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ARITHMETIC.

PART III.

DENOMINATE NUMBERS.

234. A Denominate Number is one in which the unit of value is established by law or custom. For example, 7 pounds, 6 feet, 9 inches.

When a denominate number is composed of units of but one denomination, as, for example, 3 gallons, 5 feet, 4 tons, it is called a simple denominate number. If it contains units of more than one denomination that are related to each other, as 6 feet 10 inches, or 7 pounds 5 ounces, it is called a compound denominate number.

235. The reduction of denominate numbers is the process of changing them from one denomination to another without changing their value. We may change from a higher to a lower denomination, or from a lower to a higher denomination.

NOTE. In decimals the law of increase and decrease is by the uniform scale of 10, but in compound numbers the scale varies.

MEASURES. 236. A unit of measure is a standard established by law or custom by which a quantity--such as extent, dimension, capacity, amount of value-is measured. For example, the length of a piece of cloth is ascertained by applying the yard measure; the capacity of a cask by the use of a gallon measure; that of a bin by the use of the bushel; the weight of a body by the use of the pound; etc.

237. Every unit of measure belongs to one of the following six classes:

Extension
Capacity
Weight

Time
Angles and Arcs
Money, or Commercial Value.

MEASURES OF EXTENSION.

238. Extension is that property of a body by virtue of which it occupies space and has one or more of the dimensions---length,

breadth, and thickness. The standard of extension, whether linear, surface, or solid, is the yard. Thus, the unit of length is the yard, the unit of surface is a square each side of which measures a yard, and the unit of solid measure is a cube each edge of which measures a yard.

LINEAR MEASURE. 239. Linear, or Long Measure as it is sometimes called, is used in measuring lines or distances. A line has but one dimension, and that is length.

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240. In reducing a number from a higher to a lower denomination, we use multiplication; when we reduce from a lower to a higher we use division.

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643 = number of in. in 17 yds. 2 ft. 7 in.

Study the following table of equivalents, carefully working out each result. The table may then be used for reference.

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The use of the second table is more convenient. 17 yds.=17 x 36 in., or 612 in. (since 1 yd. = 3 ft. = 36 in.); and 2 ft. = 2 x 12 in., or 24 in.

Therefore in 17 yds. 2 ft. 7 in. there are (612 + 24 + 7) or 643 inches.

Now suppose that we wished to change 643 in. to yards, feet, and inches, that is, to reverse the preceding process.

12 inches 1 foot.
643 inches = as many feet as 12 is contained times in 613:

12) 643

53 7-12 We find that there is a remainder of 7, which indicates the number of inches more than those exactly contained in 53 feet.

3 feet = 1 yard.

53 Then in 53 feet there are yds. or 17 yds., with a remainder

3 of 2 ft.; that is, there are 2 feet more than the number of feet exactly contained in 17 yards. We have now 17 yds. 2 ft. 7 in. The process is usually performed as shown below.

12) 643 in.
3 ) 53 (ft.)-7 in.

17 (yds.)-2 ft. Ans. 17 yds. 2 ft. 7 in. 241. The inch is generally divided into halves, quarters, eighths, sixteenths, and sometimes into tenths and twelfths. Civil and mechanical engineers frequently use decimal divisions of the foot and the inch.

SQUARE MEASURE.
242. A surface has two dimensions, length and breadth.

243. The area of a surface is the numerical value of its ratio to another surface called the "unit of surface;" or, in other words,

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