3. A compound fraction is the fraction of a fraction, coupled by the word, of, thus ; of, of off 4. A mixed number is composed of a whole number and a fraction joined together, thus ; 84, 162, 193, &c. A whole number may be expressed as a fraction by drawing a line under it and putting 1 for a denominator ; thus, 6=f, and 12=1, &c. PROBLEM I. To reduce Fractions to their lowest terms. The numerator and denominator together are called terms of a fraction; and may often be changed without altering the value of a fraction; thus, take an orange, or any thing, and divide it into 2 equal parts, and 1 of these parts will be of the orange: again, if we divide it into 4 equal parts, it is evident that 2 of those parts (3) will be just of the orange : and if we divide it into 8 equal parts, 4 of these parts (f) will be just equal to of the orange : and the fractions 1, 1, and , are all equal in value, but expressed in different terms. Hence the terms of fractions may be changed without altering the value of the fraction ; for if we multiply both the terms of the fraction by 2 it becomes, which is equal to 1 : again, if we divide the terms by 2, the fraction will be }, which is expressed in its lowest terms possible. RULE. 1. Divide the terms of the fraction by any number that will divide them both without a remainder. 2. Divide these quotients again in the same manner, and 80 on, until no number greater than 1 will divide them: EXAMPLES. Thus =, and 5) =}, the answer. Ans. 264 3. Reduce 432 to its lowest terms. Ans. 4. Reduce 1 47 to its lowest terms. Ans. 5. Reduce to its lowest terms. Ans. 343 6. Reduce 168 to its lowest terms. Ans: 7. Reduce to its lowest terms. 168 Ans. It 4 1 8. Reduce 168 to its lowest terms. 252 328 Ans.. Ans. Ans. Ans. 9 PROBLEM II. To change a Whole or Mixed Number to an Improper Fraction. RULE. Multiply the whole number by the denominator of the fraction and to the product add the numerator; this sum written over the denominator will form the fraction required. EXAMPLES. Ans. 405. 1. In 27 dollars how many fourths of a dollar ? Operation. $1=4 fourths of a dollar, 273 and 27 dollars=27 times 4 7.4=fourths in 1 dollar. or 108 fourths, and 3 fourths 108=fourths in 27 dollars. added to 108 fourths make +3=fourths in ili fourths=112 the Ans. 111 =-fourths= Ans. 111. 2. In 36 dollars, how many eighths of a dollar ? Ans. 223 3. Reduce 45 to ninths. 4. Reduce 8á to an improper fraction, that is, reduce it to sixths. Ans. 53 5. Reduce 33} to an improper fraction. Ans. 10. 6. Reduce 28 to a fraction having 12 for a denominator, that is, reduce it to twelfths. Ans. 336 12. 7. Reduce 45% to fifths. Ans.20. 8. Reduce 619,5 to an improper fraction. Ans. 8 635 140. 9. Reduce 84 to an improper fraction. Ans. 933 11 10. What improper fraction is equal to 5648 ? Ans. 2250 11. What improper fraction is equal to 148%? . 10 12. What improper fraction is equal to 2253 ? Ans. 1693. Ans. 1489 PROBLEM III. To change an Improper Fraction to a Whole or Mixed Number. RULE. Divide the nunterator by the denominator, and the quotient will be the value of the fraction. EXAMPLES. 1. In 4 of a dollar, how many dollars ? Operation. of a dollar åre equal to 1 dollar, and $1=6)45 6 is contained in 45, 7 times and of another time; therefore the answer is 7 dollars=71 dollars. 2. Find the value of 1.go of a cent. Ans. 33 cts. 3. Find the value of 249 of a cwt. Ans. 494cwt. 4. Reduce 5 to a mixed number. Ans. 86. 5. Reduce 405 to a whole number. Ans. 45 6. Reduce to a whole number. Ans. 28. 7. Find the value of 450. Ans. 1778 8. Find the value of $835. Ans. 61 95 140 RULE. 1. Divide the whole number by the denominator of the fraction, (when it can be done without a remainder,) and multiply the quotient by the numerator ; or, II. Multiply the whole number by the numerator of the fraction and divide the product by the denominator. EXAMPLES. 1. What is the product of 48 multiplied by ? 1st mothod. 2d method. 4)48 48 12 is of 48 3 3 4) 144=3 times 48 Ans. 36=3 times of 48, 36 is of 3 times 48, which is 1 of 48. which is the same as of 48 By this example we see, there are two ways of multiplying a whole number by a fraction, and that both methods produce the same result. Thus, by the first method, we get of 48, and this repeated 3 times is evidently equal to ,, for 3 times of any number is equal to 1 of that number. By the second method, we repeat 48, 3 times, and then take of that product, which is the same as 3 times of 48. 2. At 25 dollars per acre, what is the cost of já of an acre of land ? Ans. 231.dolls. 3. If a ship sail 246 miles a day, how far will she sail in z of a day? Ans. 191] miles. 4. How much is of $1845,56 ? Ans. $1537,96%. 5. Multiply 400 by s. Ans. 150. 6. Multiply 750 by š. Ans. 450 7. The interest of $750 for 1 year, is $45; what is the interest on the same sun for 5 months, or is of a year ? Ans. $18,75. Note. If the multiplier of any sum be greater than a unit or 1, the multiplicand will be increased as many times as the multiplier is greater than a unit; that is, the multiplicand will be taken as many times as the multiplier contains units. But when the multiplier is a fraction or part of a unit, the product will be only a part of the multiplicand. Hence in multiplying by, a proper fraction, the product is always less than the multiplicand, as will be seen by the preceding examples. PROBLEM V. To Multiply a Fraction by a Whole Number: RULE. Multiply the whole number and the numerator of the fraction together, and write the product over the denominator; and if it produce an improper fraction, change it to a whole or mixed number, by Prob. 3. EXAMPLES. 1. If a man spend á of a dollar a day, how much will he spend in 11 days? If he spend in 1 day, he will spend 11 times 5= in 11 days, and 5 of a dollar =91 dollars, the answer. 2. If 1 yard of cloth cost of a dollar, what will 15 yards cost ? Ans. $9. 3. If a bushel of oats cost fa of a dollar, what will 23 bushels cost? Ans. $975 4. A certain lot contains of an acre of land; how much land would 37 such lots contain ? Ans. 273 acres. 5. If a bushel of potatoes cost % of a dollar, what will 56 bushels cost? Ans. $164 Note. The process of multiplying a fraction by a whole number, may be shortened, thus: Divide the denominator of the fraction by the whole number, (when it can be done without a remainder,) and over the quotient write the numerator. 6. If a pound of sugar cost do of a dollar, what will 20lb. cost. Divide the denominator, 100, by 20, and the quotient 5, is a new denominator ; then write the numerator over it, and it becomes of a dollar=2} dollars, the answer. 7. If a pound of nails cost of a dollar, what will 11lb. cost? Ans. $1. 8. If a pound of butter cost of a dollar, what will 5lb. cost? Ans. $ 9. At of a dollar per pound; what will i1lb. raisins come to ? Ans. $ PROBLEM VI. To divide a Whole Number by a Fraction. RULE. Multiply the whole number by the denominator of the fraction, and divide the product by the numerator. EXAMPLES. 1. How many times is of a dollar contained in $9? 1 dollar is 4, and 9 dollars is 9 times as many; 9x4=; and is contained in 0 as many times as 3 is contained in 35. Ans. 12 2. How many times is a contained in 16 ? Thus, 16 x6=denominator. Numerator=5)96=sixths in 16 Ans. 191 |