Prealgebra: Measurements

Systems of Measurement

There are a number of different systems of measurement. Here we will look at the two most common, the English system and the Metric system.

The English System

The units of measurement in the English system are:

Length/Area

1 foot (ft) = 12 inches (in)
1 yard (yd) = 3 feet
1 mile (mi) = 1,760 yards
1 square foot (sq ft) = 144 square inches (sq in)
1 square yard (sq yd) = 9 square feet
1 acre = 4,840 square yards = 43,560 square feet
1 square mile (sq mi) = 640 acres

Capacity

1 tablespoon (T) = 3 teaspoons (t)
1 cup (c) = 16 tablespoons
1 pint (pt) = 2 cups
1 quart (qt) = 2 pints
1 gallon (gal) = 4 quarts

Weight

16 ounces (oz) = 1 pound (lb)
1 ton = 2,000 pounds

The Metric System

The units of measurement in the metric system are:

Length

1 millimeter (mm) = 0.001 meters (m)
1 centimeter (cm) = 0.01 m
1 decimeter (dm) = 0.1 m
1 meter (m) = 1 m
1 decameter (dam) = 10 m
1 hectometer (hm) = 100 m
1 kilometer (km) = 1,000 m

Capacity

1 milliliter (ml) = 0.001 liters (l)
1 centiliter (cl) = 0.01 l
1 deciliter (dl) = 0.1 l
1 liter (l) = 1 l
1 decaliter (dal) = 10 l
1 hectoliter (hl) = 100 l
1 kiloliter (kl) = 1,000 l

Weight

1 milligram (mg) = 0.001 grams (g)
1 centigram (cg) = 0.01 g
1 decigram (dg) = 0.1 g
1 gram (g) = 1 g
1 decagram (dag) = 10 g
1 hectogram (hg) = 100 g
1 kilogram (kg) = 1,000 g

Converting Measurements

We can convert measurements from one unit to another unit within the same system (English or Metric) or between the two systems. Let's say a piece of land is 2.3 square miles, but the contractor needs to know the area in acres. How would he convert the measurement?

To convert measurements, it is necessary to know conversion factors between measurements. A conversion factor is a clever way of writing 1 as a fraction in which the numerator is equal to the denominator but the numerator and the denominator have different units. For example, (1,000 m)/(1 km) is a conversion factor because 1,000 m = 1 km. (1 ft)/(12 in) is a conversion factor because 1 ft = 12 in.

An important fact to remember is that when fractions are multiplied, numbers in the numerator and numbers in the denominator cancel out, as shown in Fractions. Also, units in the numerator and the denominator cancel out: if a unit appears in both the numerator and the denominator, we can cross both units out. For example:

× = × = 5, 000 m

Using these two pieces of knowledge, here are the steps to converting a measurement from one unit into another, using the example problem above of converting 2.3 square miles into acres:

Step 1. Write down the given units as a fraction over 1.

Step 2. To the far right of this fraction, write an "=" sign and a fraction line. Put the units of the final answer in the numerator. Except for a "1" in the numerator, no numbers should be in this fraction yet.

=

Step 3. Write down conversion factors. Since we know the relationship between square miles and acres (1 sq. mi. = 640 acres), we can write down 2 conversion factors:

and

Step 4. Multiply a conversion factor into the equation. In order to cancel out, the denominator of the conversion factor must have the same units as the numerator of the given fraction (the fraction on the left of the equation). In this case, the conversion factor must have "sq mi" in the denominator. Multiply the fraction on the left by the conversion factor:

× =

Step 5. Cancel out the units and multiply:

× = 1, 472 acres

Step 6. Round the answer to the same number of significant digits as the original number (for now, we will treat our conversion factors as if they are precise). 2.3 has 2 significant digits. 1,472 acres = 1,500 acres. Thus, 2.3 sq mi = 1,500 acres.

To convert between English and Metric units, it is useful to know the following conversion factors:

Length/Area

1 in = 2.540 cm
1 ft = 0.3048 m
1 mi = 1.609 km
1 sq ft = 0.0929 sq m
1 sq mi = 2.59 sq km

Capacity

1 fl oz = 29.575 ml
1 gal = 3.785 l

Weight

1 oz = 28.35 g
1 lb = 0.4536 kg

Here is an example of a conversion from metric units to English units: if Bob ran a 5.0 km race, how many miles did he run?

Step 1. Write down the given units as a fraction over 1.

Step 2. To the far right of this fraction, write an "=" sign and a fraction line. Put the units of the final answer in the numerator. Except for a "1" in the numerator, no numbers should be in this fraction yet.

=

Step 3. Write down conversion factors. Since we know the relationship between kilometers and miles (1 mi = 1.609 km), we can write down 2 conversion factors:

and

Step 4. Multiply a conversion factor into the equation. In order to cancel out, the denominator of the conversion factor must have the same units as the numerator of the given fraction (the fraction on the left of the equation). In this case, the conversion factor must have "km" in the denominator. Multiply the fraction on the left by the conversion factor:

× =

Step 5. Cancel out the units and multiply:

× = = 3.1075 mi

Step 6. Round the answer to the same number of significant digits as the original number. 5.0 has 2 significant digits. 3.1075 mi = 3.1 mi. Thus, 5.0 km = 3.1 mi.

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