there are no two numbers in the line that can be thus divided; the product of all the divisors and remaining numbers in the last (undivided) dividend is the least common denominator, or multiple sought. EXAMPLE. What is the least common denominator of 6, 5, and, or of 20, 25, and 50? 5) 20.25.50 2) 4.5.10 5) 2.5.5 2.1.1 5 X 2 X 5 X 2: = 100. Ans. ADDITION OF VULGAR FRACTIONS. Sum of the products of each numerator with all the denominators except that of the numerator involved, forms numerator of sum. Product of all the denominators forms denominator of sum. RULE. Arrange the several fractions to be added, one after another, in a line from left to right; then multiply the numerator of the first by the denominator of the second, and the denominator of the first by the numerator of the second, and add the two products together for the numerator of the sum; then multiply the two denominators together for its denominator; bring down the next fraction, and proceed in like manner as before, continuing so to do until all the fractions have been brought down and added. Or, reduce all to a common denominator, then add the numerators together for the numerator of the sum, and write the common denominator beneath. EXAMPLES. Add together,,, and . 1 × 3 = 7× §. = £§ × 1 = 234 = 13 = 34. Ans. } = 2 + 2 = {= 15, and 3 += §= 21, and 15+ 24 = 12 1231. Ans. SUBTRACTION OF VULGAR FRACTIONS. Product of numerator of minuend and denominator of subtrahend, forms numerator of minuend, for common denominator. Product of numerator of subtrahend and denominator of minuend, forms numerator of subtrahend, for common denominator. Product of denominators forms common denominator. Difference of new found numerators forms the numerator, and common denominator the denominator, of the difference, or remainder sought. RULE. Write the subtrahend to the right of the minuend, with the sign (—) between them; then multiply the numerator of the minuend by the denominator of the subtrahend, and the denominator of the minuend by the numerator of the subtrahend; subtract the latter product from the former, and to the remainder or difference affix the product of the two denominators for a denominator; the sum thus formed is the answer, or true difference. DIVISION OF VULGAR FRACTIONS. Product of numerators of dividend and denominators of divisor, forms numerator of quotient. Product of denominators of dividend and numerators of divisor, forms denominator of quotient; therefore, RULE. - Write the divisor to the right of the dividend with the sign() between them; then multiply the numerator of the dividend by the denominator of the divisor, for the numerator of the quotient, and the denominator of the dividend by the numerator of the divisor, for the denominator of the quotient. Or, invert the divisor, and multiply as in multiplication of fractions. Or, proceed by cancellation, when practicable. EXAMPLES. Divide by; by ; by 1; and of of of by of of of. NOTE. The foregoing example can be cancelled to the extent of leaving but a 4 and a 5 (20) numerators, and a 3 denominator. Units, or 1's, in the expressions, are valueless, as a sum multiplied by I is not increased. MULTIPLICATION OF VULGAR FRACTIONS. Product of numerators of multiplier and multiplicand, forms numerator of product. Product of denominators of multiplier and multiplicand, forms denominator of product. RULE. Multiply the numerators together for a numerator, and the denominators together for the denominator. EXAMPLES.Multiply by; by 7; 1 by 12; of of by 2 of of. UNIV. OF MULTIPLICATION AND DIVISION OF FRACTIONS, COMBINED. It has been seen that a compound fraction is converted into an equivalent simple one, by multiplying the numerators together for a numerator, and the denominators together for a denominator; and it has also been seen that a series of simple fractions are converted into a product, by the same process. It is therefore evident that compound fractions and simple, or a series of compound and a series of simple, may be multiplied into each other, for a product, by multiplying all the numerators of both together for a numerator, and all the denominators of both together for a denominator; and that the product will be the same as would be obtained, if the compound were first converted into an equivalent simple fraction, and the simple fractions into a product or factor, and these multiplied together for a product. It has also been seen that a fraction is divided by a fraction by multiplying the numerator of the dividend by the denominator of the divisor, for the numerator of the quotient, and the denominator of the dividend by the numerator of the divisor, for the denominator of the quotient; and that this multiplication becomes direct as in multiplying for a product, if the divisor is inverted. And it is clear that a compound divisor, or a series of simple divisors, or both, may be used instead of their simple equivalent, and with the same result, if all are inverted. It is therefore evident that any proposition, or problem, in fractions, consisting of multiplications and divisions both, and these only, no matter how extensive and numerous, or whether in compound fractions, or simple, or both, may be solved, and the true result obtained, as a product, by simply multiplying all the numerators in the statement together for a numerator, and all the denominators in the statement for a denominator, all the divisors in the statement being inverted; that is, all the numerators of the divisors being made denominators in the statement, and all the denominators of the divisor being made numerators in the statement. And it is further evident that a proposition stated in this way, admits of easy cancellation as far as cancellation is practicable, which is often to great extent. EXAMPLE. -It is required to divide 12 by } of; to multiply the quotient by the product of 4 and 8; to divide that product by of of 8; to multiply the quotient by of of; and to divide that product by the product of 5 and 9. 9 * STATEMENT. (Dividends read from right to left, divisors from left to right.) The answer to the above proposition is 14, and the proposition as stated may be readily cancelled to its lowest terms. It may be cancelled to the extent of leaving but 4, 4, 2 in the numerator, and 7, 3, in the denominator, 42 = 32 = 127. 7X3 To reduce a fraction in a higher denomination to an equivalent fraction in a given lower denomination. numerators into by a fraction whose RULE. Multiply the fraction to be reducednumerator and denominators into denominator numerator represents the number of parts of the lower denomination, required to make ONE of the denomination to be reduced. EXAMPLE. - Reduce of a foot to an equivalent fraction in inches. 12 = 84 = 21. Ans. To reduce a fraction in a lower denomination to an equivalent fraction in a given higher denomination. -- RULE. Multiply the fraction to be reduced denominator and denominator into numerator numerator into by a fraction whose numerator represents the number of parts required of the lower denomination to make 1 of the higher. EXAMPLE. Reduce 21 inches to an equivalent fraction in feet. 31 ÷ 42 = 21 = 7. Ans. Or, 21 X = }}=}. Ans. 40 Or, XX=80. Ans. To reduce a fraction in a higher to whole numbers in lower denominations. RULE. Multiply the numerator of the given fraction by the number of parts of the next lower denomination that make ONE of the given fraction, and divide the product by the denominator. Multiply the numerator of the fractional part of the quotient thus obtained by the number of parts in the next lower denomination that make 1 of the denomination of the quotient, and divide by its denominator for whole numbers as before; so proceed until the whole numbers in each denomination desired are obtained. EXAMPLE. day? - How many hours, minutes, and seconds, in of a 9X24-216 = 15, 4×60 - 18925, 3×60-300 = 42 §, = 429 sec. 15 h., 25 m., Ans. How many minutes in of a day? Ans. To reduce fractions, or whole numbers and fractions, in lower denominations, to their value in a higher denomination. RULE. Reduce the mixed numbers to improper fractions, find their common denominator, and change each whole number and numerator to correspond therewith. Then reduce the higher numbers to their values in the lowest denomination, add the value in the lowest denomination thereto, and take their sum for a numerator. Multiply the common denominator by the number required of the lowest denomination to make ONE of the next higher, that product by the number required of that denomination to make 1 of the next higher, and so on, until the highest denomination desired is reached, and take the product for a denominator, and reduce to lowest terms. EXAMPLE. Reduce 51 oz., 3 dwts., 2 grs., troy, to lbs. 16. 16.5=160.96.75; therefore, 30 |