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

As a fracture is a break or division into parts, so a fraction is literally a piece broken off; a part of the whole.

The figures which are generally used to express a fraction show what portion of the whole, or of an integer, the fraction is: for example, let the line A B, (Fig. 274), be divided into five equal parts, then the line A C, containing three of those parts, will be three fifths of the whole line A B, and may be expressed by the figures 3 and 5, placed thus, 3/5, which is known as a fraction and is read, three fifths. The number 5 below the line denotes the number of parts into which an integer or unit, A B, is supposed to be divided; it is therefore called the denominator, and expresses the denomination or kind, whether fifths, sixths, ninths, or any number, into which a unit is supposed to be divided. The number 3 above the line, denoting the number of parts contained in the fraction, is termed the numerator, and expresses the number of parts taken, as 2, 3, 4, or any other number.

Fig. 274.

380. - Graphical Representation of Fractions: Effect of Multiplication. - In Fig. 275, let the line A B be divided into three equal parts; the line CD into six equal parts; the line EF into nine equal parts; the line GH into twelve equal parts, and the line JK into fifteen equal parts. The lines A B, CD, E F, G H, and J K, being all of equal length.

Then the parts of these lines, A L, CM, EN, etc., may be expressed respectively by the fractions 1/3, 2/6, 3/9, 4/12 and 5/15. In each case the figure below the line, as, 3, 6, 9, 12, or 15, expresses the number of parts into which the whole is divided, and the figure above the line, as 1,2, 3, 4, or 5, the number of the parts taken; and, as the lines A L, CM, EN, etc., are all equal to each other, therefore these fractions are all equal to each other. If the numerator and denominator of the first fraction be each multiplied by 2, the products will equal the numerator and denominator of the second fraction: thus -

Fig. 275.

1 | x | 2 | = | 2 | |

3 | x | 2 | = | 6 | |

1 | x | 3 | = | 3 | |

so, also, | 3 | x | 3 | = | 9 |

1 | x | 4 | = | 4 | |

and | 3 | x | 4 | = | 12 |

1 | x | 5 | = | 5 | |

and | 3 | x | 5 | = | 15 |

Thus it is shown that when the numerator and denominator of a fraction are each multiplied by the same factor, the product forms a new fraction which is of equal value with the original.

In like manner we have, 2/3, 3/12, 4/16, 5/20, etc each equal to one fourth; and which may be found by multiplying the numerator and denominator of 1/4 successively by 2, 3,4, 5, etc.

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