Six cows for 4 months are the same as 24 cows for one month, for 6X4=-24 ; and 12 cows for 3 months are the same as 36 cows for 1 month, for 12x3=36. Hence the question is the same as if A had put in 24 cows for 1 month, and B 36 cows for 1 month. 6x4=24 12 X3=36 60 : 24 :: 24: 9,60 A's. Sum of the products 60 60 : 24 :: 36 : 14,40 B's. 2. Three merchants enter into a partnership; B puts in $300 for 4 months, C $150 for 5 months, and D $200 for 8 months. They gain $400; what is each man's share ? B's. D's. 3. Three persons hire a pasture in company for which they pay $96. C put in 100 sheep for 2 months, D put in 300 sheep for 5 months, and E 450 sheep for 4 months , tyhat must each pay ? C. D. E. 4. Two men, B and C, gained by trading $250. B's stock was $350 for 8 months, C's $640 for 5 months; required cach man's share of the gain? B's share C's share. 5. Three persons received $665 interest. B's principal was $4000 for 12 months, C's $3000 for 15 months, and D's $5000 for 8 months; what is each man's share ? Ans. B's $240. C's $225. D's $200. 6. B and C enter into partnership for 16 months. B at first put in $500 and at the end of 12 months he took out $300; C at first put in $300, and at the end of eight months he put in $500 more. They gained $525; required each man's particular share of the gain? B's $228,841 C's $296,1515 7. A commenced trade on the first of January with $850, and on the first of March he took in B with $560, and on the first of May he took in C with $650 ; at the end of the Ans. { year they find they have gained $950; what must each man share of the gain? Ans. {B $153,33 C $235,231 Questions. What is Fellowship? What is Compound, or Double FelWhat is Single Fellowship? lowship? What is the Rule for Single Fellow What is the Rule for Double Fellow ship? ship? VULGAR FRACTIONS: 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 trial 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 denomi. nator, 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 We divide the greatest 91)117,(1 number by the least, and the 91 remainder is 26; therefore, 26)91(3 91 is not a factor of 117; 78 then 26 in 91 will go 3 times, and 13 remains. Hence 26 13)26(2 is not a factor of 91 ; then 26 13 will go in 26, 2 times, and nothing remains. Hence, 13 is the greatest common divisor of 91 and 117, or the greatest number that 91 and 117 can be divided by, without a reinainder. 2. What is the greatest common measure of 48 and 56? Ans. 8. 3. What is the greatest common divisor of 132 and 144? Ans. 12. 4. What is the greatest common divisor of 148 and 236? Ans. 4. 5. What is the greatest common divisor of 1224 and 1080 ? 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 ? Thus, 144)192(1 48=greatest cominon divi144 sor of 144 and 192. Then 48)456(9 48)144(3 432 144 24)4812 48 Ans. 24. 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 19, and by it reduce the fraction to its lowest terms. 8. Reduce to its lowest terms. Ans. Ans. it 3 9. Reduce 96 to its lowest terms. 544 5136 Ans. i PROBLEM II. To find the least common multiple of two or more numbers. RULE. 1. Divide by any nu ber that will divide two or more of the given numbers without a remainder, and set the quotients, together with the undivided numbers, in a line beneath. Divide this line as before, and so on until no two of the numbers can be divided by the same divisor; the continued product of all the divisors and numbers in the last line, will be the least common multiple required. EXAMPLES. 2) 2 1. What is the least common multiple of 2, 4, 3 and 6 ? Operation. In dividing the numbers 2, 4 3 6 4, 3 and 6 by 2, we find that 2 3) 1 2 3 3 is not contained in 3 even ; 1 2 1 1 therefore we bring the 3 down with the quotients. We then Ans. 2 X3 X2=12 divide again by 3, and bring down the 2, which 3 will not divide, and the divisors and quotients are factors, which multiplied together produce a common multiple ; and since these factors are expressed in the least numbers, it is evident that it is their least common multiple. 2. What is the least common multiple of 6 and 8 ? Ans. 24. 3. What is the least common multiple of 6 and 18 ? Ans. 18. 4. What is the least common multiple of 15 and 20 ? Ans. 60, 5. What is the least number that 3, 5, 8 and 10 will measure? Ans. 120. 6. What is the least number that can be divided by the 9 digits separately without a remainder ? Ans. 2320 REDUCTION OF VULGAR FRACTIONS. PROBLEM I. To reduce a compound Fraction to a simple Fraction. If of an orange be divided equally among 3 boys, what part of the whole orange will one boy receive ?- of l is equal to what part of 1 ?-Thus, x}=1'a. Illustration. If i be divided into three equal parts, it will take 12 such parts to make a whole one; therefore of is 1a, Answer. RULE. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator, and they will form the fraction required, which reduce to its lowest terms. EXAMPLES. Ans. 33 3 11 15 123 276 13 1. How much is of ?-Thus, i x =35, Ans. If į be divided into 4 equal parts, it will take 32 such parts to make a whole one. Hence, 1 of } is zz; then of is 5 times as much Hence, 1 of is, s, and of is 3 times as much, that is =ž, Ans. 2. Reduce } of į to a simple fraction. Ans. 4. Reduce of of }} to a simple fraction. Ans. 55 5. Reduce of jof to a simple fraction. Ans. 1&g. 6. How much is of 5] ? Thus, 5=u and ox}=li, or 2, Ans. 7. How much is 3 of 223? Note. If the denominator of a fraction be equal to the numerator of another in a compound fraction, they may both be dropped, and the continual multiplication of the other members will produce the fraction in lower terms. 8. Reduce of of is to a simple fraction. Thus, 3 XFXY==zy, Ans. 9. Reduce } of of į of iî of 5) to a simple fraction. Ans. 77= 10. Reduce { of of 12; to a simple fraction. Ans. H=41 Ans. 88 |