EXAMPLES. 1. What is the greatest common measure of 1836, 3996, and Operation. 1836)3996(2 1044 ? Here it is seen that 108 is the greatest common measure of 1836, and 3996. The next process is, to find the common measure of 108, and 1044, as follows, viz.: Here it is seen that 36 is the last common measure, and is therefore the greatest common measure of the three given numbers, and is the answer required. 3672 2. What is the greatest common measure of 182, and 392 ? Ans. 14. 3. What is the greatest common measure of 1224, and 1080 ? Ans. 72. 4. What is the greatest common measure of 1440, 672, and 3472 ? Ans. 16. 5. What is the greatest number that will divide 288, and 480, without a remainder? Ans. 96. 6. What is the greatest number that will divide 128, and 160, without a remainder? Ans. 32. CASE SECOND. Q. What is the second case in Vulgar Fractions? A. It is to find the common multiple of several given numbers? Q. What is meant by the common multiple of several numbers? A. It is such a number as can be divided by each of the given numbers, without a remainder. If it be the least number that can be so divided, it is called the least common tiple. Q. What is the RULE for finding the least common multiple of any given numbers? A. Write down all the given numbers in a row, with a separatrix 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. 5. What is the least common multiple of 2, 4, 6, 5, 7, and Ans. 1260. 9? 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 another, 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 est 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. 16x8+1=129 Ans. 2. Reduce 248 to am improper fraction, and also to its low est terms. 'Ans.. 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 374314 to its equivalent improper fraction. Ans. 278196 743 · 6. Reduce 14613 to an improper fraction, and also to its lowest terms. Ans. 2349 7. What is the least fractional expression of 416317? 16. Ans. 324743 78 8. Bring 3648 to its least fractional expression. Ans. 1459. 9. How many eighths are contained in 7635? Ans. 6109. 10. How many sixteenths are contained in 73618 ? 8 Ans. 11779 16 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 frac tion, 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. 2. Reduce to a whole or mixed number. 297 368 Ans. 46. 11779 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. 12 Ans. 384 24 3. What fraction, having 48 for a denominator, will be equal to 36 ? Ans. 1728 48 • 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. Ans. 200 6. Reduce 13 to a fraction, having 7 for a denominator. 36 Ans.. 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 3 of of to a simple fraction. Operation. 1×2×3×4= 24 new numerator. 2×3×4x5=120 new denominator. 2. Reduce of of 16 to a simple fraction. 3. What is the least simple expression of 81 Ans. 51. 4. What is the value of 4 of of of 18? 5. A merchant bought 1 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 of his share to D.; what part of the ship did each man buy, and how much did he of the ship for $4500. of it for $3000. of it for $1800. D. bought of it for $1125. 32 |