gallon. A cup is 8 fluid ounces, a pint 2 cups, a quart 2 pints and a gallon 4 quarts.
(a) Gallons, since a tub holds many milk jugs. (b) Fluid ounces, since a juice box holds less than a cup. (c) Quarts, a few cartons’ worth. (d) Fluid ounces, or a part of one. (e) Gallons for a big bowl, or quarts for a small one.
(a) 1 gallon. (b) 1 cup. (c) 40 gallons. (d) 16 fluid ounces. Each other choice is far too small or far too big next to its benchmark.
(a) 2 gallons: 2 cups is only a mugful. (b) 6 fluid ounces: 6 gallons is six milk jugs. (c) 40 gallons: 40 pints is only 5 gallons, a bucketful or two.
(a) 2 quarts and 4 pints, since a half gallon is half of 4 quarts and each quart is 2 pints. (b) 16 fluid ounces, since a pint is 2 cups of 8 fluid ounces.
The half gallon is 8 squares, \displaystyle \frac{1}{2} of the gallon. The quart is 4 squares, \displaystyle \frac{1}{4}. The pint is 2 squares, \displaystyle \frac{1}{8}. The gallon is 16 squares.
24 times with a cup, since each pint is 2 cups and 12 \times 2 = 24. 6 times with a quart, since each quart is 2 pints and 12 \div 2 = 6.
8 pint jars, since a gallon is 8 pints. 2 half gallon cartons.
13 cups. 8 + 4 + 1 = 13, since a half gallon is 8 cups and a quart is 4 cups.
16 cartons, since a gallon is 16 cups.
No. A fluid ounce measures room, not weight. The two cups fill the same space, but honey is heavier than water for the same space.
(a) 3 pints, since 1 quart is 2 pints. (b) 2 quarts, since 2 quarts is 2 \times 4 = 8 cups. (c) 9 pints, since 1 gallon is 8 pints. (d) 20 fluid ounces, since 2 cups is 2 \times 8 = 16 fluid ounces.
Yash, by 2 pints, which is 1 quart. A gallon is 8 pints and 8 - 6 = 2.
3 quarts, 13 cups, 7 pints, 1 gallon. In cups: 3 \times 4 = 12, 13, 7 \times 2 = 14 and 16.
1 pint, 3 cups, 30 fluid ounces, 1 quart. In fluid ounces: 16, 3 \times 8 = 24, 30 and 32.
300 quarts. Each gallon is 4 quarts, so 75 \times 4 = 300.
It does not fit in a quart carton, which holds only 4 cups. It fits in the half gallon carton, which holds 8 cups, with 8 - 6 = 2 cups of room left, which is 1 pint.
(a) A capacity. (b) The volume of a solid. (c) The volume of a solid (the rock), measured in a capacity unit by the rise of the water.
6 gallons, since 40 - 34 = 6. That is 6 \times 4 = 24 quarts, so 24 pitchers.
The half gallon, the quart and the pint: 8 + 4 + 2 = 14 cups.
Halve a gallon and get 2 half gallons; halve again for 4 quarts, again for 8 pints, and once more for 16 cups. Each halving doubles the number of parts, so four halvings make 2 \times 2 \times 2 \times 2 = 16 equal parts, and each part is a cup.
The numbers count different units. A quart is smaller than a gallon: a gallon is 4 quarts. So 3 quarts is less than a gallon, by 1 quart. Numbers can be compared directly only when the units are the same.
Every quart of water in the bucket is the same as 4 cups. So each quart counted turns into 4 cups counted, and the count of cups is 4 times the count of quarts.
Both are partly right. Capacity is a volume: the volume of the empty space inside a container. But the pool as a whole takes up more space than it holds, because its walls and floor take up space too. So the capacity is the volume inside, not the volume of the whole pool.