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Customary conversions

The bike race is 3 miles long, but the course map is marked in feet. How do you say the same length in another unit, and how do you know whether the number should get bigger or smaller?

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When you reach for it

Emre, Mykola and Aminata are helping at a bike race. The race is 3 miles long, but the course map is marked in feet. Mykola sets out cones by the yard. At the food tent the punch is mixed by the quart and poured by the cup, and at the butcher counter across the street meat is sold by the pound and weighed in ounces.

Each time, one amount is named in two different units. The amount itself does not change: the race is just as long whether you count it in miles or in feet. Only the unit you count with changes. You reach for a conversion whenever two measurements of the same kind come in different units and you need them in one unit: to compare them, to add them, or to give an answer in the unit someone asked for.

This lesson changes one unit into one other unit, inside one kind of measure: length to length, weight to weight, capacity to capacity. What the units are, and why the steps between them are not all the same size, is in The customary system. These are the facts used here. Every one is exact: a foot is defined to be 12 inches, not “about 12”.

LengthWeightCapacity
1 foot = 12 inches1 pound = 16 ounces1 cup = 8 fluid ounces
1 yard = 3 feet1 ton = 2{,}000 pounds1 pint = 2 cups
1 mile = 5{,}280 feet1 quart = 2 pints
1 gallon = 4 quarts

The procedure

Part 1: from a larger unit to a smaller one, by multiplying

A foot is 12 inches. So 2 feet is two twelves, 3 feet is three twelves, and every foot you add brings 12 more inches. A number of feet is that many equal groups of 12 inches, and equal groups are multiplication:

\text{number of inches} = \text{number of feet} \times 12

A two-column table lays the groups out. Each row is one more foot.

FeetInches
112
224
336
448
560
672

Read down, and the inches column climbs by 12 each time the feet column climbs by 1. Read across, and every row says “feet times 12”. The table is the twelves from the Multiplication facts, with units written on.

A double number line shows the same thing as a ruler would. It is one line with two scales: feet on top, inches below, and each tick names one spot on the line in both units. When you Measuring length with a ruler, this is why the 1 foot mark and the 12 inch mark are the same mark.

0 0 1 12 2 24 3 36 4 48 5 60 6 72 7 84 8 96 9 108 feet inches
Figure 1Feet and inches on one line. Each step of 1 foot along the top is a step of 12 inches along the bottom, so 9 feet sits at 108 inches.

The finish banner is 9 feet wide. In inches it is 9 \times 12 = 108 inches, the last tick on the line.

Why does the number get bigger? An inch is a much smaller unit than a foot, so it takes more inches than feet to cover the same banner. Going from a larger unit to a smaller one, the count always goes up. That gives a quick check: if you change feet to inches and your answer is smaller than the number you started with, you went the wrong way.

The same move works for every pair in the table. Yards to feet multiplies by 3, because each yard is 3 feet. Miles to feet multiplies by 5{,}280. A big number like 5{,}280 is easier in parts: split it into thousands, hundreds and tens, multiply each part, and add, as in Multiplying by tens and hundreds. One more length fact follows from the table: a mile is exactly 1{,}760 yards, because 1{,}760 yards of 3 feet each is 1{,}760 \times 3 = 5{,}280 feet.

Part 2: weight, capacity, and mixed units

Weight works the same way. A pound is 16 ounces, so a number of pounds is that many groups of 16 ounces, and pounds to ounces multiplies by 16.

PoundsOunces
116
232
348
464

A ton is 2{,}000 pounds, so tons to pounds multiplies by 2{,}000. The units of weight, and what each one feels like in your hand, are in Customary weight units.

Capacity, how much a container holds, has a chain of units, each one a set number of the next smaller one. A gallon is 4 quarts, a quart is 2 pints, a pint is 2 cups, and a cup is 8 fluid ounces. Each link is one multiplication: gallons to quarts by 4, quarts to pints by 2, pints to cups by 2, cups to fluid ounces by 8. At the punch bowl one more fact is handy: a quart is 2 pints and each pint is 2 cups, so a quart is 4 cups. The capacity units themselves are in Customary capacity units.

Many measurements come in two units at once. A height of 3 feet 7 inches means 3 whole feet and 7 more inches. To say it in inches only, change the feet to inches, then add the inches that were already there:

\begin{aligned} 3 \text{ feet } 7 \text{ inches} &= 3 \times 12 + 7 \\ &= 36 + 7 = 43 \text{ inches} \end{aligned}

Only the feet are multiplied. The 7 inches are already inches, so they are added as they are.

Part 3: from a smaller unit to a larger one, by dividing

Aminata cuts a finish-line tape 58 inches long. How long is it in feet? Now the question runs the other way: how many groups of 12 inches fit in 58 inches? Asking how many equal groups fit is division. Four groups of 12 is 48, and five groups is 60, which is more than 58. So 4 whole feet fit, and 58 - 48 = 10 inches are left over.

\begin{aligned} 58 \div 12 &= 4 \text{ R } 10 \\ 58 \text{ inches} &= 4 \text{ feet } 10 \text{ inches} \end{aligned}
12 in 1 ft 12 in 1 ft 12 in 1 ft 12 in 1 ft 10 in 58 inches
Figure 2Aminata’s tape. Four whole feet fit in 58 inches, and the shaded 10 inches are too short to make a fifth foot.

The remainder is not thrown away. It is the inches that are not enough to make another whole foot, and the answer keeps them as inches, as in Remainders and what they mean. That remainder is always less than 12: if 12 or more inches were left, one more foot would fit. Check by running Part 2 backward: 4 \times 12 + 10 = 48 + 10 = 58.

This time the count went down, from 58 to 4 (with 10 inches beside it). Fewer of the larger unit cover the same length. Larger to smaller multiplies; smaller to larger divides.

To compare two measurements in different units, first put them in the same unit. The smaller unit is usually the easier choice, because changing to it only multiplies, so there is no remainder to carry. Is 5 feet longer than 62 inches? Change the feet: 5 \times 12 = 60 inches. Now both are in inches, and 60 < 62, so 62 inches is longer, by 2 inches.

Worked examples

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

Example 1

Cones on the course

Mykola sets a cone every 7 yards along the last straight stretch of the course. Make a table of yards and feet from 1 to 7 yards, and say how many feet apart the cones are.

Solution

Each yard is 3 feet, so a number of yards is that many groups of 3 feet. Each row multiplies the yards by 3.

YardsFeet
13
26
39
412
515
618
721

The feet column climbs by 3 each row, one more yard each time. The last row is the one asked about: 7 \times 3 = 21, so the cones are 21 feet apart. A foot is smaller than a yard, so the number of feet should be larger than the number of yards, and 21 is larger than 7.

Example 2

The whole race in feet

The race is 3 miles long. How many feet long is it?

Solution

A mile is 5{,}280 feet, so 3 miles is three groups of 5{,}280 feet, which is 3 \times 5{,}280. Split 5{,}280 into 5{,}000 and 280, multiply each part by 3, and add. Multiplying each part and adding is the distributive property:

\begin{aligned} 3 \times 5{,}280 &= 3 \times 5{,}000 + 3 \times 280 \\ &= 15{,}000 + 840 = 15{,}840 \end{aligned}

Here 3 \times 280 is 3 \times 28 tens, which is 84 tens, or 840. The race is 15{,}840 feet long. As a check on size, 3 \times 5{,}000 = 15{,}000, and the answer should be a little more than that, which it is.

Example 3

A roast at the butcher counter

Emre orders a 7 pound roast at the butcher counter. The scale reads in ounces. What should it show?

Solution

A pound is 16 ounces, so 7 pounds is 7 groups of 16 ounces, which is 7 \times 16. Split 16 into 10 and 6 and use the distributive property:

7 \times 16 = 7 \times 10 + 7 \times 6 = 70 + 42 = 112

The scale should show 112 ounces. The number is bigger than 7 because an ounce is a much smaller unit than a pound.

Example 4

Glasses from the punch bowl

The punch bowl holds 6 quarts. Each glass holds 1 cup. How many glasses can be filled from a full bowl? Check the answer by going through pints.

Solution

A quart is 4 cups, so 6 quarts is 6 groups of 4 cups:

6 \times 4 = 24 \text{ cups}

Each glass takes 1 cup, so the bowl fills 24 glasses.

Check through pints. Each quart is 2 pints, so 6 quarts is 6 \times 2 = 12 pints. Each pint is 2 cups, so 12 pints is 12 \times 2 = 24 cups. Both ways give 24 cups.

Example 5

A chicken in pounds and ounces

Aminata buys a chicken that weighs 4 pounds 5 ounces. The butcher’s scale shows ounces only. What does it show?

Solution

The chicken is 4 whole pounds and 5 more ounces. Change the pounds to ounces: 4 \times 16 = 64 ounces. The 5 ounces are already ounces, so add them as they are:

\begin{aligned} 4 \text{ pounds } 5 \text{ ounces} &= 4 \times 16 + 5 \\ &= 64 + 5 = 69 \text{ ounces} \end{aligned}

The scale shows 69 ounces.

Example 6

Ounces back to pounds

Mykola’s bag of sausages weighs 91 ounces. How much is that in pounds and ounces?

Solution

Ounces are the smaller unit, so the question is how many groups of 16 ounces fit in 91 ounces. Five groups is 5 \times 16 = 80, and six groups is 6 \times 16 = 96, which is more than 91. So 5 whole pounds fit, and 91 - 80 = 11 ounces are left:

91 \div 16 = 5 \text{ R } 11

The bag weighs 5 pounds 11 ounces. The 11 ounces are fewer than 16, so they cannot make another pound, and the answer is finished. Check: 5 \times 16 + 11 = 80 + 11 = 91.

Example 7

Who rode farther

In the warm-up, Emre rides 1 mile and Mykola rides 1{,}800 yards. Who rides farther, and by how much?

Solution

The two distances are in different units, so put them in one unit first. Feet are smaller than both miles and yards, so changing both to feet only multiplies, and both answers are whole numbers.

Emre’s mile is 5{,}280 feet. Mykola’s 1{,}800 yards is 1{,}800 groups of 3 feet:

1{,}800 \times 3 = 5{,}400 \text{ feet}

Now compare: 5{,}400 > 5{,}280, so Mykola rides farther, by 5{,}400 - 5{,}280 = 120 feet.

Check in yards instead. A mile is 1{,}760 yards, and 1{,}800 - 1{,}760 = 40 yards. Forty yards is 40 \times 3 = 120 feet, the same answer.

Boundaries

  • One kind of measure at a time. A length changes only to another length, a weight to another weight, a capacity to another capacity. An ounce of weight and a fluid ounce of capacity share a name but measure different things: a cup holds 8 fluid ounces of punch, and that says nothing about how many ounces the punch weighs. No fact here turns one into the other.
  • A fact must join the two units. Each conversion here uses one fact from the table, or one fact built straight from them (a quart is 4 cups, a mile is 1{,}760 yards). Between units with no such fact, such as gallons and fluid ounces, it takes several steps, which is the work of Multi-step problems with unit conversions.
  • Whole numbers, with the leftover in the smaller unit. Every answer here is a whole number of each unit. When dividing leaves a remainder, it stays in the smaller unit, and it must be less than the number in the fact (12 for inches, 16 for ounces), or a whole larger unit is still hiding in it. Writing a leftover as a fraction of the larger unit, such as \displaystyle 4\frac{10}{12} feet, comes later.
  • Customary units only. These facts stay inside the customary system. Metric units change by tens instead, in Converting metric units by powers of ten, and no fact here goes from one system to the other.

Where this goes next

Real measurement questions often need more than one conversion, or a conversion in the middle of other work: cutting, sharing, adding up a trip. Those are taken apart in Multi-step problems with unit conversions, and in the story problems of Word problems with distance, time, volume, mass and money, where every amount is first put in one unit. Later, in Conversion factors and dimensional analysis, the facts in the table become fractions like twelve inches over one foot, so that one tidy rule decides whether to multiply or divide, even through a long chain of units.

Exercises

Part 1: larger to smaller, length

  1. Change 5 feet to inches.
  2. Change 9 yards to feet.
  3. Make a two-column table of feet and inches for 11, 13 and 15 feet.
  4. The longest race of the day is 2 miles. How many feet is that?
  5. The start corral is 25 yards long. How long is it in feet?
  6. Water stops stand every 400 yards along a course. How many feet apart are they?
  7. A mile is 1{,}760 yards. Mykola rides 4 miles. How many yards is that?

Part 2: weight, capacity and mixed units

  1. Change 9 pounds to ounces.
  2. A truck at the race carries a load of 3 tons. How many pounds is that?
  3. Change 7 quarts to pints.
  4. Change 5 gallons to quarts.
  5. A pitcher holds 9 cups. How many fluid ounces is that?
  6. Emre is 5 feet 3 inches tall. How tall is he in inches?
  7. At the butcher counter, a ham weighs 6 pounds 14 ounces. How many ounces is that?
  8. The punch recipe makes 3 gallons 2 quarts. How many quarts is that?

Part 3: smaller to larger, and comparing

  1. Change 71 inches to feet and inches.
  2. Change 53 ounces to pounds and ounces.
  3. Change 23 feet to yards and feet.
  4. Change 19 cups to pints and cups.
  5. Which is longer, 5 feet 1 inch or 64 inches? By how much?
  6. Emre wants 2 pounds of shrimp. The bag the butcher hands him weighs 35 ounces. Is that enough? How much more or less than he wants is it?
  7. Aminata has 22 pints of punch. Mykola has 10 quarts. Who has more, and by how much?

Reasoning

  1. Explain, without a table, why changing yards to feet multiplies by 3, and why the number of feet is always larger than the number of yards.
  2. Mykola changes 40 inches to feet and writes 3 feet 4 inches. Emre multiplies and writes 480 feet. Who is right? Explain how the size of the answer tells you, before doing any arithmetic.
  3. A scale shows 2 pounds 18 ounces. Explain why this is not the usual way to write a weight, and write it with fewer than 16 ounces.
Answers
  1. 60 inches. Each foot is 12 inches, so 5 \times 12 = 60.
  2. 27 feet. Each yard is 3 feet, so 9 \times 3 = 27.
  3. 132, 156 and 180 inches: 11 \times 12 = 132, 13 \times 12 = 156 and 15 \times 12 = 180. Each row is 24 inches more than the one before, because it is 2 feet more.
  4. 10{,}560 feet. Each mile is 5{,}280 feet, so 2 \times 5{,}280 = 10{,}560.
  5. 75 feet. 25 \times 3 = 75.
  6. 1{,}200 feet. 400 \times 3 = 1{,}200: 4 hundreds times 3 is 12 hundreds.
  7. 7{,}040 yards. 4 \times 1{,}760 = 4{,}000 + 2{,}800 + 240 = 7{,}040, multiplying the thousands, hundreds and tens of 1{,}760 by 4 one at a time.
  8. 144 ounces. 9 \times 16 = 9 \times 10 + 9 \times 6 = 90 + 54 = 144.
  9. 6{,}000 pounds. Each ton is 2{,}000 pounds, so 3 \times 2{,}000 = 6{,}000.
  10. 14 pints. Each quart is 2 pints, so 7 \times 2 = 14.
  11. 20 quarts. Each gallon is 4 quarts, so 5 \times 4 = 20.
  12. 72 fluid ounces. Each cup is 8 fluid ounces, so 9 \times 8 = 72.
  13. 63 inches. 5 \times 12 + 3 = 60 + 3 = 63.
  14. 110 ounces. 6 \times 16 + 14 = 96 + 14 = 110.
  15. 14 quarts. 3 \times 4 + 2 = 12 + 2 = 14.
  16. 5 feet 11 inches. 71 \div 12 = 5 \text{ R } 11, since 5 \times 12 = 60 and 71 - 60 = 11. Check: 60 + 11 = 71.
  17. 3 pounds 5 ounces. 53 \div 16 = 3 \text{ R } 5, since 3 \times 16 = 48 and 53 - 48 = 5.
  18. 7 yards 2 feet. 23 \div 3 = 7 \text{ R } 2, since 7 \times 3 = 21 and 23 - 21 = 2.
  19. 9 pints 1 cup. 19 \div 2 = 9 \text{ R } 1, since 9 \times 2 = 18 and 19 - 18 = 1.
  20. 64 inches is longer, by 3 inches. 5 feet 1 inch is 5 \times 12 + 1 = 61 inches, and 64 - 61 = 3.
  21. Yes, it is 3 ounces more than he wants. 2 pounds is 2 \times 16 = 32 ounces, and 35 - 32 = 3.
  22. Aminata has more, by 2 pints. 10 quarts is 10 \times 2 = 20 pints, and 22 - 20 = 2.
  23. Each yard is the same length as 3 feet. So a number of yards is that many groups of 3 feet, and counting equal groups is multiplying by 3. A foot is shorter than a yard, so more feet than yards are needed to cover the same length; every yard turns into 3 feet, so the count of feet is 3 times the count of yards, which is larger for any number of yards above 0.
  24. Mykola is right. Inches are smaller than feet, so a length takes fewer feet than inches: the answer must be less than 40 feet, and 480 is far more, so Emre multiplied when he should have divided. In fact 40 \div 12 = 3 \text{ R } 4, and the check 3 \times 12 + 4 = 40 works.
  25. 3 pounds 2 ounces. The ounces part is usually less than 16, because 16 ounces make a pound. Here 18 = 16 + 2, so a whole pound is hiding in the ounces. Moving it to the pounds gives 3 pounds 2 ounces. Check: 2 \times 16 + 18 = 50 and 3 \times 16 + 2 = 50.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0