year they find they have gained $950; what must each man share of the gain? A $461,4219 C $235,23. For the principal definitions of Fractions, the scholar will refer to those already given, Page 86. I. Any number that will divide two or more numbers without a remainder, is called a common measure, or divisor; and the greatest number that will do this is called the greatest common measure, or divisor. Thus, 4 is a common divisor of 8, 16 and 24, because it will divide each without a remainder; but the greatest common divisor of these numbers is 8. II. The common multiple of two or more numbers is that number which can be divided by each of those numbers without a remainder, and the least number that can be so divided is called the least common multiple. Thus, 24 is a common multiple of 3, 4 and 6; but their least common multiple is 12. III. When two or more fractions have the same denominator, it is called their common denominator. PROBLEM I. To find the greatest common divisor of two numbers RULE. Divide the greater number by the less, and this divisor by the remainder, and so on, always dividing the last divisor by the last remainder till nothing remain; and the last divisor will be the common divisor. EXAMPLES. 1. What is the greatest common measure of 91 and 117, or in other words, what is the greatest number that will divide 91 and 117 separately without a remainder? Operation. 91)117,(1 91 26)91(3 13)26(2 We divide the greatest number by the least, and the remainder is 26; therefore, 91 is not a factor of 117; then 26 in 91 will go 3 times, and 13 remains. Hence 26 is not a factor of 91; then 13 will go in 26, 2 times, and nothing remains. Hence, 13 is the greatest common divisor of 91 and 117, or the great ́est number that 91 and 117 can be divided by, without a remainder. 2. What is the greatest common measure of 48 and 56? Ans. 8. 3. What is the greatest common divisor of 132 and 144? 4. What is the greatest common divisor of 148 5. What is the greatest common divisor of 1080? Ans. 12. and 236? Ans. 4. 1224 and Ans. 72. Note. If the greatest common divisor of more than 2 numbers is required, find the common divisor of two of them first; then of that common divisor, and one of the other numbers, and so on. 6. What is the greatest common divisor of 144, 192, and 456? This Rule is also useful in finding a common divisor to divide the terms of a fraction by, in order to reduce them to their lowest terms. 7. Find the greatest common divisor of the terms of the fraction 109, and by it reduce the fraction to its lowest terms. 8. Reduce to its lowest terms. 152 9 Ans. 1 12÷3×1=4 new numerator, written over the 12= 12÷4×39 new numerator, written over the 12: 12÷6x5=10 new numerator, written over the 12-19. 2. Reduce,, and, to their least common denomiAns. 1813 15 nator. 12 6 3. Reduce, and, to their least common denomi nator. Ans. 6 45 4. Reduce 2, §, and of %, to their least common denominator. Ans. 1848. 5. Reduce,,, and, to a common denominator by Rule I. then reduce them by Rule II. 4500 6750 Ans. by Rule I, 22000 22500 220000 By Rule II, 3 30 30 30. PROBLEM III. To reduce fractions of higher denominations into those of lower denominations. RULE. Multiply the numerator of the given fraction, by the common parts of its own integer, and under the product, write the denominator; or make a compound fraction, by comparing the given fraction with all the denominations between it and the denomination you would reduce it to; then reduce the compound fraction to a simple one. EXAMPLES. 1. Reduce of a pound to the fraction of a penny. Operation. X 20 20 12 numerator 240 then 10 2. Reduce 3. Reduce 12 4. Reduce Ans. Or, by comparing the given fraction with the several denominations, and making a compound fraction, it will stand thus, of 20 of 2, then yo x20x12, and 328-3 of a penny, the answer as before. 543 240 of a pound to the fraction of a shilling. Thus, TX20-60s. Ans. of a pound, to the fraction of a farthing. Ans.. of a hogshead, to the fraction of a gallon. Ans. gal. 5. Reduce of a guinea, to the fraction of a penny. Ans. d. 6. Reduce 1920 of a pound Troy, to the fraction of a pwt. Ans. Zpwt. 7. Reduce 3520 of a mile, to the fraction of a rod. Ans. rod. 8. Reduce 10% of a cwt. to the fraction of a pound. 9. Reduce Ans. 18lb. of a day, to the fraction of a minute. Ans. 4m. 10. Reduce of a guinea, to the fraction of a pound. Compounded thus, of 28 of PROBLEM IV. 112= Ans. To reduce fractions of lower denominations, into those of higher denominations. RULE. or Multiply the denominator of the given fraction, by the common parts of an integer of the required fraction, for a new denominator, over which write the numerator; make a compound fraction by comparing the given fraction with the denominations between it and the one you would reduce it to, then reduce this compound fraction to a simple one. EXAMPLES. 1. Reduce of a penny, to the fraction of a pound. Operation. denominator 4 Or by comparing the given fraction with the several denominations and making a compound fraction, it will stand thus 2 of 1 of 20=980 and 980-320 an swer as before. 2. Reduce of a shilling to the fraction of a pound. 3. Reduce of a farthing to the fraction of a pound. Ans. 180. Ans. 1280 4. Reduce gallon to the fraction of a hogshead. Ans. 146. 5. Reduce of a penny, to the fraction of a guinea. Ans. 10. 6. Reduce of a pwt. to the fraction of a pound Troy. Ans. 1920 Ans. 3520 7. What part of a mile, is of a rod? Ans. 1064 9. Reduce of a minute, to the fraction of a day. Ans. 5040 10. Reduce of a £ to the fraction of a guinea. Compounded thus, of 20 of 10 Ans. 4. 80 ADDITION OF VULGAR FRACTIONS. RULE. Reduce compound fractions to single ones, and all of them to their least common denominator (Rule 2, page 169) then the sum of the numerators written over the common denominator, will be the sum of the fractions required. EXAMPLES. 90 63 Add together 123, 95, and 7 of 3. Thus, 7 of 21 then 3, 5, and 21, reduced to a common denominator, by Rule 2, Problem II. are equal to 120, 120, 120, and the sum of the numerators 90+ 100+ 63=253, which written over the common denominator, will be 253 which reduced to a mixed number, is equal to 2, then the whole numbers 12+9+22=23120 Ans. 120' 2. What is the whole amount of 7 and 8 yards? 3. Add together,, and . yards, 135 yards, Ans.29 20yds. Ans. 12 Ans. 17. h Ans. 31. Ans. 48. Note. In adding mixed numbers that are confpounded with other fractions, reduce them first to improper fractions, and all of them to a common denominator, by Rule 2, page 196, then add as before. |