This section is from the book "The American House Carpenter", by R. G. Hatfield. Also available from Amazon: The American House Carpenter.

The fractions 9/3, 17/5, 24/3, etc., all fractions which have the numerator larger than the denominator are termed improper fractions. They are not improper arithmetically, but they are so named because it is an improper use of language to call that a part which is greater than the whole.

As expressions of this kind, however, are subject to the same rules as those which are fractions proper, it is customary to include them all under the technical term of fractions. Expressions like these - all expressions in which one number is separated by a horizontal line from another number below it, or one set of numbers is thus separated from another set below it - may be called fractions, and are always to be understood as indicating division, or that the quantity above the line is to be divided by the quantity below the line.

* Division is indicated by this sign ÷, which is read "divided by."

Thus, 9/3, 17/3,24/3, 3x8x4/2x12, 17x82/125, etc., are all fractions, technically, although each may be greater than unity. And it is understood in each case that the operation of division is required. Thus, 9/3 = 3, 24/3 = 8, 3x8x4/2x12 = 4. When the division cannot be made without a remainder, then the fraction, by cutting the numerator into two, may be separated into two parts, one of which may be exactly divided, and the other will be a fraction proper. Thus, the fraction 17/5 is equal to .

15/5 +2/5 (for 15 + 2 = 17); and since 15/5 equals 3, therefore,

17/5 = 15/5 + 2/5 = 3 + 2/5= 3 2/5 So, likewise, the fraction

17x82/125 = 1394/125 = 1375/125 = 19/125 = 11+19/125 = 11 19/125

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