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

The quantity on each side of the sign = is called a member of the equation. If each member be multiplied by the same quantity, the equality of the two members is not thereby disturbed (Art. 369); therefore, if the two members of the equation

45/9=435/87 (Art. 367) be each multiplied by 87, or be modified thus:

45x87/9 = 435 x87/87

in which x, the sign for multiplication, indicates that the quantities between which it is placed are to be multiplied together; this ddition to each member of the equation does not destroy the equality; the members are still equal, though considerably enlarged. The equality may be easily tested by performing the operations indicated in the equation. For example: for the first member, we have 45 times 87 equals 3915, and this divided by 9 equals 435. Again, for the second member we have 435 times 87 equals 37845, and this divided by 87 equals 435, the same result as that for the first member. Thus the multiplication has not interfered with the equality of the members.

371. - Multiplying and Dividing: one Member of an Equation: Cancelling. - If a quantity be multiplied by a given number, and the product be divided by the same given number, the quotient will equal the original quantity. For example: if 8 be multiplied by 3, the product will be 24; then if this product be divided by 3, the quotient will be 8, the original quantity. Thus the value of a quantity is not changed by multiplying it by a number, provided it be also divided by the same number.

From this, also, we learn that the value of a quantity which is required to be multiplied and divided by the same number will not be changed if the multiplication and division be both omitted; one cancels the other. Therefore the number 87, appearing in the second member of the equation in the last article both as a multiplier and a divisor, may be omitted without destroying the equality of the two members. The equation thus treated will be reduced to -

45x87/9 = 435.

This expression is read: the product of 45 times 87 divided by 9 equals 435. It will be observed that we have here the four terms of the problem in Art. 365, three of them in the first member, and the fourth, the answer to the problem, in the second member.

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