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Units and measurement
A recipe in cups, a board in feet and inches, a drive in hours and minutes: word problems love to mix units. Learn the facts by heart and one rule about when to multiply and when to divide, and most unit questions become one line of arithmetic.
A measurement is a number and a unit
A unit is a fixed amount you count with: an inch, a pound, a minute. A measurement tells you how many of those units you have. 12 feet means 12 lengths of one foot, end to end. If you change the unit, the number changes, but the length stays the same.
Big unit, small number
A yard is bigger than a foot, so you need fewer yards to cover the same distance: 4 yards, but 12 feet. This one idea tells you what to do in every conversion. Going to a smaller unit, the number gets bigger, so you multiply. Going to a bigger unit, the number gets smaller, so you divide.
How to convert units
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The facts to know by heart
In the United States, everyday life uses U.S. customary units: inches and miles for length, ounces and pounds for weight, cups and gallons for liquids. Each fact below says how many of a small unit make one big unit. Learn them the way you learned your phone number. You cannot convert what you do not know.
LengthOne of these equals 1 foot (ft) 12 inches (in) 1 yard (yd) 3 feet, or 36 inches 1 mile (mi) 5,280 feet, or 1,760 yards TimeOne of these equals 1 minute (min) 60 seconds (s) 1 hour (h) 60 minutes 1 day 24 hours 1 week 7 days Next step: Multiply or divideWeightOne of these equals 1 pound (lb) 16 ounces (oz) 1 ton 2,000 pounds LiquidsOne of these equals 1 pint (pt) 2 cups 1 quart (qt) 2 pints 1 gallon (gal) 4 quarts ShortcutDoubles, then 4
Liquids go cup, pint, quart in doubles (2 and 2), then 4 quarts make a gallon. The name helps: a quart is a quarter of a gallon.
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To a smaller unit, multiply. To a bigger unit, divide.
to a smaller unit: × to a bigger unit: ÷
Before you calculate, ask one question: is the new unit bigger or smaller? A smaller unit fits more times, so the answer is a bigger number: multiply by the fact. A bigger unit fits fewer times, so the answer is a smaller number: divide by the fact.
- 6 yd
- 6 × 3 ft
- 18 ft
Each yard is replaced by 3 feet. Feet are smaller, so the number grows. Next step: Two stepsExample6 yards in feet
- A foot is smaller than a yard, so there will be more feet than 6. Multiply.
- Each yard is 3 feet: 6 × 3 = 18.
Answer 18 feet
Example80 ounces in pounds
- A pound is bigger than an ounce, so there will be fewer pounds than 80. Divide.
- Each pound is 16 ounces: 80 ÷ 16 = 5, because 5 × 16 = 80 (5 × 10 = 50 and 5 × 6 = 30).
Answer 5 pounds
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No single fact? Go through the units in between
No fact links hours and seconds directly in the table. So walk through the unit in the middle, one fact at a time. Rewrite the whole amount at each step, so you always know what unit you are holding.
- 2 hours
- 2 × 60 minutes
- 120 minutes
- 120 × 60 seconds
- 7,200 seconds
Hours to minutes, then minutes to seconds. Both steps go smaller, so both multiply. Next step: Mixed unitsExample5 gallons in cups
- Gallons to quarts: 5 × 4 = 20 quarts.
- Quarts to pints: 20 × 2 = 40 pints.
- Pints to cups: 40 × 2 = 80 cups.
Answer 80 cups
Shortcut1 gallon is 16 cups
4 quarts, each 2 pints, each 2 cups: 4 × 2 × 2 = 16. So 5 gallons is 5 × 16 = 80 cups in one step.
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Mixed units like 5 ft 9 in
A mixed unit uses two units at once: 5 ft 9 in means 5 feet plus 9 inches. To work with it, change all of it into the smaller unit first.
- 5 ft 9 in
- 5 × 12 in + 9 in
- 60 in + 9 in
- 69 in
The feet become inches; the 9 inches stay as they are. To go back, divide by the fact. The whole number of times it fits is the big unit, and what is left over, the remainder, is the small unit.
Next step: RatesExample100 inches in feet and inches
- 12 fits into 100 eight times: 8 × 12 = 96. Nine times would be 108, too much.
- Left over: 100 − 96 = 4 inches.
Answer 8 ft 4 in
Example3 h 45 min + 2 h 30 min
- Hours with hours: 3 + 2 = 5. Minutes with minutes: 45 + 30 = 75.
- 75 minutes is more than an hour. Take one hour out: 75 − 60 = 15.
- 5 h 75 min becomes 6 h 15 min.
Answer 6 h 15 min
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The conversion factor: a fraction worth 1
A conversion factor is a fact written as a fraction, like 1 hour over 60 minutes. The top and the bottom are the same amount of time, so the fraction is worth exactly 1, and multiplying by 1 never changes an amount. Put the unit you want to get rid of on the bottom: it cancels with the same unit on top.
- 90 min
- 90 min × 1 h / 60 min
- 90 ÷ 60 h
- 1.5 h
Minutes on top and minutes on the bottom cancel. Hours are left. For 90 ÷ 60: 60 fits once, with 30 minutes left, and 30 minutes is half an hour: 1.5 h. This matters most for rates, numbers with “per” in them: dollars per ounce, miles per hour. A rate has two units, and you may need to change one of them.
Next step: MetricExampleRice costs $0.15 per ounce. What is the price per pound?
- A pound is 16 ounces, so one pound costs 16 times as much as one ounce.
- Split 16 into 10 and 6: 10 × 0.15 = 1.50 and 6 × 0.15 = 0.90.
- 1.50 + 0.90 = 2.40.
Answer $2.40 per pound
Shortcut60 mph is 1 mile a minute
An hour is 60 minutes, so 60 miles per hour is exactly 1 mile per minute. Then 45 mph is 3/4 of a mile per minute, and 90 mph is 1.5 miles per minute.
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Metric units: kilo, centi and milli
The metric system is used in science, in medicine and in most countries. It has one base unit for each thing: the meter for length, the gram for weight and the liter for liquids. A prefix, a piece in front of the name, makes it bigger or smaller by a power of 10. So converting is just moving the decimal point.
Three prefixesPrefix Means Example kilo- 1,000 1 kilometer = 1,000 meters centi- 1/100 1 meter = 100 centimeters milli- 1/1,000 1 liter = 1,000 milliliters Next step: Three examplesKilometers to meters2.5 km
Smaller unit, so × 1,000: the point moves 3 places right. 2,500 m
Centimeters to meters340 cm
Bigger unit, so ÷ 100: the point moves 2 places left. 3.4 m
Milliliters to liters750 mL
Bigger unit, so ÷ 1,000: 3 places left. 0.75 L
ShortcutCount the zeros
Kilo means 1,000 and milli means 1/1,000. Both have three zeros, so move the point three places. Centi means 1/100: two zeros, two places.
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Three worked examples, from easy to test level
Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.
The test-level one needs three conversions in a row and then a division. Write the unit next to every number: if the units in a step do not match, you have found the mistake.
Example · EasyHow many ounces are in 6 pounds?
First try
6 ÷ 16 = 0.375 ounces- An ounce is smaller than a pound, so there must be more ounces than 6. The first try divided and got less than 1.
- Going to the smaller unit, multiply by 16.
- Split the 16 into 10 and 6: 6 × 10 = 60 and 6 × 6 = 36.
- 60 + 36 = 96.
Answer 96 ounces
Example · MediumA movie is 135 minutes long. How long is that in hours and minutes?
First try
135 ÷ 60 = 2.25, so 2 h 25 min- 2.25 hours is right, but the .25 is a part of an hour, not 25 minutes.
- 0.25 of an hour is a quarter of 60 minutes, which is 15 minutes.
- Or skip the decimal: 2 × 60 = 120, and 135 − 120 = 15 minutes left over.
Answer 2 h 15 min
Example · Test levelA field kitchen makes 6 gallons of soup. Each bowl holds 1 1/2 cups. How many bowls can it fill?
First try
6 × 4 = 24 quarts, so 24 ÷ 1.5 = 16 bowls- The first try divided quarts by cups. The units do not match, so convert all the way to cups.
- Gallons to quarts: 6 × 4 = 24 quarts. Quarts to pints: 24 × 2 = 48 pints. Pints to cups: 48 × 2 = 96 cups.
- Two bowls use 1 1/2 + 1 1/2 = 3 cups. 96 ÷ 3: 3 × 30 = 90 and 3 × 2 = 6, so 3 × 32 = 96: 32 pairs of bowls.
- 32 pairs is 32 × 2 = 64 bowls.
Answer 64 bowls
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Multiplying when you should divide
Changing 48 ounces to pounds:
Wrong48 × 16 = 768 pounds
Right48 ÷ 16 = 3 pounds
Reading decimal hours as minutes
Writing 3.4 hours in hours and minutes:
Wrong3.4 hours → 3 h 40 min
Right0.4 × 60 = 24 → 3 h 24 min
Writing feet and inches as a decimal
Writing 5 ft 6 in in feet:
Wrong5 ft 6 in → 5.6 feet
Right6 ÷ 12 = 0.5 → 5.5 feet
Stopping halfway
Changing 3 gallons to cups:
Wrong3 × 4 = 12 cups
Right3 × 4 × 2 × 2 = 48 cups
15 questions in three levels
Warm-up first, then core, then test level. Pick an answer, then read why.
Units show up inside Arithmetic Reasoning word problems: a recipe in cups, a board in feet and inches, a drive in hours and minutes. On the computer test that section has 15 questions in 55 minutes; on paper, 30 questions in 36 minutes. Mathematics Knowledge (15 questions in 31 minutes on the computer, 25 in 24 minutes on paper) can need units too.
No calculator. Before you multiply or divide, ask which unit is bigger. If the answer is in a smaller unit, it must be a bigger number. That one check can rule out choices before you calculate.
On the computer test you cannot go back to a question. Write the unit next to every number as you work, and change everything into the same unit before you add, subtract or divide.
How many inches are in 7 feet?
Show the answer
D. An inch is smaller than a foot, so the number must get bigger: multiply. Each foot is 12 inches: 7 × 12 = 84. 0.58 divides instead (7 ÷ 12), 19 adds 12, and 21 uses 3, the feet in a yard. 7 feet is 84 inches.
How many pounds are in 3 tons?
Show the answer
C. A ton is 2,000 pounds, and a pound is the smaller unit, so multiply: 3 × 2,000 = 6,000. 48 uses 16, the ounces in a pound; 2,003 adds instead of multiplying; and 96,000 keeps going past pounds into ounces (3 × 2,000 × 16). The question asks for pounds, so stop there. 3 tons is 6,000 pounds.
How many hours are 150 minutes?
Show the answer
D. An hour is bigger than a minute, so divide by 60. 2 × 60 = 120, and the 30 minutes left over are half an hour: 2.5 hours. 0.4 divides the wrong way round (60 ÷ 150), 1.5 treats an hour as 100 minutes, and 2.3 writes 2 h 30 min as if it were a decimal. 150 minutes is 2.5 hours.
How many quarts are in 9 gallons?
Show the answer
B. A quart is smaller than a gallon, so multiply. Each gallon is 4 quarts: 9 × 4 = 36. 13 adds 4 instead of multiplying, 72 keeps going to pints (9 × 4 × 2), and 144 keeps going to cups (9 × 16). The question asks for quarts. 9 gallons is 36 quarts.
How many centimeters are in 4.2 meters?
Show the answer
B. Centi means one hundredth, so 1 meter is 100 centimeters. Going to the smaller unit, multiply by 100: move the point two places right, and 4.2 becomes 420. 42 moves the point only one place, 4,200 uses 1,000 instead of 100 (three places instead of two), and 42,000 moves the point four places. 4.2 meters is 420 centimeters.
A shelf is 89 inches tall. How tall is it in feet and inches?
Show the answer
B. Divide by 12. 7 × 12 = 84, and 8 × 12 = 96 is too much, so it is 7 whole feet. What is left over: 89 − 84 = 5 inches. 7 ft 4 in reads 89 ÷ 12 ≈ 7.4 as 7 feet 4 inches, but 0.4 of a foot is not 4 inches. 8 ft 9 in just splits the digits, and 29 ft 2 in divides by 3, the feet in a yard. The shelf is 7 ft 5 in tall.
A soup recipe needs 3 quarts of broth. How many cups is that?
Show the answer
D. Go through pints. 1 quart is 2 pints, so 3 quarts = 3 × 2 = 6 pints. 1 pint is 2 cups, so 6 pints = 6 × 2 = 12 cups. 6 stops at pints, 1.5 divides by 2 instead of multiplying (3 ÷ 2), and 0.75 divides by 4, the quarts in a gallon. 3 quarts is 12 cups.
A running track is 1/4 mile around. How many yards is one lap?
Show the answer
A. A mile is 1,760 yards, and you need a quarter of it: 1,760 ÷ 4. Half of 1,760 is 880, and half of 880 is 440. 1,320 is a quarter of 5,280, which gives feet, not yards; 1,760 is the whole mile; and 7,040 multiplies by 4. One lap is 440 yards.
A training course takes 4 h 10 min. Lee has done 1 h 35 min of it. How much time is left?
Show the answer
A. You cannot take 35 minutes from 10, so borrow 1 hour: 4 h 10 min becomes 3 h 70 min, because 60 + 10 = 70. Now minutes: 70 − 35 = 35. Hours: 3 − 1 = 2. 3 h 25 min takes 10 from 35 instead, 2 h 75 min borrows 100 minutes instead of 60, and 3 h 35 min borrows the minutes but forgets to take the hour away. Time left: 2 h 35 min.
A roll of tape is 12 meters long. Each strip you cut is 40 centimeters. How many strips can you cut?
Show the answer
C. The units must match first. 12 meters is 12 × 100 = 1,200 centimeters. Then 1,200 ÷ 40 is the same as 120 ÷ 4 = 30. 0.3 divides 12 by 40 without converting, 3 uses 10 centimeters in a meter, and 300 uses 1,000 centimeters in a meter instead of 100. You can cut 30 strips.
A 2-pound bag of coffee costs $11.20. What is the price per ounce?
Show the answer
A. Price per ounce is dollars ÷ ounces, so first change the pounds to ounces: 2 × 16 = 32 ounces. Then halve both numbers until it is easy: 11.20 ÷ 32 is 5.60 ÷ 16, which is 2.80 ÷ 8. Think in cents: 280 ÷ 8. 8 × 30 = 240 and 8 × 5 = 40, so 8 × 35 = 280, and the answer is 35 cents, $0.35. $5.60 is the price per pound, $0.70 divides by 16 as if the bag were 1 pound, and $0.56 uses 10 ounces in a pound. The price per ounce is $0.35.
A water trailer holds 400 gallons. How many 2-quart canteens can it fill?
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C. Change gallons to quarts first: 400 × 4 = 1,600 quarts. Each canteen takes 2 quarts: 1,600 ÷ 2 = 800. 1,600 forgets to divide by 2, 200 divides 400 by 2 without converting, and 100 divides by 4 instead of multiplying. It fills 800 canteens.
A truck drives at 30 miles per hour. How many feet does it travel in one second?
Show the answer
B. Change one unit at a time. 30 mph is half of 60 mph, and 60 mph is 1 mile a minute, so the truck goes half a mile each minute. Half of 5,280: half of 5,000 is 2,500 and half of 280 is 140, so 2,640 feet per minute. A minute is 60 seconds: 2,640 ÷ 60 is the same as 264 ÷ 6. 6 × 40 = 240 and 6 × 4 = 24, and 240 + 24 = 264, so 44 feet. 158,400 is feet per hour (30 × 5,280), 2,640 is feet per minute, and 0.5 is miles per minute. Each second it travels 44 feet.
Each of 6 boxes weighs 4 lb 10 oz. What is their total weight?
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C. Multiply each part: 6 × 4 = 24 pounds and 6 × 10 = 60 ounces. 60 ounces is more than a pound, so take out whole pounds: 3 × 16 = 48, and 60 − 48 = 12. That is 3 more pounds and 12 ounces, and 24 + 3 = 27. 24 lb 10 oz multiplies only the pounds, 24 lb 6 oz writes 4 lb 10 oz as 4.10 and multiplies it like a decimal, and 30 lb treats 60 ounces as 6 pounds. The total is 27 lb 12 oz.
A jug holds 2 liters of sports drink. Each cup holds 125 milliliters. After 9 cups are poured, how many milliliters are left in the jug?
Show the answer
B. Change liters to milliliters first: 2 × 1,000 = 2,000 mL. Nine cups: 8 cups are 1,000 mL, plus one more 125, so 9 × 125 = 1,125 mL. Left: 2,000 − 1,125 = 875 mL. 1,125 is what was poured, not what is left; 1,875 takes away only one cup; and 0.875 is the answer in liters, but the question asks for milliliters. The jug has 875 mL left.
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