to those of the same; and all of them to a common denominator; then the sum of the numerators, written over the common denominator, will be the sum of the fractions required. EXAMPLES. 1. Add 3, 7, of, and 7 together. First 3-2, of &=7%, 7={• 2 29 Then the fractions are 3, 7, fa and 7;.. 5. Add 41. s. and of a penny together. 3139 Ans. 13 109 Ans. 18, or 3s. 1d. 11. 6. What is the sum of of 151. 31. of of of a pound, and of of a shilling? er. er. Ans 71. 17s. 5d. 7. Add § of a yard, of a foot, and of a mile togeth 8. Add of a week, of a day, and of an hour togethAns. 2d. 14th. tor, are entirely dissimilar, and therefore cannot be incorporated with one another; but when they are reduced to a common denominator, and made parts of the same thing, their sum or difference may then be as properly expressed by the sum or difference of the numerators, as the sum or difference of any two quantities whatever by the sum or difference of their individuals; whence the reason of the rules, both for addition and subtraction, is manifest. SUBTRACTION OF VULGAR FRACTIONS. RULE. Prepare the fractions as in addition, and the difference of the numerators, written above the common denominator, will give the difference of the fractions required. Reduce compound fractions to single ones, and mixed numbers to improper fractions; then the product of the numerators is the numerator; and the product of the denomi nators, the denominator of the product required. EXAMPLES. 1. Required the continued product of 2i, 1, 1 of £, and 2. * Multiplication by a fraction implies the taking some part or parts of the multiplicand, and therefore may be truly expressed by a compound fraction. Thus & multiplied by & is the same as of; and as the directions of the rule agree with the method already given to reduce these fractions to single ones> it is shown to be right. Ans. 9140 6. Multiply 41, of, and 18, continually together. DIVISION OF VULGAR FRACTIONS. RULE.* Prepare the fractions as in multiplication; then invert the divisor, and proceed exactly as in multiplication. 19% 2 2 • 3x3 === 38=74 the quotient required. 5 1 Suppose it * The reason of the rule may be shown thus. were required to divide by. Now 3÷2 is manifestly of of 2,/. of 2, or must be contained 5 times 4X3 as often in as 2 is; that is 3X5 the answer; which is accor ding to the rule; and will be so in all cases. NOTE. A fraction is multiplied by an integer, by dividing the denominator by it, or multiplying the numerator. And divided by an integer, by dividing the numerator, or multiplying the de* nominator, DECIMAL FRACTIONS. A DECIMAL is a fraction, whose denomiaator is an unit, or 1, with as many cyphers annexed, as the numerator has places; and is commonly expressed by writing the numerator only, with a point before it, called the separatrix. A finite decimal is that, which ends at a certain number of places. But an infinite decimal is that, which is understood to be indefinitely continued. A repeating decimal has one figure, or several figures, continually repeated, as far as it is found. As 33, &c. which is a single repetend. And 20-2424, &c. or 20' 246246, &c. which are compound repetends. Repeating decimals are also called circulates, or circulating decimals. A point is set over a single repetend, and a point over the first and last figures of a compound repetend. The first place, next after the decimal mark, is 10th parts, the second is 100th parts, the third is 1000th parts, and so on, decreasing toward the right by 10ths, or increasing toward the left by 10ths, the same as whole or integral numbers do. As in the following Cyphers on the right of decimals do not alter their value. 500 is ». 1000 But cyphers before decimal figures, and after the separating point, diminish the value in a tenfold proportion for every cypher. Too or T£T So that, in any mixed or fractional number, if the separating point be moved one, two, three, &c. places to the right, every figure will be 10, 100, 1000, &c. times greater than before. But if the point be moved toward the left, then every figure will be diminished in the same manner, or the whole quantity will be divided by 10, 100, 1000, &c. ADDITION OF DECIMALS. RULE. 1. Set the numbers under each other according to the value of their places, as in whole numbers, or so that the decimal points may stand each directly under the preceding. |