Grade 5 · 5.MD.A.1

Multi-step problems with unit conversions

A board is measured in feet, but the shelves you cut from it are measured in inches. How do you solve a problem when its amounts come in different units?

You need

Word problems with distance, time, volume, mass and moneyGrade 4

Where you are 6 in Systems of units

Measurement12

Systems of units6

The customary system The metric system and SI units Metric prefixes Conversion factors and dimensional analysis Converting square and cubic units

Multi-step problems with unit conversions

When you reach for it

Ziva is at the hardware store. She buys a board 8 feet long and wants to cut shelves that are each 18 inches long. How much board is left over?

You cannot subtract 18 inches from 8 feet as they stand. The numbers count different things: one counts feet, the other counts inches. Before you add, subtract or compare, both amounts must be in the same unit. So the problem has an extra step in the middle: change a unit, then carry on.

You reach for this move whenever a problem mixes units from one family: feet and inches, pounds and ounces, liters and milliliters, hours and minutes. The single conversion, such as 1 foot = 12 inches, is reviewed in Customary conversions and Converting metric units by powers of ten. Word problems that already sit in one unit are in Word problems with distance, time, volume, mass and money. This lesson puts the two together, in problems with several steps.

The procedure

This lesson takes four days. Day 1 works with length in feet, inches, yards and miles. Day 2 works with weight and capacity: pounds and ounces, gallons, quarts, pints and cups. Day 3 is metric units, where every change is a power of ten. Day 4 is time, answers written in two units at once, and choosing which unit to work in.

Day 1: length, and the step in the middle

These are the customary length facts. They are exact: a mile is defined as 5{,}280 feet.

bigger unit	equals
1 foot	12 inches
1 yard	3 feet, or 36 inches
1 mile	5{,}280 feet, or 1{,}760 yards

Bigger to smaller, multiply. Each foot holds 12 inches. So 3 feet is 3 groups of 12 inches:

3 \times 12 = 36, \quad \text{so } 3 \text{ feet} = 36 \text{ inches}

The number gets bigger because the unit got smaller. It takes more small units to cover the same length.

Smaller to bigger, divide. To change 50 inches to feet, ask how many groups of 12 fit in 50. Four groups use 48 inches and 2 are left:

50 \div 12 = 4 \text{ R } 2, \quad \text{so } 50 \text{ inches} = 4 \text{ feet } 2 \text{ inches}

The remainder is not thrown away. It is still a length, in inches. It can also be written as a fraction of a foot: 2 inches is \displaystyle \frac{2}{12} = \frac{1}{6} of a foot, so 50 inches is \displaystyle 4\frac{1}{6} feet.

Where the conversion goes. In a problem with several steps, convert as soon as two amounts in different units must meet. Ziva’s board is in feet and her shelves are in inches, so the board is changed to inches first. Then the rest of the problem is ordinary multiplying and subtracting, all in inches.

Day 2: weight and capacity

For weight, 1 pound = 16 ounces. Heavy loads use the ton: 1 ton = 2{,}000 pounds.

For capacity (how much a container holds), a gallon is four quarts, a quart and a pint are each twice the next smaller unit, and a cup is eight fluid ounces:

unit	equals
1 gallon	4 quarts
1 quart	2 pints
1 pint	2 cups
1 cup	8 fluid ounces

Chain the facts to skip a step. A gallon is 4 quarts, each quart is 2 pints, so a gallon is 4 \times 2 = 8 pints. Each pint is 2 cups, so a gallon is 8 \times 2 = 16 cups.

Figure 1One gallon is 4 quarts, 8 pints or 16 cups. The shaded part shows 1 quart = 2 pints = 4 cups.

The same two moves work as for length. Bigger to smaller multiplies: 3 pounds is 3 \times 16 = 48 ounces. Smaller to bigger divides, and a remainder stays in the smaller unit: 40 ounces is 40 \div 16 = 2 \text{ R } 8, so 2 pounds 8 ounces.

Day 3: metric units, powers of ten

In the metric system every unit is 10, 100 or 1{,}000 times another:

unit	equals
1 kilometer	1{,}000 meters
1 meter	100 centimeters, or 1{,}000 millimeters
1 centimeter	10 millimeters
1 kilogram	1{,}000 grams
1 liter	1{,}000 milliliters

So a metric conversion is multiplying or dividing by a power of ten, and the digits simply shift. That move is taught in Multiplying a decimal by ten, one hundred or one thousand and Dividing a decimal by ten, one hundred or one thousand. For example, 3.5 kilometers is 3.5 \times 1{,}000 = 3{,}500 meters, and 2{,}400 grams is 2{,}400 \div 1{,}000 = 2.4 kilograms.

The rule for where to convert does not change. Get every amount into one unit before you combine them. Metric answers can stay as decimals, such as 2.4 kilograms, so you rarely need two units at once.

Day 4: time, and answers in two units

Time is not counted in tens: 1 hour = 60 minutes, 1 minute = 60 seconds, 1 day = 24 hours and 1 week = 7 days. Reading clock times is reviewed in Twenty four hour time and crossing midnight; this lesson works with lengths of time.

Customary answers are often written in two units, such as 5 feet 4 inches or 2 hours 30 minutes. To add amounts like these, add each unit separately, then regroup if the smaller unit reaches a whole bigger one. 1 hour 50 minutes plus 40 minutes is 1 hour 90 minutes, and 90 minutes is 1 hour 30 minutes, so the total is 2 hours 30 minutes. This is the same regrouping as carrying a ten, except that 60 minutes, not ten, make the next unit.

To subtract, trade one bigger unit for smaller ones when the smaller part is too small. For 5 feet 4 inches minus 2 feet 9 inches, 4 inches is less than 9 inches, so trade 1 foot for 12 inches:

\begin{aligned} 5 \text{ ft } 4 \text{ in} &= 4 \text{ ft } 16 \text{ in} \\ 4 \text{ ft } 16 \text{ in} - 2 \text{ ft } 9 \text{ in} &= 2 \text{ ft } 7 \text{ in} \end{aligned}

Choosing a unit to work in. You can also turn every amount into the smallest unit, work in whole numbers, and convert back at the end. Or you can work in the bigger unit with fractions. Both give the same answer, since they measure the same length. The smallest unit usually keeps the numbers whole; the bigger unit can be quicker when the fractions are easy. Example 8 does it both ways.

Worked examples

Examples 1 and 2 go with Day 1, 3 and 4 with Day 2, 5 and 6 with Day 3, and 7 and 8 with Day 4.

Example 1

Ziva’s shelves

Ziva cuts 5 shelves, each 18 inches long, from a board 8 feet long. How much board is left? Could she cut a sixth shelf?

Solution

The board is in feet and the shelves are in inches, so change the board to inches. Each foot is 12 inches, and bigger to smaller multiplies:

8 \times 12 = 96 \text{ inches}

Now everything is in inches. Five shelves of 18 inches use 5 \times 18 = 90 inches. Subtract from the whole board:

96 - 90 = 6 \text{ inches}

Figure 2Ziva’s board. Five shelves of 18 inches use 90 of the 96 inches, and 6 inches are left.

So 6 inches are left, which is \displaystyle \frac{6}{12} = \frac{1}{2} of a foot. A sixth shelf would need 18 inches, and 6 is less than 18, so no.

Example 2

Navid’s rope by the yard

Navid needs 6 pieces of rope, each \displaystyle 2\frac{1}{2} feet long. The store sells rope by the yard, at 1.20 dollars per yard. How many yards should he buy, and what do they cost?

Solution

First find the total length in feet. Six pieces of \displaystyle 2\frac{1}{2} feet is \displaystyle 6 \times 2\frac{1}{2}. Split the mixed number, using the distributive property, as in Multiplying mixed numbers:

6 \times 2\tfrac{1}{2} = 6 \times 2 + 6 \times \tfrac{1}{2} = 12 + 3 = 15 \text{ feet}

The price is per yard, so change feet to yards. Smaller to bigger divides, and 3 feet make a yard:

15 \div 3 = 5 \text{ yards}

It comes out even, so 5 yards is exactly enough. The cost is 5 \times 1.20 = 6.00, so 6.00 dollars.

Example 3

Afia’s suitcase

At the airport, Afia’s checked bag may weigh at most 50 pounds. It weighs 47 pounds 10 ounces. She wants to add 4 jars of honey, each 14 ounces. Is she over the limit? If so, how many jars can she add and stay at or under it?

Solution

The jars are in ounces, so work in ounces. The bag holds 47 whole pounds and 10 more ounces. Each pound is 16 ounces:

47 \times 16 + 10 = 752 + 10 = 762 \text{ ounces}

The limit is 50 \times 16 = 800 ounces. Four jars weigh 4 \times 14 = 56 ounces, so the bag would weigh 762 + 56 = 818 ounces. That is 818 - 800 = 18 ounces over, which is 1 pound 2 ounces.

How many jars fit? The room left is 800 - 762 = 38 ounces. Then 38 \div 14 = 2 \text{ R } 10, so 2 jars fit, with 10 ounces to spare. A third jar would go 4 ounces over.

Example 4

Paint by the gallon and by the quart

Ziva needs \displaystyle 3\frac{1}{4} gallons of paint. A gallon can costs 32 dollars and a quart can costs 11 dollars. What is the cheapest way to buy at least enough?

Solution

The cans come in two sizes, so put the need in quarts. Each gallon is 4 quarts:

3\tfrac{1}{4} \times 4 = 3 \times 4 + \tfrac{1}{4} \times 4 = 12 + 1 = 13 \text{ quarts}

Compare the two cans for the same amount. Four quart cans hold a gallon and cost 4 \times 11 = 44 dollars, more than one gallon can at 32 dollars. So buy gallons where you can: 3 gallons cover 12 quarts. One quart is still needed. A quart can costs 11 dollars and a fourth gallon costs 32, so the quart is cheaper.

3 \times 32 + 11 = 96 + 11 = 107 \text{ dollars}

For comparison, 4 gallons would cost 128 dollars and 13 quart cans 143 dollars. The cheapest is 3 gallons and 1 quart, for 107 dollars.

Example 5

A spool of wire

A spool holds 25 meters of wire. Navid cuts 16 pieces, each 85 centimeters long. How many meters of wire are left?

Solution

The pieces are in centimeters, so find their total first: 16 \times 85 = 1{,}360 centimeters. The spool is in meters. A meter is 100 centimeters, and smaller to bigger divides, so the digits shift two places right:

1{,}360 \div 100 = 13.6 \text{ meters}

Now subtract in meters: 25 - 13.6 = 11.4 meters. Check by working in centimeters instead. The spool holds 25 \times 100 = 2{,}500 centimeters, and 2{,}500 - 1{,}360 = 1{,}140 centimeters, which is 11.4 meters. The two ways agree.

Example 6

Gifts in a metric bag

On another trip, Afia’s bag limit is 23 kilograms. Her bag weighs 19.4 kilograms, and she adds 6 gifts that weigh 450 grams each. How far under the limit is she?

Solution

The gifts weigh 6 \times 450 = 2{,}700 grams. A kilogram is 1{,}000 grams, so divide by 1{,}000:

2{,}700 \div 1{,}000 = 2.7 \text{ kilograms}

The bag now weighs 19.4 + 2.7 = 22.1 kilograms. The limit is 23, and 23 - 22.1 = 0.9, so she is 0.9 kilograms under. In grams that is 0.9 \times 1{,}000 = 900 grams, room for two more gifts but not three.

Example 7

Navid’s two flights

Navid’s first flight takes 2 hours 45 minutes. He waits 1 hour 50 minutes, then flies 3 hours 35 minutes. How long is the whole trip?

Solution

Add each unit on its own. The hours are 2 + 1 + 3 = 6. The minutes are 45 + 50 + 35 = 130. That gives 6 hours 130 minutes, which is not finished, since 130 minutes is more than an hour. Regroup: 130 \div 60 = 2 \text{ R } 10, so 130 minutes is 2 hours 10 minutes.

6 \text{ h } 130 \text{ min} = 8 \text{ h } 10 \text{ min}

Check in minutes only: 165 + 110 + 215 = 490, and 490 = 8 \times 60 + 10. The trip is 8 hours 10 minutes.

Example 8

Rope in yards, pieces in feet and inches

A coil of rope is 10 yards long. Ziva cuts 4 pieces, each 5 feet 8 inches. How much rope is left? Answer in feet and inches, and work it two ways.

Solution

In inches. A yard is 36 inches, so the coil is 10 \times 36 = 360 inches. A piece is 5 \times 12 + 8 = 68 inches, and four pieces are 4 \times 68 = 272 inches. What is left is 360 - 272 = 88 inches. Since 88 \div 12 = 7 \text{ R } 4, that is 7 feet 4 inches.

In feet. The coil is 10 \times 3 = 30 feet. A piece is \displaystyle 5\frac{8}{12} = 5\frac{2}{3} feet. Four pieces are

4 \times 5\tfrac{2}{3} = 20 + \tfrac{8}{3} = 22\tfrac{2}{3} \text{ feet}

and \displaystyle 30 - 22\frac{2}{3} = 7\frac{1}{3} feet. A third of a foot is 12 \div 3 = 4 inches, so this is 7 feet 4 inches again.

Both ways give 7 feet 4 inches, which is also 2 yards 1 foot 4 inches. Inches kept every number whole; feet needed thirds. Either is correct, and the choice is about which arithmetic is easier.

Boundaries

One system at a time. Every fact here stays inside the customary system or inside the metric system. Changing inches to centimeters needs a fact that links the two, 1 inch = 2.54 centimeters, and is done in Conversion factors and dimensional analysis.
Only the same kind of amount converts. Feet change to inches because both measure length. Gallons cannot be changed to pounds: how much a gallon weighs depends on what is in it. The fluid ounce measures capacity and the ounce measures weight, so they are different units with a shared name.
A count of pieces is a whole number. When cutting, the leftover is shorter than one more piece. When something is sold only in whole units, round up to have enough, as in exercise 6.
Time is not base ten. 1 hour 30 minutes is \displaystyle 1\frac{1}{2} hours, or 1.5 hours, not 1.30 hours, because an hour has 60 minutes, not 100.
A two-unit answer is finished only when the smaller part is less than one bigger unit. 2 feet 15 inches is a true length but not a finished answer; it regroups to 3 feet 3 inches.

Where this goes next

Next in this unit, Line plots with fractions collects measurements such as \displaystyle \frac{1}{4} and \displaystyle \frac{3}{8} of an inch and solves problems with them. Later, Conversion factors and dimensional analysis writes each conversion fact as a fraction equal to one, which settles whether to multiply or divide by writing the units down, and also carries amounts between the two systems. Changing the units of a rate uses the same idea for speeds, changing feet per minute into miles per hour. Changing square and cubic units, where the length fact is used two or three times, is in Converting square and cubic units.

Exercises

Day 1: length

A board is 6 feet long. Ziva cuts it into pieces 9 inches long. How many pieces does she get?
Navid buys 4 yards of rope and cuts off 7 feet. How many feet are left?
A walking loop is 880 yards long. Afia walks it 6 times. How many miles does she walk?
Chain costs 3 dollars per foot. Ziva needs \displaystyle 2\frac{1}{3} yards. What does it cost?
Ziva cuts 3 pieces, each 16 inches, from a 5-foot board. How much is left? Can she cut a fourth piece?
Navid needs 8 tent lines, each 28 inches long. The store sells rope only in whole feet. How many feet must he buy, and how much will be left over?

Day 2: weight and capacity

A box of nails weighs 12 ounces. How many pounds do 20 boxes weigh?
Afia’s bag may weigh at most 50 pounds and it weighs 44 pounds. How many bottles of syrup, 12 ounces each, can she add?
Navid has 3 gallons of paint. He uses 5 quarts on a door and pours the rest into pint jars. How many jars does he fill?
A sprayer holds 1 gallon. Ziva puts in 3 cups of cleaner and fills the rest with water. How many cups of water does she add?
One suitcase weighs 22 pounds 9 ounces and another weighs 18 pounds 15 ounces. How much heavier is the first?
Grass seed is sold loose for 0.25 dollars per ounce. What does a \displaystyle 2\frac{1}{2}-pound bag cost?

Day 3: metric units

A spool holds 50 meters of cord. Navid cuts 12 pieces, each 175 centimeters. How many meters are left?
A box holds 250 screws, each weighing 4 grams. The empty box weighs 150 grams. What does the full box weigh in kilograms?
A can holds 4 liters of paint. Ziva uses 650 milliliters on each of 5 doors. How much paint is left, in milliliters?
Afia packs 6 small bottles of lotion, 85 milliliters each. Is that more than half a liter? By how much?
Navid walks 2.3 kilometers through the airport and then rides 4 moving walkways, each 120 meters long. How far does he travel, in kilometers?
A stack holds 40 floor tiles, each 9 millimeters thick. How tall is it in centimeters, and in meters?

Day 4: time and two-unit answers

Afia drives 1 hour 25 minutes to the airport, waits 2 hours 50 minutes, and flies 4 hours 15 minutes. How long is her whole trip?
A rope is 12 feet long. Navid cuts 2 pieces, each 4 feet 10 inches. How much is left, in feet and inches?
Ziva paints a door in 45 minutes. How long do 7 doors take, in hours and minutes?
Boards 7 feet 3 inches and 5 feet 11 inches long are laid end to end. How long is the row, in feet and inches? In yards, feet and inches?

Reasoning

Is 3 feet 5 inches the same length as 3.5 feet? Explain, and write 3 feet 5 inches in feet.
Explain why changing gallons to cups multiplies and gives a bigger number, using 2 gallons as the example.
Adding 2 hours 40 minutes and 1 hour 35 minutes unit by unit gives 3 hours 75 minutes. Is that the right amount of time? Write it as a finished answer. Is it the same as 3.75 hours? Explain.

Answers

8 pieces. 6 feet = 6 \times 12 = 72 inches, and 72 \div 9 = 8.
5 feet. 4 yards = 4 \times 3 = 12 feet, and 12 - 7 = 5.
3 miles. 6 \times 880 = 5{,}280 yards, and 5{,}280 \div 1{,}760 = 3.
21 dollars. \displaystyle 2\frac{1}{3} yards \displaystyle = 2\frac{1}{3} \times 3 = 6 + 1 = 7 feet, and 7 \times 3 = 21.
12 inches, or 1 foot. The board is 5 \times 12 = 60 inches and the pieces use 3 \times 16 = 48, so 60 - 48 = 12. No fourth piece: 12 is less than 16.
19 feet, with 4 inches left over. The lines need 8 \times 28 = 224 inches, and 224 \div 12 = 18 \text{ R } 8, which is 18 feet 8 inches. Rope comes in whole feet, so buy 19 feet = 228 inches, and 228 - 224 = 4.
15 pounds. 20 \times 12 = 240 ounces, and 240 \div 16 = 15.
8 bottles. There is room for 50 - 44 = 6 pounds = 6 \times 16 = 96 ounces, and 96 \div 12 = 8.
14 jars. 3 gallons = 12 quarts, and 12 - 5 = 7 quarts. Each quart is 2 pints, so 7 \times 2 = 14.
13 cups. 1 gallon = 16 cups, and 16 - 3 = 13.
3 pounds 10 ounces. 22 \times 16 + 9 = 361 ounces and 18 \times 16 + 15 = 303 ounces, so the difference is 58 ounces, and 58 \div 16 = 3 \text{ R } 10.
10 dollars. \displaystyle 2\frac{1}{2} pounds \displaystyle = 2\frac{1}{2} \times 16 = 32 + 8 = 40 ounces, and 40 \times 0.25 = 10.
29 meters. 12 \times 175 = 2{,}100 centimeters = 21 meters, and 50 - 21 = 29.
1.15 kilograms. The screws weigh 250 \times 4 = 1{,}000 grams, the full box 1{,}000 + 150 = 1{,}150 grams, and 1{,}150 \div 1{,}000 = 1.15.
750 milliliters. 5 \times 650 = 3{,}250 milliliters used, 4 liters = 4{,}000 milliliters, and 4{,}000 - 3{,}250 = 750.
Yes, by 10 milliliters. 6 \times 85 = 510 milliliters, half a liter is 500 milliliters, and 510 - 500 = 10.
2.78 kilometers. The walkways are 4 \times 120 = 480 meters, 2.3 kilometers is 2{,}300 meters, and 2{,}300 + 480 = 2{,}780 meters = 2.78 kilometers.
36 centimeters, or 0.36 meters. 40 \times 9 = 360 millimeters, 360 \div 10 = 36 centimeters, and 36 \div 100 = 0.36 meters.
8 hours 30 minutes. Hours: 1 + 2 + 4 = 7. Minutes: 25 + 50 + 15 = 90, which is 1 hour 30 minutes, so 7 + 1 = 8 hours and 30 minutes.
2 feet 4 inches. The rope is 144 inches, a piece is 4 \times 12 + 10 = 58 inches, two pieces are 116 inches, and 144 - 116 = 28 inches, which is 2 feet 4 inches.
5 hours 15 minutes. 7 \times 45 = 315 minutes, and 315 \div 60 = 5 \text{ R } 15.
13 feet 2 inches, which is 4 yards 1 foot 2 inches. Feet: 7 + 5 = 12. Inches: 3 + 11 = 14, which is 1 foot 2 inches, so 13 feet 2 inches. Then 13 feet is 4 yards 1 foot, since 13 \div 3 = 4 \text{ R } 1.
No. 3 feet 5 inches is \displaystyle 3\frac{5}{12} feet. The .5 in 3.5 means 5 tenths of a foot, but a foot has 12 inches, not 10, so 5 inches is \displaystyle \frac{5}{12} of a foot. In fact 3.5 feet is \displaystyle 3\frac{1}{2} feet, which is 3 feet 6 inches.
A cup is a much smaller unit than a gallon, and each gallon holds 16 cups. So 2 gallons is 2 groups of 16 cups, 2 \times 16 = 32 cups. The amount of liquid is the same; it takes more of the small unit to hold it, so the number is bigger.
Yes, 3 hours 75 minutes is the right amount, but it is not finished. 75 minutes is 1 hour 15 minutes, so the answer is 4 hours 15 minutes. As a decimal that is \displaystyle 4\frac{15}{60} = 4\frac{1}{4} = 4.25 hours. So it is not 3.75 hours: 3.75 hours is 3 hours 45 minutes, because the .75 means \displaystyle \frac{3}{4} of an hour, not 75 minutes.