Q. What is the RULE for finding the least common multiple of any given numbers? with a sep A. Write down all the given numbers in a row, aratrix between them; then divide them by any number that will divide two or more of them, without a remainder; set the quotients and the undivided figures in a line beneath; divide this line again as before, and thus continue, till there are no two numbers that can be divided by the same figures; then multiply all the figures in the quotients and divisors continually together, and their product will be the least common multiple. EXAMPLES. 1. What is the least common multiple of 5, 8, 12, and 16? Operation. 4)5, 8, 12, 16 2)5, 2, 3, 4 5, 1, 3, 2 No two numbers can be divided. Therefore 4×2×5×3×2=240, least common multiple. 2. What is the least number that can be divided by 12, 14, 16, and 18? Ans. 1008. 3. What is the common multiple of 3, 6, 9, 12, and 15? Ans. 180. 4. What is the least number that can be divided by the nine digits? Ans. 2520. 9? 5. What is the least common multiple of 2, 4, 6, 5, 7, and Ans. 1260. 6. What is the least common multiple of 3, 5, 7, 9, 11, 13, 15, 17, 19, and 21? Ans. 14, 549, 535. REDUCTION OF VULGAR FRACTIONS. Q. What are you taught by Reduction of Vulgar Fractions? A. We are taught to bring them out of one form into an other, without altering their value, and also, to prepare them for addition, subtraction, multiplication and division. Q. Under how many different cases is reduction of Vulgar Fractions performed? A. Ten different operations are performed in reduction of vulgar fractions, and it is, threfore, arranged under ten different cases. CASE FIRST. Q. What is the first case? A. It is to abbreviate or reduce a fraction to its least or lowest term. Q. What is the RULE in this case? A. Divide each term of the fraction, by any number that will divide them both, without a remainder, and these quotients again, in the same manner, and thus continue, till there is no number, greater than 1, that will divide both the terms without a remainder, and the fraction will then be in its least term; or, divide both the terms of the fraction, by their greatest common measure, and these quotients will be the least terms of the fraction. Q. What is the second case in Reduction of Vulgar Fractions ? A. To bring mixed numbers to improper fractions. Q. What is the RULE in this case? A. Multiply the whole number by the denominator of the fraction, and add in the numerator, then write this product over the denominator of the fraction, and it will form the improper fraction required. EXAMPLES. 1. Reduce 16 to an improper fraction? Operation. 16×8+1=129 129 Ans. 2. Reduce 248 to am improper fraction, and also to its low 3. Reduce 9 to its lowest terms, in an improper fraction. 4. What is the least fractional expression of 12814? Ans. 29. Ans. 1031. 5. Reduce 374214 to its equivalent improper fraction. Ans. 278196 743 • 6. Reduce 1461 to an improper fraction, and also to its lowest terms. Ans. 2349 7. What is the least fractional expression of 4163174? 78 16 Ans. 324743 8. Bring 3648 to its least fractional expression. Ans. 6109. 8 Ans. 11779 18 CASE THIRD. Q. What is the third case in Reduction of Fractions? A. To reduce an improper fraction to an equivalent whole or mixed number. Q. What is the RULE in this case? A. Divide the numerator by the denominator of the fraction, and the quotient will be the whole number: if there be a remainder, it will be a numerator to the given denominator, and must be placed at the right hand of the quotient, which will express the same value, as the given fraction. EXAMPLES. 1. Reduce 129 to a whole or mixed number. 297 2. Reduce to a whole or mixed number. 3. What is the value of 116 980 6. What is the value of 986 ? 368 ? 7. What whole number is equal to CASE FOURTH. Q. What is the fourth case in Reduction of Fractions? A. To reduce any whole number to an equivalent fraction, having a given denominator. Q. What is the RULE in this case? A. Multiply the whole number, by the given denominator, and under this product, write the given denominator, and you will have the fraction required. EXAMPLES. 1. Reduce 8 to a fraction, whose denominator shall be 12. Operation. 8×12-96; then 96 is the answer. 2. Reduce 16 to a fraction, whose denominator shall be 24. Ans. 34 3. What fraction, having 48 for a denominator, will be equal to 36 ? 48. Ans. 1128 4. What numerator must be written over 19, to make a fraction equal in value to 9? Ans. 171. 5. Reduce 75 to a fraction, having 36 for a denominator. 36 Ans. 2700 6. Reduce 13 to a fraction, having 7 for a denominator. CASE FIFTH. Q. What is the fifth case in Reduction of Fractions? A. To reduce compound fractions to simple ones of equal value. Q. What is the RULE in this case? A. First, reduce all whole or mixed numbers to equivalent improper fractions; then multiply all the numerators together, for a new numerator, and all the denominators together, for a new denominator, and you will have the fraction required. EXAMPLES. 1. Reduce of of of to a simple fraction. Operation. 1x2×3×4=24 new numerator. 2×3×4x5=120 new denominator. Ans. 20 or 24 2. Reduce of of 16 to a simple fraction. 3. What is the least simple expression of Ans. 924 or 23 of of 3 of 3 of Ans. 14 12 81 Ans. 58. 4. What is the value of 4 of 4 of of 18? 5. A merchant bought of a ship, for 9000 dollars, and sold to A. of his share; A. sold to B. of his share; B. sold of his share to C.; and C. again, sold what part of the ship did each man buy, give for his purchase? Ans. 6. What simple fraction is equal in value to of § of } ? Ans. 12 7. If I buy of of of a manufacturing establishment, the whole of which is worth 25,000 dollars; what must I pay for my share? Ans. 4375 dollars. CASE SIXTH. Q. What is the sixth case in Reduction of Fractions? A. To reduce fractions, of different denominators, to frac tions having the same, or a common denominator. Q. What is the RULE in this Case? A. First, reduce all compound fractions to simple ones, and all whole, or mixed numbers, to improper fractions; then multiply each numerator into all the denominators, except its own, for new numerators; then multiply all the denominators together for the common denominator, and this new denomina. tor, written under the several new numerators, will form the fractions required. EXAMPLES. 1. Reduce,,, and, to fractions having a common denominator. Operation. 1x4x8x7=224) 3×2×8x7=336 New numerators. 5×2×4×7-280 6×2x4x8=384 2x4x8x7=448 Common denominator. The fractions, thus reduced, stand 224, 336, 280, 384. Ans. 2. Reduce, 448 448 448' 448. of, and, to a common denominator. Ans. 480 378 384 576 576 576. Ans. 120 1184 160. 3. Reduce,, and 73, to a common denominator. 4. Reduce, tor. 5. Reduce of, nominator. 60 160 160 of, and of 143, to a common denomi Ans. 13860 29568 11520 181720 36960 36960 36960' 36960. of, of, and of, to a common deAns. 60480 86016 100800 69120 161280 161280161280 16 1280* 6. Reduce of 7, 4 of 9, of 12, and 4 of 16. Ans. 1960 945 3024 3840 420'420' 420' 420⚫ CASE SEVENTH. Q. What is the seventh case in Reduction of Fractions? A. To bring fractions of one denomination to fractions of another, but greater, retaining the same value. |