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

Each couple is also termed a Ratio, and the two the Equality of Ratios. Thus the ratio

45/9is equal to the ratio 435/87. If the division indicated in these two ratios be actually performed, the equality between the two will at once be apparent, for the quotient in each case is 5. The resolution of each couple into its simplest form by actual division is shown thus:

45/9 = 5

435/87 = 5.

These are read: 45 divided by 9 equals 5; and 435 divided by 87 equals 5.

If two equal quantities be each multiplied by a given quantity, the two products will be equal. For example, the fractions 2/4 and 3/6 are each equal to 1/2, and are therefore equal to each other. If these two equal quantities be each multiplied by any given number, say, for example, by 4, we shall have 4 times 2/4 equals 8/4, and 4 times 3/6 equals 12/6; these products, 8/4 and 12/4 are each equal to 2, and therefore equal to each other.

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