Math 6 · 5.MD.A.1 · 6.RP.A.3D

Conversion factors and dimensional analysis

A bag of rice is labeled in pounds and the recipe asks for kilograms. How do you change the unit without changing the amount, and know which way to go?

You need

Multiplying decimalsGrade 5 Rate and unit rateGrade 6 Customary conversionsGrade 4

Unlocks

Converting square and cubic unitsMath 7 Estimating a huge quantity from small factsAlgebra 1 Units, quantities and a sensible level of accuracyAlgebra 1 The derivative as a rate of changeCalculus 1 +1 more

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

Why this exists

Every measurement is a number with a unit, and the same amount has many names. A relay leg of 100 meters is about 109 yards. A 50 pound sack of rice is 800 ounces, or about 23 kilograms. The amount stays put; only its name changes.

Single steps you already know. Customary conversions say a foot is 12 inches, and metric conversions say a meter is 100 centimeters. For one step, common sense picks between multiplying and dividing: inches are small, so there are more of them. But a chain of steps, such as yards to inches to centimeters to meters, or a unit made of two units, such as square feet or dollars per pound, leaves too many choices to guess. This lesson writes each fact as a fraction equal to 1, and then the units themselves show which way the fraction goes.

Definition

A conversion factor is a fraction whose top and bottom are the same amount written in two different units. Because the top and the bottom are equal, the fraction is equal to 1.

Each fact gives two conversion factors, one the other turned upside down. From 1\text{ ft} = 12\text{ in}:

\frac{12\text{ in}}{1\text{ ft}} = 1 \qquad\qquad \frac{1\text{ ft}}{12\text{ in}} = 1

Dimensional analysis means carrying the units through a calculation along with the numbers: multiplying by conversion factors, and canceling a unit that sits both on top and on the bottom, just as a number on the top and the bottom of a fraction cancels.

A compound unit is a unit made from two others. A square foot, \text{ft}^2, is a foot times a foot. A price in dollars per pound is dollars divided by pounds.

Intuition

Figure 1One length with two names. A fraction with the same length on top and bottom is equal to 1.

In Figure 1 the two bars are the same length. One is called 1 foot and the other 12 inches, so the fraction \displaystyle \frac{12\text{ in}}{1\text{ ft}} has the same length on top and on the bottom. Any amount divided by itself is 1, just as \displaystyle \frac{5}{5} = 1.

Multiplying by 1 never changes anything. So a length of 3 feet can be multiplied by this particular 1:

3\,\cancel{\text{ft}} \times \frac{12\text{ in}}{1\,\cancel{\text{ft}}} = 36\text{ in}

The feet on top and the feet on the bottom cancel, and inches are left. The number went up from 3 to 36 while the unit went down from feet to inches, by the same factor of 12. A smaller unit needs more of itself to cover the same length.

Which way up? The unit you have must cancel, so it goes on the bottom of the factor, and the unit you want goes on top. Turned the other way, the units refuse to cancel:

3\text{ ft} \times \frac{1\text{ ft}}{12\text{ in}} = 0.25\,\frac{\text{ft}\cdot\text{ft}}{\text{in}}

That is true, but “feet times feet per inch” is not a length anyone wants. The leftover units show the factor was upside down before any number is checked.

Results

A conversion factor is equal to 1. Multiplying by it changes the number and the unit together, never the amount.
Units cancel like factors. The same unit on the top and on the bottom divides out.
The unit you have goes on the bottom; the unit you want goes on top.
Factors chain. Going through a middle unit uses one factor per step. Each is 1, and 1 \times 1 = 1, so the chain is still a name for 1. For example, \displaystyle \frac{3\text{ ft}}{1\text{ yd}} \times \frac{12\text{ in}}{1\text{ ft}} = \frac{36\text{ in}}{1\text{ yd}}.
A bigger unit takes a smaller number, and a smaller unit a bigger number.
A square unit needs the length factor twice, once for each length it is made of.

The customary facts used here are 1\text{ ft} = 12\text{ in}, 1\text{ yd} = 3\text{ ft}, 1\text{ mi} = 5{,}280\text{ ft}, 1\text{ lb} = 16\text{ oz} and 1\text{ ton} = 2{,}000\text{ lb}, with 1\text{ min} = 60\text{ s} and 1\text{ h} = 60\text{ min} for time. These bridge the two systems:

Fact	Exact or about
1\text{ in} = 2.54\text{ cm}	exact: the inch is defined this way
1\text{ yd} = 0.9144\text{ m}	exact, since 36 \times 2.54 = 91.44
1\text{ mi} \approx 1.609\text{ km}	about
1\text{ kg} \approx 2.2\text{ lb}	about

Why it is true

Two facts carry the whole method.

The factor is 1. A fraction means its top divided by its bottom. In \displaystyle \frac{12\text{ in}}{1\text{ ft}} the top and the bottom are one and the same length, and any amount divided by itself is 1. Multiplying any number by 1 leaves it unchanged, which is the identity property of multiplication. So the amount after converting is the amount before.

A unit behaves like a factor. 3\text{ ft} means 3 times one foot. Write the product with that in mind:

\begin{aligned} &3\text{ ft} \times \frac{12\text{ in}}{1\text{ ft}} \\ &\quad = \frac{3 \times 12}{1} \times \frac{\text{one foot} \times \text{one inch}}{\text{one foot}} \end{aligned}

Multiplication can be regrouped and reordered (the associative and commutative properties), so the numbers and the units can be gathered separately like this. One foot divided by one foot is 1, for the same reason 5 \div 5 = 1, and what remains is 36 times one inch.

A chain of factors is a product of several 1s, which is 1, so a chain never changes the amount either. The one place this needs care is a factor built from a rounded fact. One kilogram is closer to 2.2046 pounds than to 2.2, so \displaystyle \frac{2.2\text{ lb}}{1\text{ kg}} is only about 1, and an answer found with it is only about right.

Worked examples

The lesson is taught in two parts. Examples 1 to 3 belong to the first: choosing which way up a factor goes, and converting through a middle unit inside one system. Examples 4 to 7 belong to the second: converting between metric and customary units, and a first look at compound units.

Example 1

Pounds to ounces and back

Hector buys a 5\text{ lb} bag of rice. (a) How many ounces is that? (b) He scoops 40\text{ oz} of it into a jar. How many pounds are in the jar?

Solution

The fact 1\text{ lb} = 16\text{ oz} gives two factors, \displaystyle \frac{16\text{ oz}}{1\text{ lb}} and \displaystyle \frac{1\text{ lb}}{16\text{ oz}}.

(a) Pounds must disappear, so they go on the bottom:

5\,\cancel{\text{lb}} \times \frac{16\text{ oz}}{1\,\cancel{\text{lb}}} = 80\text{ oz}

(b) Now ounces must disappear, so use the other factor, with ounces on the bottom:

40\,\cancel{\text{oz}} \times \frac{1\text{ lb}}{16\,\cancel{\text{oz}}} = \frac{40}{16}\text{ lb} = 2.5\text{ lb}

The same fact served both questions; the unit that had to cancel chose the factor.

Check. An ounce is smaller than a pound, so a weight is more ounces than pounds: 80 is more than 5, and 40 is more than 2.5.

Example 2

A relay leg through a middle unit

Zuri marks out a relay leg of 110 yards, but her tape measure reads in inches. How many inches long is the leg?

Solution

Use the two facts 1\text{ yd} = 3\text{ ft} and 1\text{ ft} = 12\text{ in}, with feet as the middle unit. The first factor cancels yards and brings in feet; the second cancels feet and brings in inches:

\begin{aligned} 110\text{ yd} &= 110\,\cancel{\text{yd}} \times \frac{3\,\cancel{\text{ft}}}{1\,\cancel{\text{yd}}} \times \frac{12\text{ in}}{1\,\cancel{\text{ft}}} \\ &= 110 \times 3 \times 12\text{ in} \\ &= 3{,}960\text{ in} \end{aligned}

The two factors together make one factor: \displaystyle \frac{3\text{ ft}}{1\text{ yd}} \times \frac{12\text{ in}}{1\text{ ft}} = \frac{36\text{ in}}{1\text{ yd}}, which is the fact 1\text{ yd} = 36\text{ in}.

Check. Using that single factor, 110 \times 36 = 3{,}960. The leg is 3{,}960 inches.

Example 3

Seconds to hours for a bird in flight

Dorian follows a tagged migrating bird. The tag shows it flew for 9{,}000 seconds without landing. How many hours is that?

Solution

Hours are far bigger than seconds, so go up through minutes, with 1\text{ min} = 60\text{ s} and 1\text{ h} = 60\text{ min}. Seconds must cancel, so seconds go on the bottom of the first factor; then minutes must cancel, so minutes go on the bottom of the second:

\begin{aligned} 9{,}000\text{ s} &= 9{,}000\,\cancel{\text{s}} \times \frac{1\,\cancel{\text{min}}}{60\,\cancel{\text{s}}} \times \frac{1\text{ h}}{60\,\cancel{\text{min}}} \\ &= \frac{9{,}000}{60 \times 60}\text{ h} = \frac{9{,}000}{3{,}600}\text{ h} \\ &= 2.5\text{ h} \end{aligned}

Here each factor has the bigger unit on top with a 1, so multiplying by it works as dividing by 60. The bird flew for 2.5 hours, which is 2 hours 30 minutes.

Check. An hour is 3{,}600 seconds, so 9{,}000 seconds should be between 2 hours (7{,}200 s) and 3 hours (10{,}800 s). It is.

Example 4

Is a 440-yard track longer than 400 meters?

The track at Alba’s old school is 440 yards around. The new one is 400 meters around. Which is longer, and by how much?

Solution

The exact bridge between the systems is 1\text{ in} = 2.54\text{ cm}, so the path runs from yards to inches, across to centimeters, and up to meters. One factor at a time, each canceling the unit just reached:

440\,\cancel{\text{yd}} \times \frac{36\text{ in}}{1\,\cancel{\text{yd}}} = 15{,}840\text{ in}

15{,}840\,\cancel{\text{in}} \times \frac{2.54\text{ cm}}{1\,\cancel{\text{in}}} = 40{,}233.6\text{ cm}

40{,}233.6\,\cancel{\text{cm}} \times \frac{1\text{ m}}{100\,\cancel{\text{cm}}} = 402.336\text{ m}

Every fact used is exact, so this answer is exact: 440 yards is 402.336 meters. The old track is longer, by 402.336 - 400 = 2.336 meters, a little over 2 meters.

The three factors multiply to \displaystyle \frac{36 \times 2.54}{100} = 0.9144 meters per yard, the fact in the table. As a check, 440 \times 0.9144 = 402.336.

Figure 2A meter is a little longer than a yard, so the same distance takes more yards than meters.

Example 5

Miles and kilometers on a migration

Dorian’s bird flew 150 miles on Monday and 400 kilometers on Tuesday. On which day did it fly farther? Use 1\text{ mi} \approx 1.609\text{ km}.

Solution

Two distances can be compared only in one unit. Change Monday’s into kilometers; miles must cancel, so miles go on the bottom:

150\,\cancel{\text{mi}} \times \frac{1.609\text{ km}}{1\,\cancel{\text{mi}}} \approx 241.35\text{ km}

Tuesday’s 400 km is more than 241.35 km, so the bird flew farther on Tuesday, by about 400 - 241.35 = 158.65, or roughly 159 kilometers.

Changing Tuesday’s distance into miles instead gives the same verdict. Kilometers must cancel:

\begin{aligned} 400\,\cancel{\text{km}} \times \frac{1\text{ mi}}{1.609\,\cancel{\text{km}}} &= \frac{400}{1.609}\text{ mi} \\ &\approx 248.6\text{ mi} \end{aligned}

and 248.6 miles is more than 150 miles.

Check. A mile is a bit more than 1.6 km, and 400 \div 1.6 = 250, close to 248.6. Each answer says “about” because 1.609 is itself rounded.

Example 6

Square feet to square inches

Vida paints a sign for the finish line of the relay. Its area is 8\text{ ft}^2. How many square inches is that?

Solution

A square foot is a foot times a foot, so 8\text{ ft}^2 carries two feet. One factor of \displaystyle \frac{12\text{ in}}{1\text{ ft}} cancels one of them; it takes two factors to cancel both:

\begin{aligned} 8\text{ ft}^2 &= 8\,\cancel{\text{ft}} \cdot \cancel{\text{ft}} \times \frac{12\text{ in}}{1\,\cancel{\text{ft}}} \times \frac{12\text{ in}}{1\,\cancel{\text{ft}}} \\ &= 8 \times 144\text{ in}^2 \\ &= 1{,}152\text{ in}^2 \end{aligned}

Using the factor only once would leave 96\text{ ft}\cdot\text{in}, an area measured half in feet and half in inches: true, but not what was asked.

Check. A square 1 foot on each side holds 12 rows of 12 one-inch squares, which is 144 square inches, and 8 \times 144 = 1{,}152. More on area and volume units is in Converting square and cubic units.

Example 7

Rice priced per pound and per kilogram

Florin’s shop sells rice for 1.50 dollars per pound. A shop across the border sells it for 3.50 dollars per kilogram. Which is cheaper? Use 1\text{ kg} \approx 2.2\text{ lb}.

Solution

A price per pound is a compound unit, dollars divided by pounds: \displaystyle \frac{1.50\text{ dollars}}{1\text{ lb}}. To compare, write it per kilogram. Pounds sit on the bottom of the price, so they cancel only against a factor with pounds on top:

\frac{1.50\text{ dollars}}{1\,\cancel{\text{lb}}} \times \frac{2.2\,\cancel{\text{lb}}}{1\text{ kg}} \approx \frac{3.30\text{ dollars}}{1\text{ kg}}

Florin’s rice costs about 3.30 dollars per kilogram, so it is cheaper than 3.50 dollars per kilogram, by about 0.20 dollars on every kilogram.

Check. A kilogram is more than 2 pounds, so a kilogram of rice should cost more than twice the price of a pound, and 3.30 is. With the closer fact 1\text{ kg} \approx 2.2046\text{ lb} the price is about 3.31 dollars per kilogram, and the verdict is the same. A price per pound is a Rates and unit rates, and changing the units of a rate in general is its own skill.

Boundaries

The top and the bottom must be the same amount. \displaystyle \frac{12\text{ in}}{1\text{ ft}} is 1; \displaystyle \frac{12\text{ ft}}{1\text{ in}} is not, because 12 feet is far longer than 1 inch.

Both units must measure the same kind of thing. Length converts to length and weight to weight. No conversion factor turns feet into pounds, or square feet into feet. A fraction such as 8 ounces per bag can still be multiplied, and its units still cancel, but it is a rate, not a name for 1: it changes what is being counted, from ounces to bags.

Units cancel only when they are the same unit. Feet cancel feet. Feet do not cancel inches, and one foot on the bottom of a factor cancels only one of the two feet in a square foot, which is why Example 6 needs two factors.

A rounded fact gives a rounded answer. A factor built from 1\text{ kg} \approx 2.2\text{ lb} or 1\text{ mi} \approx 1.609\text{ km} is only about 1, so the answer is “about”, and digits far beyond the precision of the fact mean nothing. Exact facts such as 1\text{ in} = 2.54\text{ cm} give exact answers.

Some scales are not a single multiplication. Temperature is the usual case: 0 degrees Celsius is 32 degrees Fahrenheit, not 0, because the two scales start from different zeros. Changing between them needs an addition as well, so no conversion factor alone can do it. That change has its own lesson, Converting between Celsius and Fahrenheit.

Where this goes next

A conversion factor is the tool behind every change of unit from here on. A speed in feet per second has two units to change at once, and changing the units of a rate uses one factor for the distance and one for the time. Areas and volumes use the length factor two or three times, worked out in full in Converting square and cubic units.

Dimensional analysis also checks work. Reasoning with units reads the units of a set-up to see whether it can possibly give the quantity asked for. Money follows the same pattern with an exchange rate in place of a fixed fact, in Currency exchange, with the twist that the rate changes from day to day. And estimating a huge quantity strings rough factors together exactly as Example 3 strings exact ones.

Exercises

Exercises 1 to 11 go with the first part of the lesson (Examples 1 to 3), Exercises 12 to 21 with the second (Examples 4 to 7), and the reasoning group draws on both.

Part 1: names for one

Which of these fractions are equal to 1? (a) \displaystyle \frac{16\text{ oz}}{1\text{ lb}} (b) \displaystyle \frac{1\text{ yd}}{3\text{ ft}} (c) \displaystyle \frac{12\text{ ft}}{1\text{ in}} (d) \displaystyle \frac{1{,}000\text{ g}}{1\text{ kg}}
Write both conversion factors that come from 1\text{ h} = 60\text{ min}. Which one changes 3 hours into minutes, and what is the answer?
Convert 7\text{ lb} of rice to ounces.
Convert 54\text{ in} to feet.
Convert \displaystyle 3\frac{1}{2}\text{ yd} to feet.
Convert 2{,}500\text{ g} of rice to kilograms.

Part 1: through a middle unit

Convert 2\text{ mi} to inches, going through feet.
A migrating bird glides for 14{,}400 seconds. How many hours is that?
A shipment of rice weighs 2 tons. How many ounces is that?
Four runners each run 440 yards of a relay. Show, going through feet, that together they run exactly 1 mile.
Florin splits a 50\text{ lb} sack of rice into bags of 8\text{ oz} each. How many bags does he fill?

Part 2: between the two systems

A ruler is 12 inches long. How long is it in centimeters?
Convert 100 yards to meters, using 1\text{ in} = 2.54\text{ cm}.
A 25\text{ kg} sack of rice is about how many pounds? Use 1\text{ kg} \approx 2.2\text{ lb}.
A migrating bird flies 60 miles. About how many kilometers is that, to the nearest tenth? Use 1\text{ mi} \approx 1.609\text{ km}.
A 10\text{ lb} bag of rice is about how many kilograms, to the nearest tenth? Use 1\text{ kg} \approx 2.2\text{ lb}.
Convert 254\text{ cm} to inches, and then to feet.
Alba’s relay team runs four legs of 100 meters. How far is 400 meters in yards, to the nearest yard? Use 1\text{ in} = 2.54\text{ cm}.

Part 2: compound units

A relay banner has an area of 2.5\text{ ft}^2. How many square inches is that?
A tray for drying rice has an area of 1{,}008\text{ in}^2. How many square feet is that?
Rice costs 2.75 dollars per kilogram. About what is that per pound? Use 1\text{ kg} \approx 2.2\text{ lb}.

Reasoning

Multiplying 3\text{ ft} by \displaystyle \frac{12\text{ in}}{1\text{ ft}} turns the number 3 into 36. Explain why the length has not changed.
Someone multiplies 6\text{ ft} by \displaystyle \frac{1\text{ ft}}{12\text{ in}}. What number and what units come out, and how do the units show the factor was the wrong way up? What should it have been?
Show that converting 1 yard to meters through inches and centimeters gives exactly 0.9144 meters, and explain why the answer is exact.
A meter is a little longer than a yard (Figure 2). Without working out the exact number, is a 1{,}500 meter race more or fewer than 1{,}500 yards? Explain.

Answers

(a), (b) and (d). In each, the top and the bottom are the same amount: 16 oz is 1 lb, 1 yd is 3 ft, and 1{,}000 g is 1 kg. (c) is not: 12 feet is 144 inches, so the fraction is 144, not 1.
\displaystyle \frac{60\text{ min}}{1\text{ h}} and \displaystyle \frac{1\text{ h}}{60\text{ min}}. Hours must cancel, so use the first: \displaystyle 3\text{ h} \times \frac{60\text{ min}}{1\text{ h}} = 180\text{ min}.
112\text{ oz}. \displaystyle 7\text{ lb} \times \frac{16\text{ oz}}{1\text{ lb}} = 112\text{ oz}.
4.5\text{ ft}. \displaystyle 54\text{ in} \times \frac{1\text{ ft}}{12\text{ in}} = \frac{54}{12}\text{ ft} = 4.5\text{ ft}.
\displaystyle 10\frac{1}{2}\text{ ft}. \displaystyle \frac{7}{2}\text{ yd} \times \frac{3\text{ ft}}{1\text{ yd}} = \frac{21}{2}\text{ ft}.
2.5\text{ kg}. \displaystyle 2{,}500\text{ g} \times \frac{1\text{ kg}}{1{,}000\text{ g}} = 2.5\text{ kg}.
126{,}720\text{ in}. \displaystyle 2\text{ mi} \times \frac{5{,}280\text{ ft}}{1\text{ mi}} \times \frac{12\text{ in}}{1\text{ ft}} comes to 2 \times 63{,}360 inches.
4\text{ h}. \displaystyle 14{,}400\text{ s} \times \frac{1\text{ min}}{60\text{ s}} \times \frac{1\text{ h}}{60\text{ min}} comes to \displaystyle \frac{14{,}400}{3{,}600} hours.
64{,}000\text{ oz}. \displaystyle 2\text{ tons} \times \frac{2{,}000\text{ lb}}{1\text{ ton}} \times \frac{16\text{ oz}}{1\text{ lb}} comes to 2 \times 2{,}000 \times 16 ounces.
Together they run 4 \times 440 = 1{,}760 yd. Then \displaystyle 1{,}760\text{ yd} \times \frac{3\text{ ft}}{1\text{ yd}} \times \frac{1\text{ mi}}{5{,}280\text{ ft}} comes to \displaystyle \frac{5{,}280}{5{,}280}\text{ mi} = 1\text{ mi}.
100 bags. \displaystyle 50\text{ lb} \times \frac{16\text{ oz}}{1\text{ lb}} = 800\text{ oz}, and \displaystyle 800\text{ oz} \times \frac{1\text{ bag}}{8\text{ oz}} = 100 bags. The last fraction is a rate, not a name for 1, but the ounces still cancel and leave bags.
30.48\text{ cm}. \displaystyle 12\text{ in} \times \frac{2.54\text{ cm}}{1\text{ in}} = 30.48\text{ cm}.
91.44\text{ m}. 100 yd is 3{,}600 in, which is 3{,}600 \times 2.54 = 9{,}144 cm, and \displaystyle 9{,}144\text{ cm} \times \frac{1\text{ m}}{100\text{ cm}} = 91.44\text{ m}.
About 55\text{ lb}. \displaystyle 25\text{ kg} \times \frac{2.2\text{ lb}}{1\text{ kg}} = 55\text{ lb}.
About 96.5\text{ km}. \displaystyle 60\text{ mi} \times \frac{1.609\text{ km}}{1\text{ mi}} = 96.54\text{ km}.
About 4.5\text{ kg}. \displaystyle 10\text{ lb} \times \frac{1\text{ kg}}{2.2\text{ lb}} is \displaystyle \frac{10}{2.2}\text{ kg} = 4.545\ldots\text{ kg}.
100\text{ in}, which is \displaystyle 8\frac{1}{3}\text{ ft} (8 ft 4 in). \displaystyle 254\text{ cm} \times \frac{1\text{ in}}{2.54\text{ cm}} = 100\text{ in}, and \displaystyle 100\text{ in} \times \frac{1\text{ ft}}{12\text{ in}} = \frac{100}{12}\text{ ft}. The extra \displaystyle \frac{1}{3} ft is 4 in.
437 yards. \displaystyle 400\text{ m} \times \frac{100\text{ cm}}{1\text{ m}} \times \frac{1\text{ in}}{2.54\text{ cm}} \times \frac{1\text{ yd}}{36\text{ in}} comes to \displaystyle \frac{40{,}000}{91.44}\text{ yd} \approx 437.4\text{ yd}. All three factors are exact, so the only rounding is at the end.
360\text{ in}^2. Two factors of \displaystyle \frac{12\text{ in}}{1\text{ ft}}: 2.5 \times 144 = 360.
7\text{ ft}^2. Two factors of \displaystyle \frac{1\text{ ft}}{12\text{ in}}: \displaystyle \frac{1{,}008}{144} = 7.
About 1.25 dollars per pound. Kilograms sit on the bottom of the price, so the factor has kilograms on top: \displaystyle \frac{2.75\text{ dollars}}{1\text{ kg}} \times \frac{1\text{ kg}}{2.2\text{ lb}} = \frac{2.75}{2.2} dollars per pound, which is 1.25.
The factor has 12 inches on top and 1 foot on the bottom, which are the same length, so it is equal to 1, and multiplying by 1 changes nothing. The number grew by 12 because the unit shrank by 12: an inch is a twelfth of a foot, so it takes 12 times as many of them.
0.5, in feet times feet per inch. \displaystyle 6 \times \frac{1}{12} = 0.5, and the feet did not cancel: a second foot joined the first and inches appeared on the bottom. A length should come out in a single length unit. The factor should have been \displaystyle \frac{12\text{ in}}{1\text{ ft}}, giving 72\text{ in}.
\displaystyle 1\text{ yd} \times \frac{36\text{ in}}{1\text{ yd}} \times \frac{2.54\text{ cm}}{1\text{ in}} \times \frac{1\text{ m}}{100\text{ cm}} comes to \displaystyle \frac{36 \times 2.54}{100}\text{ m} = \frac{91.44}{100}\text{ m} = 0.9144\text{ m}. Each fact used is exact, so each factor is exactly 1, and so is their product.
More than 1{,}500 yards. A yard is shorter than a meter, and a shorter unit needs more of itself to cover the same distance. (It comes to about 1{,}640 yards.)