Before you begin
A unit is the name of the piece you are counting in. Twelve inches and one foot are one length wearing two names, the way 3/6 and 1/2 in Quiz 8 were one amount wearing two. Converting a measurement changes the name and the count together, and never the thing being measured. That is the whole topic, and every rule below follows from it.
Quiz 24 already solved every conversion on this page. One foot is 12 inches, so 4 feet is how many inches? Feet on top, inches underneath, both sides: 1/12 = 4/x, and x = 48. The proportion works every time, but it is slow, and nearly every conversion is one multiplication or one division. The new work is deciding which of the two.
Which way does the number move?
One yard, three feet, thirty-six inches: the same bar three times, cut finer each time. The finer the pieces, the more of them it takes to make up the length. That is the whole rule, and it is worth saying in both directions.
To a bigger unit, the count goes down: divide. Inches to feet, ounces to pounds, minutes to hours. A film that runs 180 minutes runs 180 ÷ 60 = 3 hours.
Before you touch the arithmetic, say which way the number should move. If the answer comes out on the wrong side, you multiplied where you should have divided, and no care with the digits will fix it. Quiz 23 ended on an ant: inches per second became inches per minute by multiplying and feet per minute by dividing, and each step was decided by which unit was getting bigger. This quiz is built on that observation.
A fraction that equals one
There is a second way to see the same move, and it does the deciding for you. Sixty minutes and one hour are the same stretch of time, so the fraction 60 minutes1 hour equals 1, and so does 1 hour60 minutes. Quiz 7 said that multiplying by 1 changes nothing, and it is still true: multiplying a measurement by one of these fractions changes its name and leaves its size alone.
Write the units in and watch them cancel, the way factors canceled in Quiz 11:
165 minutes × 1 hour60 minutes = 2.75 hours. Minutes cancel this time, and 165 ÷ 60 is what remains. The number shrank, because hours are the bigger unit.
If the units do not cancel, the fraction is upside down and the number is about to move the wrong way. That is the direction test written into the arithmetic itself, and it is why this way of setting out a conversion is the one to use whenever a problem takes more than one step.
The chain
Some conversions take more than one step, because no single fact joins the two units. Gallons to cups goes through quarts and pints. Write each fact as a fraction equal to 1 and chain them, and the units cancel down the line:
Any chain works the same way, and the only thing that can go wrong is stopping early. Three gallons is 12 quarts, and 24 pints, and 48 cups, and a question that asked for cups is not answered by either of the first two.
The conversions on this page
| Length | 1 foot = 12 inches · 1 yard = 3 feet · 1 mile = 5,280 feet · 1 kilometer = 1,000 meters · 1 kilometer ≈ 0.621371 miles · 1 mile ≈ 1.609 kilometers · 1 meter ≈ 3.28 feet |
|---|---|
| Weight | 1 pound = 16 ounces · 1 gram = 1,000 milligrams |
| Volume | 1 gallon = 4 quarts · 1 quart = 2 pints · 1 pint = 2 cups |
| Time | 1 hour = 60 minutes · 1 minute = 60 seconds |
| Area | 1 hectare = a square 100 meters on each side · 100 hectares = 1 square kilometer |
The GED formula sheet does not list these. The everyday ones, inches and feet and ounces and pounds, you are expected to know. A conversion between the two systems, such as kilometers to miles, is normally given in the question that needs it, and the sign ≈ on the table means about: the factor has been rounded.
The trap: a square unit
Now the one that catches careful people. A square kilometer is not a kilometer. It is a square one kilometer on each side, and it is written km², read square kilometers; a square mile is written mi² the same way. The small 2 says the unit has been multiplied by itself: a length across times a length down, which is the rectangle from Quiz 4. To convert it, convert the side, and then remember that a square has two of them.
A kilometer is 0.621371 miles. So a square one kilometer on each side is a square 0.621371 miles on each side, and its area is 0.621371 × 0.621371 = 0.386 square miles. One square kilometer is a bit more than a third of a square mile, not five-eighths of one. The factor is used twice because the square has two sides, exactly as the squares figure in Quiz 25 showed with a 2: double the side and the area goes up four times. Here the side is multiplied by a decimal instead, and the area is multiplied by that decimal twice.
Here it is on a real number. The Okapi Wildlife Reserve, in the Ituri forest of the Democratic Republic of the Congo, is listed by UNESCO at 13,726 square kilometers. Convert with the factor once and you get 8,528.9 square miles. Convert with it twice and you get 5,299.6. The first answer is too big by 8,528.9 ÷ 5,299.6 = 1.609, which is the number of kilometers in a mile: the very factor that was left out. The error names itself, and the estimate guard catches it too, because a square mile is bigger than a square kilometer, so the count in square miles has to come out well under half of 13,726.
Three to study before you start
A job took 165 minutes. Hours are the bigger unit, so the count comes down: 165 ÷ 60 = 2.75 hours. As a fraction equal to 1: 165 minutes × 1 hour60 minutes, the minutes cancel, and 165 ÷ 60 = 2.75 is what remains. As a proportion, the slow way: 60/1 = 165/x, so 60x = 165 and x = 2.75. Three roads, one number, and the first question on all three was which way the number should move.
A curtain rod is 2 yards long. How many inches? No single fact joins yards to inches, so go through feet: 2 yards × 3 feet1 yard × 12 inches1 foot. Yards cancel, feet cancel, and 2 × 3 × 12 = 72 inches is left. Guard: an inch is a much smaller unit than a yard, so the count should climb a long way, and it did.
The Okapi Wildlife Reserve is listed at 13,726 square kilometers. A kilometer is 0.621371 miles, but that is the factor for a side. For an area it is 0.621371 × 0.621371 = 0.386102, and 13,726 × 0.386102 = 5,299.6 square miles. Using the factor once would give 8,528.9, which is question 8's error in advance. Guard: a square mile is bigger than a square kilometer, so the count must fall to well under half. It fell to 39 percent of 13,726, and 0.386 is exactly what it should be.
Now you
Work without a calculator, on paper. Questions 3 and 8 are multiple choice: choose the one best answer. Before every conversion, say which way the number should move.
- Convert 4 feet to inches.
- Convert 180 minutes to hours.
- Marisol measures a shelf at 54 inches and needs the length in feet. Before doing any arithmetic, what must be true of the answer?
- A) It will be a larger number, because feet are bigger than inches.
- B) It will be a smaller number: a foot is bigger than an inch, so 54 inches make fewer feet.
- C) It will be the same number with a different word after it.
- D) There is no way to know until the arithmetic is done.
- Rosa's time card shows 7 hours 45 minutes. Payroll needs it as a decimal number of hours. Write it.
- A prescription is written as 0.5 grams. The tablets are labeled in milligrams, and a gram is 1,000 milligrams. How many milligrams is the prescription?
- A cooler holds 3 gallons of water. The paper cups beside it hold 1 cup each. How many cups can be filled?
ReadingA reading rest stop, for the stubborn ones.
The reserve in question 8 has a page of its own on UNESCO's World Heritage list. Start at the paragraph headed Outstanding Universal Value, then go further down to the part headed Protection and management requirements, where the reserve is divided into zones measured in hectares. Every figure there is one you can now convert.
The photograph above is not from that page. It is Plate 44 of Lankester's 1910 atlas: the specimen Sir Harry Johnston presented to the British Museum in 1901.
The page also says that about 30,000 okapi survive in the wild. That figure is older than the page. The IUCN, which keeps the international list of threatened species, lists the okapi as Endangered and says that its numbers cannot be reliably counted.
- UNESCO lists the Okapi Wildlife Reserve at 1,372,625 hectares. A hectare is a square 100 meters on each side, and 100 hectares make one square kilometer. Write the reserve's area in square kilometers.
- Brian converted the reserve's area, 13,726 square kilometers, to square miles. He multiplied by 0.621371, the number of miles in a kilometer, and wrote 8,528.9 square miles. What went wrong?
- A) Nothing; 8,528.9 square miles is correct.
- B) He should have divided by 0.621371 instead of multiplying.
- C) A square kilometer has two sides to convert, so the factor is needed twice: 13,726 × 0.621371 × 0.621371, which is about 5,300 square miles.
- D) He should have multiplied by 1.609, the number of kilometers in a mile.
- The New York City Marathon is 26.2 miles. A mile is about 1.609 kilometers. How long is the marathon in kilometers, to the nearest tenth?
- An okapi stands about 1.5 meters at the shoulder. A meter is about 3.28 feet. How tall is that in feet, to the nearest tenth?
Check your work
Open the key: after you've finished all ten
Reading your results
| Questions | The skill they test | If they gave trouble |
|---|---|---|
| 1, 2, 5 | One-step conversions | Which way does the number move? Then one multiplication or one division. Say the direction out loud first. |
| 3 | Knowing the direction before the arithmetic | A bigger unit means fewer of them. The yard bar in the Guide shows it without a single digit. |
| 4, 9, 10 | Decimal factors and rounding | Set the fraction equal to 1 so the units cancel, multiply with care, and round only at the end. |
| 6, 7 | Chains, and powers of ten | Write every fact as a fraction and cancel down the line. Dividing by 100 slides the point two places. |
| 8 | Square units | The signature error of the topic. A square has two sides, so the factor is used twice. |
Eight or more right: measurement is in hand, and Quiz 27 turns the square you just converted into the subject. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. One habit for a week: before any conversion, say out loud whether the answer should be a bigger number or a smaller one, and why. That single sentence catches most of the errors on this page before they happen. Then come back to these same ten.
Next in the Okapi Series: Quiz 27 — Perimeter and Area. The square kilometer you converted here becomes the subject: what area is, why it grows faster than length, and why that is the reason a giraffe cannot simply be a large okapi.
Known before it was named
The okapi lives in the Ituri forest of what is now the Democratic Republic of the Congo, and the peoples of that forest, the Mbuti and the Efe among them, had hunted it and named it for as long as anyone can say. In 1890 the explorer Henry Morton Stanley heard about it from Mbuti hunters and wrote down a word for it. The word he wrote down was most likely their name for a different animal.
Ten years later Sir Harry Johnston, the British commissioner for Uganda, met a group of Mbuti who had been taken from the forest by a showman to be exhibited at the Paris fair of 1900. He sent them home, and on that journey, at a Belgian post in the Congo Free State, the colony King Leopold of Belgium then held as his personal property, he was given two belts cut from the animal's striped hide. He posted them to London. From two strips of skin, a zoologist there named the animal a horse, with a question mark: Equus (?) johnstoni. Later that year a whole skin and two skulls arrived, obtained by Johnston from a lieutenant in the Congo Free State's administration, and Ray Lankester saw that it was no horse at all but the giraffe's only living relative. The horse was dropped. Johnston's name stayed.
Johnston had gone looking for the animal himself, with Mbuti guides. They showed him its tracks: cloven, two-toed, the print of a giraffe's cousin. He would not follow them, because he had decided he was hunting a horse, and a horse leaves one round hoof. The evidence was on the ground in front of him and he read it through a habit. If that sounds like Denise in Quiz 24, adding where the relationship multiplied, it should.
The name is worth counting. The word okapi was taken down through an interpreter from a neighboring language of the forest; it was not the Mbuti's word. By 1907 the published record already held at least six names for the animal, from six languages of the region: M'Boota, Kenghe, M'Dumba, O'api, Dumla, Boole. Europe chose one of them, spelled it its own way, and bolted a British surname onto the back. The fifteen skulls Lankester measured for this atlas sat in London, Kent, Tring, Edinburgh, Paris, Rome, Madrid, Genoa and Tervuren, which is the Congo's museum and is in Belgium. Not one was in Africa.
None of that changes the arithmetic of this quiz. It is here because it is the same lesson the arithmetic teaches: the thing was known, measured and named before the people who now hold the name arrived, and the record, read plainly, says so in numbers you can count.
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