ner; and so on; till it appears that there is no number greater than 1 which will divide them; then the fraction will be in its lowest terms. Or, Divide both the terms of the fraction by their greatest common measure, and the quotients will be the terms of the fraction required, of the same value as at first. EXAMPLES. 1. Reduce to its least terins. 140 = 7% = #8 = £8 = &, the answer. To reduce a mixed number to its equivalent improper fraction. RULE.* Multiply the whole number by the denominator of the fraction, and add the numerator to the product; then set that sum above the denominator for the fraction required. 9. Ali prime numbers, except 2 and 5, have either 1, 3, 7, or 9, in the place of units; and all other numbers are composite, or can be divided. 2 =5+4-2=7 10. When numbers, with the sign of addition or subtraction between them, are to be divided by any number, then each of those numbers must be divided by it. Thus, 10+8-4 11. But if the numbers have the sign of multiplication between them, only one of them must be 10 X 4 X 3 10 X 4 X 1 divided. Thus, 10 × 8 × 3 6X1 2 X 1 6 X 2 = 10 X 2 X 1 20 = This is no more than first multiplying a quantity by some number, and theu dividing the result back again by the same, which it is evident does not alter the value: for any fraction represents a division of the numerator by the denominator To reduce an improper fraction to its equivalent whole or mixed number. RULE.*—Divide the numerato by the denominator, and the quotient will be the whole or mixed number sought. To reduce a whole number to an equivalent fraction, having a given denominator. RULE.-Multiply the whole number by the given denominator, then set the product over the said denominator, and it will form the fraction required. EXAMPLES. 1. Reduce 9 to a fraction whose denominator shall be 7. Here 9 X 7 : 63, then 63 is the answer. For 636379, the proof. 2. Reduce 13 to a fraction whose denominator shall be 12. 3. Reduce 27 to a fraction whose denominator shall be 11. * This rule is evidently the reverse of the former; and the reason of it is manifest from the nature of Common Division. + Multiplication and Division being here equally used, the result must be the same as the quantity first proposed. CASE V. To reduce a compound fraction to an equivalent simple one. RULE.*—Multiply all the numerators together for a numerator, and all the denominators together for the 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. To reduce fractions of different denominators to equivalent fractions, having a common denominator. RULE.-Multiply each numerator into all the denominators except its own for the new numerators; and multiply all the denominators together for a co.nmon 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 be reduced, by their proper rules, to the form of simple fractions. • The truth of this rule may be shown as follows: Let the compound fraction be 3 of §. Now of is÷3, which is; consequently of will be 2 or 1; that is the numerators are multiplied together, and also the denominators, as in the rule.-When the compound 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. + This is evidently no more than multiplying each numerator and its denominator by the same quantity, and consequently the value of the fraction is not altered. Therefore the equivalent fractions are 14 14, and 12. 16 Or the whole operation of multiplying may be very well performed mentally, and only set down the results and given fractions thus:,,, = 14, 14, 14 12, 12, 12, by abbreviation. 35 Ans., 3. Ans. 40, 60, ỗ• 36 45 2. Reduce and to fractions of a common denominator. Ans. 78 3, 38, 120. 30. Note 1. 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. 2. When the less denominator of two fractions exactly divides the greater, multiply the terms of that which hath the less denominator by the quotient. 3. When more than two fractions are proposed; it is sometimes convenient, first to reduce two of them to a cominon denominator; then these and a third ; and so on till they be all reduced to their least common denominator. CASE VII. To find the value of a fraction in parts of the integer. RULE.-Multiply the integer by the numerator, and divide the product by the denominator, by Compound Multiplication and Division, 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 Compound Division, or the valuation of remainders in the Rule of Three, before explained. To reduce a fraction from one denomination to another. RULE.* 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, or the denominator, if to a greater. 4. Reduce of a farthing to the fraction of a pound. 5. Reduce cwt. to the fraction of a lb. 7. Reduce § of a crown to the fraction of a guinea. Ans. 32d. Ans. 1440 Ans. 32 Ans. Ans. 25 168. Ans. 2. ADDITION OF VULGAR FRACTIONS. To add fractions together that have a common denominator. RULE.—Add all the numerators together, and place the sum oyer the common denominator, and that will be the sum of the fractions required. If the fractions proposed have not a common denominator, they must be reduced to one. Also compound fractions must be reduced to simple ones; and mixed numbers to improper fractions; also fractions of different denominations to those of the same denomination.† This is the same as the rule of Reduction in whole numbers, from one denomination to another. ↑ 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 these 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. |
