EXAMPLES. For 63 1. Reduce 9 to a fraction whose denominator shall be 7. 63 = 7 = 9, the Proof. Ans. 3. Reduce 27 to a fraction whose denominator shall be 11. Aos. ? 15 G CASE V. To Reduce a Compound Fraction to an Equivalent Simple One: MULTIPLY all the numerators together for a numerator, and all the denominators together for a denominator, and they will form the simple fraction sought. When part of the compound fraction is a whole or mixed number, it must first be reduced to a fraction by one of the former cases. And, when it can be done, any two terms of the fraction may be divided by the same number, and the quotients used instead of them. Or, when there are terms that are common, they may be omitted, or cancelled. EXAMPLES. 1. Reduce of of 2 to a simple fraction. 1 X2 X3 6 1 Here the Answer. 2 X 3 X 4 24 4 1 X2 X3 1 Or, by cancelling the 2's and 3's. 2X 8 X 4 4 The truth of this Rule may be shown as follows: Let the compound fraction be of . Now fof=3, which is consequently of $ will be tix l'or that is the numerators are multiplied together, and also the denominators, as in the Rule. When the com. pound fraction consists of more than two single ones ; having first reduced two of them as above, then the resulting fraction and a third will be the same as a compound fraction of two parts; and so on to the last of all. 2. Reduce Vol. I. I Ог, , 3. Redure {of of ji to a simple fraction. 2 X 3 X 10 60 12 4 Here the Answer. .3 X 5 X 11 165 33 11 2x8 x 18 4 the same as before, by cancelling 8 * 5x11 11 the 3's, and dividing by 5's. 3 Reduce of 10 a simple fraction. 4. Reduce of of to a simple fraction. 5. Reduce off of 3} to a siniple fraction. 6. Reduce jobs of 1 of 4 to a simple fraction. 7. Reduce 2 and off to a traction. Ans. 11 Ans, Ans. Ansa Ans. . CASE VI. . To Reduce Fractions of Different Denominators, to Equivalent Fractions having a Common Denominator. * MULTIPLY each numerator by all the denominators except its own, for the new numerators : and multiply all the denominators together for a common denominator. Note, It is evident that in this and several other operations, when any of the proposed quantities are integers, or mixed numbers, or compound fractions, they must first be reduced, by their proper Rules, to the form of simple fractions. EXAMPLES. 1. Reduce }, ş, and is to a common denominator. 1 X 3 X 4 = 12 the new numerator for } di to ditto 2 X 3 X 4 24 the common denominator. Therefore the equivalent fractions are , and Or the whole operation of multiplying may be best perförmed mentally, only setting down the results and given fractions thus : , , $ = 44, s, it = s is by abbreviation. 2. Reduce and to fractions of a common denominator. Ans. 35 * This is evidently no more than multiplying each numerator and its denominator by the same quantity, and conseqdently the value of the fraction is not altered. 3. Reu ce 3. Reduce } }, and }, to a common denominator. Ans. 46 36 45 4. Reduce , 23, and 4 to a common denominator. Ans. 36, 38. 38 Nore I. When the denominators of two given fractions have a common measure, let them be divided by it ; then multiply the terms of each given fraction by the quotient arising from the other's denominator. Eras, and = it's and it's, by multiplying the former by 7, and the latter by 5. 2. When the less denominator of two fractions exactly divides the greater, multiply the terms of that which has the less denominator by the quotient. Ex and this = 1% and o's, by mult. the former by 2; 3. When more than two fractions are proposed, it is sometimes convenient, first to reduce two of them to a common denominator ; then these and a third ; and so on till they be all reduced to their least common denominator. Ex. į and and } = { and f and į = land and 31. CASE VII. To find the value of a Fraction in Parts of the Integer. MULTIPLY the integer by the numerator, and divide the product by the denominator, by Compound Multiplication and Divison, if the integer be a compound quantity. Or, if it be a single integer, multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator. Then, if any thing remains, multiply it by the parts in the next inferior denomination, and divide by the denominator as before ; and so on as far as necessary; so shall the quotients, placed in order, be the value of the fraction required* The numerator of a fraction being considered as a remainder, in Division, and the denominator as the divisor, this rule is of the same nature as Componnd Division, or the valuation of remainders in the Rule of Three, before explained. EXAMPLES. EXAMPLES. 1. What is the of 21 68 ? 2 What is the value of of 12? By the former part of the Rule By the 2d part of the Rule, 21 68 2 20 3. Find the value of of a pound sterling. Ans. 78 6d. 4. What is the value of į of a guinea ? Ans. 48 8d. 5. What is the value of of half crown ? Ans. 18 10 d. 6 What is the value of of 48 100 ? Ans. 1$ 11 d. 7. What is the value of Ib troy? Ans. 9 oz 12 dwis. 8. What is the value of its of a cwt ? Ans. I qr 7 lb 9. What is the value of of an acre ? Ans 3 10. 20 po. 10. What is the value of jo of a day? Ans. 7 hrs 12 min. CASE VIII. To Reduce a Fraction from one Denomination to another. * CONSIDER how many of the less denomination make one of the greater; then multiply the numerator by that number, if the reduction be to a less name, but multiply the denominator, if :o a greater. EXAMPLES. 1. Reduce of a pound to the fraction of a penny.. * X XY = 480 160, the Answer. * This is the same as the Rule of Reduction in whole nnmbers from one denomination to another. 2. Reduce 2. Reduce of a penny to the fraction of a pound. { x = yg the Answer. 3. Reduce gl to the fraction of a penny. Ans. d. 4. Reduce q to the fraction of a pound. Ans rito o " Ans. 31. 5. Reduce cwt to the fraction of a lb. Ib 6. Riduce dwt to the fraction of troy. Ans. 7. Raduce crown to the fraction of a guinea. 8. Reduce half-crown to the fract. of a shilling. Ans. 21 9. Reduce 28 6d to the fraction of a £.. Ans. 10. Reduce 178 7d 3'q to the fraction of a £. a Ans. aoo ADDITION OF VULGAR FRACTIONS. If the fractions have a common denominator ; add all the numerators together, then place the sum over the common denominator, and that will be the sum of the fractions required. If the proposed fractions have not a common denominator, they must be reduced to one. Also compound fractions must be reduced to simple ones, and fractions of different denominations to those of the same denomination. Then add the numerators as before. As to mixed numbers, they may either be reduced to improper fractions, and so added with the others; or else the fractional parts only added, and the integers united afterwards. Before fractions are reduced to a common denominator, they are quite dissimilar, as much as shillings and pence are, and therefore cannot be incorporated with one another, any more than there can. But when they are reduced to a common denominator, and made parts of the same thing, their sum, or difference, may then be as properly expressed by the sum or difference of the numerators, as the sum or difference of any two quantities whatever, by the sum or difference of their individuals whence the reason of the Rule is manifest, both for Addition and Subtraction, When several fractions are to be collected, it is commonly best first to add two of them together that most easily reduce to a common denominator; then add their sum and a third, and so on. EXAMPLES. |