Grade 4 · 4.MD.A.2

Word problems with distance, time, volume, mass and money

At field day the relay legs are in meters and kilometers, the water coolers in liters and milliliters, and the snacks in dollars and cents. How do you answer a question that mixes them, and how do you know the answer is the right size?

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Converting metric units by powers of tenGrade 4 Customary conversionsGrade 4

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Multi-step problems with unit conversionsGrade 5

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▴ 9 above Measurement is approximate: precision of a tool Significant figures Story problems with grams, kilograms and liters

Word problems with distance, time, volume, mass and money

Measurement error

When you reach for it

Radovan, Yindi and Narelle are helping at their school’s field day. The relay track is marked in meters, but the long run is posted in kilometers. The water coolers hold liters, and the paper cups hold milliliters. The obstacle course is timed in minutes and seconds. At the snack table, prices are in dollars and cents.

Every question they ask that day is a story with measurements in it. How far does the relay team run in all? How many cups does a cooler fill? Who finished the course faster? How much change comes back from a five-dollar bill? You reach for this skill whenever a story gives you measurements, asks for a new measurement, and the two do not come in the same unit.

The conversions themselves are not new. Changing one unit to another is in Customary conversions and Converting metric units by powers of ten, and the units are described in The customary system, Customary weight units, Customary capacity units and Units of time. Story problems that keep every amount in one unit are in Story problems with grams, kilograms and liters. What is new here is putting the pieces together: reading what a story is asking, getting every amount into one unit, choosing the operation from what happens in the story, and checking that the answer makes sense.

These are the facts used in this lesson. Each one is exact.

Measure	Facts
Length	1 km = 1{,}000 m, 1 m = 100 cm, 1 mile = 5{,}280 feet, 1 yard = 3 feet, 1 foot = 12 inches
Mass and weight	1 kg = 1{,}000 g, 1 pound = 16 ounces
Liquid volume	1 L = 1{,}000 mL, 1 gallon = 4 quarts, 1 quart = 2 pints, 1 pint = 2 cups
Time	1 hour = 60 minutes, 1 minute = 60 seconds
Money	1 dollar = 100 cents

Here km is kilometer, m is meter, cm is centimeter, kg is kilogram, g is gram, L is liter and mL is milliliter.

The procedure

Day 1: one unit first, then the operation the story calls for

Start by asking two questions about the story. What is the answer measuring: a length, a mass, a liquid volume, a time, an amount of money? And in what unit does the story want it? The answer to the first question tells you which facts can matter. A distance is never added to a time, and a mass is never added to a liquid volume, so a story about the relay needs only length facts.

Next, put every amount in one unit. Usually the smaller unit is the easiest choice. Changing a larger unit to a smaller one only multiplies, so every amount becomes a whole number and there is no remainder to carry. For example, 1 km 350 m becomes 1{,}000 + 350 = 1{,}350 meters.

Only then choose the operation, and choose it from what happens in the story, not from a single word in the question. Four kinds of things happen in measurement stories:

What happens in the story	The operation
parts are put together into a whole	add
one amount is taken from another, or you ask how much more one is than another	subtract
one amount is repeated a number of times	multiply
an amount is split into equal parts, or you ask how many of one size fit	divide

A word alone does not decide this. “How much is left?” can follow water poured out of a cooler, which is subtraction, or ribbon cut into equal pieces, where the leftover is a remainder from division. Picture what is happening to the amounts, and the operation follows.

Here is the whole move once. The bean bags for the toss weigh 150 grams each, and a box holds 12 of them. How much do the bean bags in a full box weigh, in kilograms and grams? The story repeats one mass 12 times, so it is multiplication:

12 \times 150 = 1{,}800 \text{ grams}

Then change back to the units the question asked for. A kilogram is 1{,}000 grams, so 1{,}800 grams is 1 kilogram and 800 more grams: 1 kg 800 g.

Day 2: fractions and decimals in a measurement, and money with change

Measurements are not always whole numbers of a unit. A walking loop might be \displaystyle \frac{1}{4} of a mile, an event might last \displaystyle \frac{3}{4} of an hour, and a bag of ice might weigh 2.5 kilograms. The same plan works: put the amount into a smaller unit, where it becomes a whole number.

A fraction of a unit is that fraction of the number in the fact. One fourth of an hour is one fourth of 60 minutes, which is 15 minutes, since 4 \times 15 = 60. Three fourths of an hour is three of those fourths:

\frac{3}{4} \text{ hour} = 3 \times 15 = 45 \text{ minutes}

Taking a fraction of a whole number this way is A fraction times a whole number. When a story repeats a fraction, such as walking a \displaystyle \frac{1}{4} mile loop several times, the fractions can also stay as they are and be added or multiplied first, as in Adding and subtracting fractions with like denominators: they are all fourths, so only the number of fourths changes.

A decimal measurement reads the same way. The digit after the point counts tenths, as in Tenths and hundredths. So 2.5 kilograms is 2 kilograms and 5 tenths of a kilogram. A tenth of 1{,}000 grams is 100 grams, so 5 tenths is 500 grams:

2.5 \text{ kg} = 2{,}000 \text{ g} + 500 \text{ g} = 2{,}500 \text{ g}

Money is a measurement too, with dollars as the larger unit and cents as the smaller one, as in Money and Money written as decimals. A price of 2.50 dollars is 2 dollars and 50 cents, or 250 cents. Working in cents turns every price into a whole number, and the answer changes back to dollars at the end.

Change is a comparison: how much more is the money handed over than the price? One way to find it is to count up from the price, the way a cashier hands back coins. From a price of 2.50 dollars to a five-dollar bill, add 50 cents to reach 3.00 dollars, then 2 dollars to reach 5.00 dollars. The change is the two jumps together, 2.50 dollars. Counting up is the same subtraction, 500 - 250 = 250 cents, done in friendly steps.

Day 3: pictures that help, and checking the answer’s unit and size

When a story has several amounts, a picture shows how they fit before any arithmetic. Two pictures do most of the work.

A bar model draws the whole as one long bar and the parts as pieces of it, as in Bar models for story problems. If the parts are known and the whole is not, the pieces are added. If the whole and some parts are known, the missing piece is what is left after taking the known parts away. Equal pieces show repeated amounts, which is multiplication or division.

A measurement number line is a ruler, a scale or a clock laid out flat, as in The number line. Each move along it is a jump of some amount. Distances walked one after another, or events that follow one another in time, are jumps added end to end. The distance between two points is a difference. On a time line, jumping to the next whole hour first is often the easiest path, because an hour is 60 minutes, not 100; Elapsed time works this way.

Last, check the answer. Two checks fit every measurement story.

The unit. The answer must be the kind of measure the question asked about, in the unit it asked for. A question about how much water is left wants liters or milliliters, never minutes.
The size. Round the amounts and work the story roughly. Three jugs that each hold a few liters cannot hold 300 liters in all. A number far from the rough answer usually means a unit was missed or the wrong operation was used.

Worked examples

Examples 1, 2 and 3 go with Day 1, Examples 4 and 5 with Day 2, and Examples 6 and 7 with Day 3.

Example 1

The relay in one unit

The relay has three legs. Radovan runs 800 meters, Yindi runs 1 kilometer 50 meters, and Narelle runs 650 meters. How far does the team run in all? Give the answer in meters, and then in kilometers and meters.

Solution

The answer measures a distance. The amounts are in two units, so first put all of them in meters, the smaller unit. A kilometer is 1{,}000 meters, so Yindi’s leg is 1{,}000 + 50 = 1{,}050 meters.

The three legs are run one after another and put together into one whole distance, so add. The order of adding may change (the commutative property), so pair 800 and 650 first:

\begin{aligned} 800 + 1{,}050 + 650 &= 800 + 650 + 1{,}050 \\ &= 1{,}450 + 1{,}050 = 2{,}500 \end{aligned}

The team runs 2{,}500 meters. Each 1{,}000 meters is a kilometer, and 2{,}500 holds two thousands and 500 more, so the distance is 2 km 500 m.

Check the size: the legs are about 800, 1{,}000 and 700 meters, which make about 2{,}500.

Example 2

Cups from a cooler

A water cooler holds 12 liters. Each paper cup holds 250 milliliters. How many cups can be filled from a full cooler?

Solution

The question asks how many cups of one size fit in the cooler, so it is division. The cooler is in liters and the cup in milliliters, so first put the cooler in milliliters: 12 \times 1{,}000 = 12{,}000 milliliters.

Dividing 12{,}000 by 250 is easier one liter at a time. Four cups of 250 milliliters make 1{,}000 milliliters, since 4 \times 250 = 1{,}000. So every liter fills 4 cups, and 12 liters fill 12 groups of 4:

12 \times 4 = 48 \text{ cups}

Check by multiplying back: 48 \times 250 = 12{,}000 milliliters, which is 12 liters. The answer is a count of cups, and it is much bigger than 12 because a cup is much smaller than a liter.

Example 3

Who was faster on the obstacle course

Narelle finishes the obstacle course in 2 minutes 15 seconds. Radovan’s time is 148 seconds. Who was faster, and by how much?

Solution

The answer measures time. Put both times in seconds, the smaller unit. A minute is 60 seconds, so 2 minutes is 2 \times 60 = 120 seconds, and Narelle’s time is

2 \times 60 + 15 = 120 + 15 = 135 \text{ seconds}

Only the minutes are multiplied; the 15 seconds are already seconds.

Comparing two amounts is subtraction: 148 - 135 = 13. Faster means less time on the clock, and 135 < 148, so Narelle was faster, by 13 seconds.

Example 4

Laps on the walking loop

The walking loop around the field is \displaystyle \frac{1}{4} of a mile. Yindi walks 3 laps and Narelle walks 5 laps. How far does each girl walk? How much farther does Narelle walk, in miles and in feet?

Solution

Each lap repeats the same \displaystyle \frac{1}{4} mile, so each distance is a whole number times \displaystyle \frac{1}{4}. Yindi walks three fourths and Narelle walks five fourths:

3 \times \frac{1}{4} = \frac{3}{4} \text{ mile}, \qquad 5 \times \frac{1}{4} = \frac{5}{4} \text{ miles}

Four fourths make 1 whole mile, so \displaystyle \frac{5}{4} miles is \displaystyle 1\frac{1}{4} miles.

Comparing the two is subtraction. Both are fourths, so subtract the numbers of fourths:

\frac{5}{4} - \frac{3}{4} = \frac{2}{4} = \frac{1}{2} \text{ mile}

The difference is just Narelle’s 2 extra laps, which is another way to see \displaystyle \frac{2}{4}. In feet, half a mile is half of 5{,}280 feet. Half of 5{,}000 is 2{,}500 and half of 280 is 140, so Narelle walks 2{,}640 feet farther. Check: 2{,}640 + 2{,}640 = 5{,}280, a whole mile.

Example 5

Change at the snack table

At the snack table Radovan buys 2 bags of popcorn at 1.25 dollars each and a juice for 0.80 dollars. He pays with a five-dollar bill. How much change does he get?

Solution

Work in cents, so every price is a whole number. A bag of popcorn is 125 cents. Two bags repeat that price, so multiply: 2 \times 125 = 250 cents. The juice is 80 cents. Putting the costs together is addition:

250 + 80 = 330 \text{ cents, or } 3.30 \text{ dollars}

The change is how much more 5.00 dollars is than 3.30 dollars, so subtract: 500 - 330 = 170 cents. Counting up gives the same thing: 10 cents takes 3.30 to 3.40, 60 cents takes it to 4.00, and 1 dollar takes it to 5.00, and 10 + 60 + 100 = 170 cents. Radovan gets 1.70 dollars in change.

Check the size: the snacks cost a little more than 3 dollars, so the change should be a little less than 2 dollars, and it is.

Example 6

What is left in the cooler

A cooler starts the morning with 18 liters of water. Each of 4 classes fills a jug that holds 3 liters 500 milliliters. How much water is left in the cooler?

Solution

Draw the cooler as a bar of 18 liters. Four equal jugs come out of it, so the bar has four equal pieces, and the rest of the bar is the unknown.

Figure 1The cooler as a bar. Four jugs of 3 liters 500 milliliters come out of 18 liters, and the shaded piece is what is left.

The picture shows two steps. The four jugs are one amount repeated 4 times, which is multiplication. Then the jugs are taken from the whole, which is subtraction. Put everything in milliliters first: each jug is 3{,}000 + 500 = 3{,}500 milliliters, and the cooler is 18{,}000 milliliters.

\begin{aligned} 4 \times 3{,}500 &= 14{,}000 \text{ mL} \\ 18{,}000 - 14{,}000 &= 4{,}000 \text{ mL} \end{aligned}

4{,}000 milliliters is 4 liters. Check with the bar: the four jugs and the leftover piece should make the whole, and 14{,}000 + 4{,}000 = 18{,}000. The leftover is also a sensible size, a little more than one jug.

Example 7

The morning schedule

Field day starts at 9{:}15. The first three events are the sack race, which lasts 25 minutes, the relay, 40 minutes, and the water balloon toss, 35 minutes. They run one right after another. At what time does the balloon toss end? How long is the morning, in hours and minutes?

Solution

The events follow one another, so their times are jumps added end to end along a time line.

Figure 2The morning schedule as jumps along a time line. Each jump is one event, and the last dot is the end time.

Jump 25 minutes from 9{:}15 to reach 9{:}40. The next jump is 40 minutes. Take 20 of them to reach the hour, 10{:}00, and the other 20 to reach 10{:}20. The last jump of 35 minutes lands at 10{:}55. The balloon toss ends at 10{:}55.

The whole morning is the three jumps together:

25 + 40 + 35 = 100 \text{ minutes}

An hour is 60 minutes, so 100 minutes is 1 hour and 40 more minutes: 1 hour 40 minutes. Check: one hour after 9{:}15 is 10{:}15, and 40 more minutes is 10{:}55, the same end time as the jumps.

Boundaries

One kind of measure in a sum or difference. Amounts are added or subtracted only when they measure the same kind of thing: lengths with lengths, times with times, money with money. Liters of water and kilograms of ice cannot be added into one total, and no fact here turns one into the other. Multiplying or dividing by a plain count, such as 4 classes or 12 bean bags, is always allowed.
A known fact must join the units. Every conversion here uses one fact from the table, or a fact built straight from it. Between units with no fact joining them, such as miles and kilometers, these facts give no answer.
The remainder means what the story says. When a division does not come out even, the leftover might be thrown away (a part piece of ribbon is not a piece), kept in the smaller unit (20 centimeters left), or force one more (one more jug to hold the rest). The arithmetic cannot decide; the story does, as in Remainders and what they mean.
Simple fractions and decimals. The fractions here are tenths, fourths, thirds, halves and the other fractions whose denominators divide the number in the fact, so the amount becomes a whole number of the smaller unit. The decimals have one or two digits after the point.
An answer is only as exact as its measurements. A cooler marked 18 liters holds about 18 liters. The arithmetic is exact, but the measurements it starts from carry Measurement error.

Where this goes next

Many real questions take several of these moves in a row, with a conversion in the middle of the work rather than only at the start or end. Those longer problems are taken apart in Multi-step problems with unit conversions. The same habits, a picture first, one unit, and a check on size, come back in perimeter and area problems with units, and later in Conversion factors and dimensional analysis, where the conversion facts become fractions equal to one so that a single rule decides when to multiply and when to divide.

Exercises

Day 1: one unit first

The relay has four legs of 400 meters each. How far does the team run in all? Is that more or less than 2 kilometers, and by how much?
Cooler A holds 9 liters. Cooler B holds 6 liters 400 milliliters. How many milliliters more does cooler A hold?
In the long jump, Narelle jumps 2 meters 35 centimeters and Yindi jumps 251 centimeters. Who jumps farther, and by how much?
Each round of the hoop game lasts 3 minutes 20 seconds. There are 4 rounds. How long is the game in seconds, and in minutes and seconds?
A crate holds 24 juice boxes, and each full juice box weighs 250 grams. How many kilograms do the juice boxes weigh?
Yindi pours 6 liters of lemonade equally into 8 pitchers. How many milliliters go in each pitcher?
The finish line tape is 5 yards long. Radovan cuts it into pieces 18 inches long for the team flags. How many pieces does he get?
Field day lasts 5 hours, and the time is split equally among 6 stations. How many minutes does each station get?

Day 2: fractions, decimals and money

Radovan walks 7 laps of the \displaystyle \frac{1}{4} mile loop. How far does he walk? Is it more than 1 mile, and by how much?
Each shift at the face painting booth lasts \displaystyle \frac{2}{3} of an hour. Narelle works 2 shifts. How long does she work, in minutes and in hours and minutes?
A bag of oranges for the snack table weighs 1.5 kilograms. A bag of grapes weighs 800 grams. How much do the two bags weigh together, in grams and in kilograms and grams?
Each team gets 6 bottles of water, and each bottle holds 0.5 liter. How many milliliters is that in all, and how many liters?
Yindi buys a granola bar for 0.65 dollars and a lemonade for 1.50 dollars. She pays with 3 dollars. How much change does she get?
Frozen fruit bars cost 0.60 dollars each. Radovan buys 5 of them for his team. How much does he spend?
The class pays 9.50 dollars for 10 juice boxes, all the same price. What is the price of one juice box?
The fun run is 1 mile long. Narelle has run \displaystyle \frac{5}{8} of a mile so far. How much farther does she have to go, in miles and in feet?

Day 3: pictures and checks

A cooler holds 20 liters. In the morning 7 liters 400 milliliters are used, and at lunch 8 liters 900 milliliters. Draw a bar model, and find how much water is left.
The tug of war starts at 1{:}35 in the afternoon and lasts 50 minutes. Use a time number line to find when it ends.
The three-legged race starts at 10{:}50 and ends at 11{:}35. How long does it last?
Four runners each run 300 meters. Radovan says the team ran 12 kilometers in all. Check the size of his answer, and find the right distance.
The snack table starts the day with 40 dollars in the cash box for making change. It sells 36 ice pops at 0.50 dollars each. How much money is in the cash box at the end?
Narelle’s wagon can carry 13 kilograms. She loads 3 bags of sand for the jumping pit, each weighing 4 kilograms 250 grams. How much do the bags weigh in all? How much more could the wagon carry?

Reasoning

A cooler holds 2 liters of water, and Yindi pours in 300 milliliters more. She writes that the cooler now holds 302 liters. Is that right? Explain, and give the right amount.
Narelle has 3 meters of ribbon and cuts pieces 40 centimeters long for prize bows. How many bows can she make? Explain why the answer is not 8, and say what the leftover means.
Radovan says that whenever a story says “each”, you multiply. Write a short field day story that uses the word “each” and is solved by dividing, solve it, and explain what tells you to divide.

Answers

1{,}600 meters, or 1 km 600 m. That is less than 2 kilometers, by 400 meters. Four legs repeat 400 meters, so 4 \times 400 = 1{,}600; 2 km is 2{,}000 m, and 2{,}000 - 1{,}600 = 400.
2{,}600 milliliters more. Cooler A is 9{,}000 mL and cooler B is 6{,}000 + 400 = 6{,}400 mL, and comparing is subtraction: 9{,}000 - 6{,}400 = 2{,}600, which is 2 L 600 mL.
Yindi, by 16 centimeters. Narelle’s jump is 2 \times 100 + 35 = 235 cm, and 251 - 235 = 16.
800 seconds, or 13 minutes 20 seconds. One round is 3 \times 60 + 20 = 200 seconds, and 4 \times 200 = 800. Then 800 \div 60 = 13 \text{ R } 20, since 13 \times 60 = 780.
6 kilograms. 24 \times 250 = 6{,}000 grams, because 4 boxes make 1{,}000 grams and 24 boxes are 6 groups of 4. 6{,}000 g is 6 kg.
750 milliliters. 6 L is 6{,}000 mL, and splitting into 8 equal parts is division: 6{,}000 \div 8 = 750. Check: 8 \times 750 = 6{,}000.
10 pieces. 5 yards is 5 \times 3 = 15 feet, which is 15 \times 12 = 180 inches, and 180 \div 18 = 10.
50 minutes. 5 hours is 5 \times 60 = 300 minutes, and 300 \div 6 = 50.
\displaystyle \frac{7}{4} miles, which is \displaystyle 1\frac{3}{4} miles. That is more than 1 mile, by \displaystyle \frac{3}{4} mile. \displaystyle 7 \times \frac{1}{4} = \frac{7}{4}, and four of the fourths make 1 mile, leaving 3 fourths.
80 minutes, or 1 hour 20 minutes. One third of 60 minutes is 20 minutes, so \displaystyle \frac{2}{3} hour is 40 minutes, and 2 \times 40 = 80. That is 60 + 20 minutes.
2{,}300 grams, or 2 kg 300 g. 1.5 kg is 1{,}000 + 500 = 1{,}500 g, and 1{,}500 + 800 = 2{,}300.
3{,}000 milliliters, which is 3 liters. 0.5 L is 5 tenths of 1{,}000 mL, which is 500 mL, and 6 \times 500 = 3{,}000.
0.85 dollars. The cost is 65 + 150 = 215 cents, and 300 - 215 = 85 cents. Counting up: 5 cents to 2.20, 80 cents to 3.00, and 5 + 80 = 85.
3.00 dollars. 5 \times 60 = 300 cents, which is 3 dollars.
0.95 dollars. 9.50 dollars is 950 cents, and sharing equally among 10 is division: 950 \div 10 = 95 cents. Check: 10 \times 95 = 950.
\displaystyle \frac{3}{8} mile, which is 1{,}980 feet. 1 mile is \displaystyle \frac{8}{8}, and \displaystyle \frac{8}{8} - \frac{5}{8} = \frac{3}{8}. One eighth of 5{,}280 feet is 660 feet, since 8 \times 660 = 5{,}280, so \displaystyle \frac{3}{8} mile is 3 \times 660 = 1{,}980 feet.
3 liters 700 milliliters. The bar is 20 L with two known pieces and an unknown piece. In milliliters the two uses are 7{,}400 + 8{,}900 = 16{,}300, and 20{,}000 - 16{,}300 = 3{,}700 mL.
2{:}25 in the afternoon. Jump 25 minutes from 1{:}35 to 2{:}00, then the other 25 minutes to 2{:}25, since 25 + 25 = 50.
45 minutes. From 10{:}50 to 11{:}00 is 10 minutes, and from 11{:}00 to 11{:}35 is 35 more, so 10 + 35 = 45.
1{,}200 meters, or 1 km 200 m. Four runners going a few hundred meters each cannot reach 12 kilometers, which is 12{,}000 meters. In fact 4 \times 300 = 1{,}200 meters, so his answer is ten times too big: 1{,}200 \times 10 = 12{,}000.
58 dollars. The ice pops bring in 36 \times 50 = 1{,}800 cents, which is 18 dollars, and adding it to the starting money gives 40 + 18 = 58.
12 kilograms 750 grams in all, and the wagon could carry 250 grams more. Each bag is 4{,}250 g, and 3 \times 4{,}250 = 12{,}750 g. The wagon holds 13{,}000 g, and 13{,}000 - 12{,}750 = 250.
No. The cooler holds 2 liters 300 milliliters, which is 2{,}300 milliliters. Yindi added 2 and 300 as if they were the same unit, but 2 counts liters and 300 counts milliliters. Putting both in milliliters first gives 2{,}000 + 300 = 2{,}300 mL. Her 302 liters is also far too big a size for water poured into a cooler.
7 bows, with 20 centimeters of ribbon left over. 3 m is 300 cm, and asking how many pieces of 40 fit is division: 300 \div 40 = 7 \text{ R } 20, since 7 \times 40 = 280. Eight bows would need 8 \times 40 = 320 cm, more ribbon than she has. The 20 cm is too short for a bow, so it is not another piece; it stays as leftover ribbon, in centimeters.
One story: 18 liters of water are shared equally among 6 teams. How much does each team get? Each team gets 18 \div 6 = 3 liters. The word “each” appears, but what happens is a whole being split into equal parts, and splitting into equal parts is division. A story where one amount is repeated, such as 6 teams that each get 3 liters, would be multiplication. The action in the story decides, not the word.