| MEASURING DISTANCE, WEIGHT, & CAPACITY |
Introduction and History
People have measured and weighed all sorts of things in this world throughout history. Every society that ever existed developed measuring units and a system of equivalences -- so that buildings could be built to specific standards, goods could be traded or sold for a fair price, and everyone would have a common reference with which to operate their affairs. In our early history, before the invention of rulers and weigh scales, our ancestors used different parts of their body as measuring devices. It was convenient, since everyone took their body with them everywhere they went, however, not all thumbs, palms, hands and feet are the same size.
For instance, almost every culture has used the human foot as a unit of measurement. The natural foot, an ancient unit based on the length of actual feet, is about 9.8 inches. It was replaced in early civilizations of the Middle East by a longer foot which was the size of the modern foot, because this longer length was conveniently expressed in terms of other natural units.
Back then, math students learned these equivalences:
1 foot = 3 hands = 4 palms = 12 inches (thumb widths) = 16 digits (finger widths)
Both ancient Greece and Rome used the foot as a unit of measure; the Greek foot is estimated at 12.1 inches, and the Roman foot at 11.7 inches. In northern Europe, however, there was a competing unit known as the manual foot. It was equal to 2 shaftments, the distance from the tip of the outstretched thumb to the opposite side of the palm of the hand. The manual foot was measured "by hand," by grasping a rod with both hands, thumbs extended and touching. The manual foot is estimated at 13.1 inches. All these different lengths for a foot created land measurement and therefore tax collection problems.
Other units of measure, such as the rod -- which may have originated as the length of a pole used to control a team of 8 oxen (4 yokes) -- were based on the length of the foot, so a standard foot was established in the twelfth century, but the British royal government didn't want to change the length of the rod, since it was the basis of land measurement, land records, and taxes. Therefore the rod was redefined to equal 16.5 of the new feet and is still defined that way today.
(For more history of units of measure visit: http://www.unc.edu/~rowlett/units/index.html)
As international communication, trade and commerce increased between formerly isolated societies, there arose a need to develop a common system of measurement so that a buyer from one country could understand the seller from another. That's why all countries today have legally defined and standardized systems of measurement.
We measure distance, weight, volume, area, time, and temperature. We create units of measure that suit the dimensions of the things we're measuring. We measure mushroom spores with teeny-tiny units called microns – and we measure distances across the universe with unimaginably huge units called light years. In this lesson, we study units of measure for length, weight (mass), and capacity or volume of dry goods or liquids.
Measuring Length and Distance
What we call the US Customary System of Units of Length and Distance today, used to be called the Imperial or British System of Units of Length, since it was the standard for the British Empire (that's where Imperial comes from). In modern times, it has almost disappeared. Today, all countries of the world except the United States, have officially adopted the metric system of measurement for all things – length, weight, dry goods and liquids.
The customary units of length and distance used in the USA today are:
| 12 inches (in) = 1 foot (ft) | 3 ft or 36 in = 1 yard (yd) | 5280 ft or 1760 yd = 1 mile (mi) |
The rod (rd) = 5.5 yards = 16.5 feet and the
furlong (fur) = 40 rd or 220 yd or 660 ft are no longer used.
They are obsolete.
memory hint: if we learn the number of yards in a mile as seventeen-sixty, instead of one thousand, seven hundred and sixty, it is easier to remember. And, once we know 1760 yards in a mile, we multiply it by 3 to get 5280 (fifty-two-eighty) feet in a mile.
Questions on measurement usually ask us to change a length from one size unit to another.
| To change a length from small units to bigger ones, we divide the length by the number of small units in each bigger one. |
Example: How many feet are there in 60 inches?
There are 12 inches in a foot -- so we get 60 ÷ 12 or 5 feet.
| To change a length from big units to smaller ones, we multiply the length by the number of smaller units in each big one. |
Example: After a long run, Harry said he'd covered 7040 yards. How many miles did he run?
There are 1760 yards per mile so we divide (we're changing from small to big) 7040 by 1760
to find that Harry ran 4 miles.
Metric Units of Length or Distance
The Metric System, used throughout the world in trade and commerce, like our number system, is based on powers of 10. This makes it very easy to change units since division by a power of 10 simply moves the decimal point to the left and multiplication moves it to the right.
The prefixes of the units tell us what power of 10 to use.
If the prefix ends in " i " -- it indicates a small unit. Millimeters, centimeters and decimeters are fractions (divisions) of a meter. It takes a lot of them to make a meter which is 3.37 inches longer than a yard.
If the prefix ends in " a " or " o "-- it indicates a big unit. Decameters, hectometers and kilometers are multiples of a meter.
This ruler is marked off in inches (top) and metric units -- millimeters and centimeters -- below.
The teeny units (bottom) are millimeters -- milli means thousandth . The larger units (10 millimeters) are centimeters -- centi mean hundredth. It takes 2.54 centimeters to make an inch.
The metric units of length and distance used in most countries today are:
| 1000 millimeters (mm) = | 100 centimeters (cm) = | 10 decimeters (dm) = | 1 meter (m) |
| 1000 meters (m) = | 10 hectometers (hm) = | 100 decameters (dam) = | 1 kilometer(km) |
Just as rods and furlongs are rarely used any more in the British system, decimeters (tenths of a meter), decameters (10 meters), and hectometers (100 meters) are rarely used in metric measure today. Small objects are measured in millimeters and centimeters, medium objects are measured in meters, and long distances are measured in kilometers. Eight kilometers is the same as 5 miles so 1 mile = 8/5 = 1.6 kilometers. Maps in Canada, which uses the metric system exclusively since 1981, show distances in both miles and kilometers.
Examples:
| 12 mm = 1.2 cm = 0.012 m | 375 cm = 3.75 m = 0.375 km | 7 km = 7 000 m = 70 000 dm |
Notice how easily we convert:
To go from small units to larger ones, move the decimal to the left
the correct number of places to make the number smaller.
To go from big units to smaller ones, move the decimal to the right
the correct number of places to make the number bigger.
Though standardization has solved many problems, it has caused another, because many European countries (especially France and Belgium) use the comma ( , ) rather than the point ( . ) to indicate the decimal. So, if we write 3, 507 to mean three thousand five hundred and seven, it could be interpreted by a metric user as 3.507. To avoid confusion, those of us who still use the decimal point no longer use the comma to mark off groups of 3 digits. We now use a space where the comma would be, like this: 19 201.3 instead of 19, 201.3
Measuring Weight or Mass
Many items such as food and nails are priced and sold by weight. At the market, we see signs like, Bananas 29¢ / lb , Tomatos 93¢ / lb . The check out cash is programmed with price codes so that when we buy 7 pounds (lb) of bananas, it will multiply 7 × 29¢ to find the total.
US Customary Units of Weight:
| 16 ounces (oz) = 1 pound (lb) | 2000 pounds (lbs) = 1 Ton (T) | 2 200 (lbs) = 1 Metric Ton |
Metric System Units of Weight
| 1000 milligrams (mg) = | 100 centigrams (cg) = | 10 decigrams (dg) = | 1 gram (g) |
1000 grams (g) = 1 kilogram (kg) = 2.2 pounds 454 grams (g) = 1 pound (lb) = 16 ounces (oz) |
1000 kg = 2 200 lbs = 1 metric ton | ||
Centigrams and decigrams are rarely used.
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Measuring Capacity or Volume
CAPACITY is usually called VOLUME. It is the measure of how much of something fits into a container of a specific size. We use these units to measure dry materials and liquids. These are the units most often used by cooks, grocers, and food shoppers. We can buy a quart of strawberries and a quart of strawberry syrup at the market, but though the name of the measurement unit (quart) is the same, in the case of the syrup, we're using liquid measure not dry measure. Though many of these units are used for both liquid and dry measure, others are not. For example, we can buy a bushel of apples but we can't buy a bushel of apple juice -- since a bushel is a unit of measure exclusively for dry materials, not liquids.
US Customary Units of Liquid or Dry Measure
3 teaspoons (t) = 1 tablespoon (tbsp or T) = 16 tablespoons = 8 fluid ounces (fl. oz.) = 1 cup
2 cups = 16 fluid ounces = 1 pint
2 pints = 4 cups = 32 fl. oz. = 1 quart (qt)
4 quarts = 8 pints = 16 cups = 128 fl. oz. = 1 gallon (g)
Exclusively Dry Measure
2 pints (pt) = 1 quart (qt)
8 quarts (qt) = 1 peck (pk)
4 pecks (pk) = 1 bushel (bu)
Metric Liquid Measure
1 teaspoon (t) = 5 milliliters (mL)
1000 milliliters (mL) = 100 centiliters (cL) = 10 deciliters = 1 liter (L)
1000 liters (L) = 1 kiloliter (kL)
A dry quart and dry liter are almost identical. A dry US quart = 1.1 dry liters.
A liquid quart holds slightly more than a liquid liter. A liquid US quart = 0.945 liters.
Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.
Practice Exercises
1) Which units -- inches, feet, yards, miles, or kilometers do we use to measure:
| a) a man's height | b) distance between European cities | c) a fence | d) a pencil |
2) Change these small units of length to the larger ones indicated: show your work.
| a) 18 in = ____ ft | b) 25 ft. = _____ yds | c) 3520 yds = ___ mi | d) 720 in = ___ yds |
| e) 1250 mm = ___ m | f) 427 cm = ___ m | g) 1 120 m = ___ km | h) 42 mm = ___ cm |
3) Change these big units of weight to the smaller ones indicated: show your work.
| a) 16 tons = ____ lbs | b) 25 lbs = _____ oz | c) 3 Kg = ___ g | d) 15 dg = ___ mg |
4) Change these units of capacity or volume to the ones indicated: show your work.
| a) 6 qts = ____ c | b) 24 c = _____ fl oz | c) 15 pks = ___ bu | d) 1 500 mL = ___ L |
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Solutions
1) Which units -- inches, feet, yards, miles, or kilometers do we use to measure:
| a) a man's height feet |
b) distance between European cities kilometers |
c) a fence yards |
d) a pencil inches |
2) Change these small units of length to the larger ones indicated: show your work.
| a) 18 in = ____ ft 18 ÷ 12 = 1½ ft |
b) 25 ft. = _____ yds 25 ÷ 3 = |
c) 3520 yds = ___ mi 3520 ÷ 1760 = 2 mi |
d) 720 in = ___ yds 720 ÷ 36 = 20 yds |
| e) 1250 mm = ___ m 1250 mm = 1.25 m |
f) 427 cm = ___ m 427 cm = 4.27 m |
g) 1 120 m = ___ km 1 120 m = 1.12 km |
h) 42 mm = ___ cm 42 mm = 4.2 cm |
3) Change these big units of weight to the smaller ones indicated: show your work.
| a) 16 tons = ____ lbs 16 × 2000 = 32 000 lb |
b) 25 lbs = _____ oz 25 × 16 = 400 oz |
c) 3 Kg = ___ g 3 × 1000 = 3000 g |
d) 15 dg = ___ mg 15 × 100 = 1500 mg |
4) Change these units of capacity or volume to the ones indicated: show your work.
| a) 6 qts = ____ c 6 qts × 4 = 24 cups |
b) 24 c = _____ fl oz 24 c × 8 = 192 fl oz |
c) 15 pks = ___ bu 15 pks ÷ 4 = 3¾ bu |
d) 1 500 mL = ___ L 1 500 mL = 1.5 L |
yds