= a b a2 + 2ab+ b2 a2 - b2 and 8 + d 8-d 14 2 2 and d. If we suppose s to be greater than d, so that 8d is positive, then, since of s and d, and s + d since 2 2 d is the half difference of s and d, and =d, we see that the half difference of two quantities, when subtracted from their half sum, gives Remarks.-Positive quantities having the sign + either expressed or understood, and negative quantities (or quantities to be subtracted) having the sign, it is evident that we may reduce the rules of Addition and Subtraction under the general rule of Addition, as follows; viz., collect the quantities into one sum (or whole) by adding them according to their signs; and that whether they are integral or fractional, or if some are integral and others fractional. Generally, in the addition of mixed quantities, it will be advisable to add the integral quantities into one sum, and then to add the fractional quantities into another, after they have been reduced to a common denominator; then the sum of the in tegral quantities and that of the fractional quantities being written together with their proper signs will be the sum required. Thus, suppose it is required to subtract 11 from 17, or, 4 4 2 and - 11. Here we which is the same thing, to add 17 and Also, if we have to add 3a, 5b,, and from the sum sub Observing the rule of signs, the product of the numerators, when written over that of the denominators, in the form of a fraction, will be the required product. Before the rule is applied, it will generally be advisable to reduce mixed quantities to improper fractions, and integral quantities to a fractional form, by writing 1 under them; also, to erase any factor which is common to the numerator of any fraction and the denominator of any other fraction; and to reduce each fraction to its lowest terms by erasing all the factors that are common to its numerator and denominator. This rule has been proved to be true in Division. We have drawn a line across the factors that are to be erased, since they will be common factors to the numerator and denominator of the product. We have also in the first 2 1 result reduced to by erasing the factor 2 that is common to its terms; then the product of 1 and 7 gives 7 for the numerator, and that of 3 and 9 gives 27 for the denominator of the sought product. 8. Find the product of 8(a+b)3, 4(a+b)3 (a−b)3 (a - b)3 (c — d)2 6pq 13mn 5m2n Ans. 9. Find the product of a +63 x yð + y3 8a3 — 2763 a2 + b23 13(x + y3) 14(a2 + b2) 25+ y3 13(a3 + b3)' 14(8a3 — 2763) Ans. 15(a + b) 15(a + b) Remarks.-A compound fraction is reduced to a single one by the rule of Multiplication. Thus, if we wish to take 2 4 of it is clearly the same as to multiply the numerators 3 5' together for a new numerator, and the denominators together for a new denominator, and we shall have of = a compound fraction, and we shall have correct result. If we wish to put an integral form under the form of a fraction having a given denominator, it can be done in much the same way. Thus if we would put 6 under the form of a fraction having 13 for its denominator, we multiply 6 by ner, if we would put a x under the form of a fraction having a + for its denominator, we have a − x = (a — = x) and so on in like cases. So that we multi ply the given quantity by the given denominator, and take the product for the numerator of the sought fraction, and the given denominator for its denominator. CASE VIII. To divide one fractional quantity by another. RULE. If the dividend or divisor is mixed, reduce it to an improper fraction; or, in case either of them is integral, reduce it to the form of a fraction, having 1 for its denominator. Then, if the numerator of the dividend is exactly divisible by that of the divisor, and the denominator of the dividend is exactly divisible by that of the divisor, take the quotient of the division of the numerator of the dividend by that of the divisor for the new numerator, and the quotient of the division of the denominator of the dividend by that of the divisor for a new denominator, and the resulting fraction will be the sought quotient. If the numerator of the dividend is not an exact multiple of that of the divisor, etc., we may invert the divisor, and then proceed as in Multiplication; which process is applicable in all cases. Observing in all instances, that it will be advisable to erase all the factors that are common to the numerators of the dividend and divisor, and also those that are common to their denomi nators, before the rule is applied. This rule was proved in Division. Here 4 ÷ 2 = 2, 21 ÷ 7 = 3, and a2 ÷ a = a, b2÷b=b; Here 422, and 2 x 7=14; so that as re which is the same as to invert the divisor, or change it to |