SECTION VI.-Vulgar Fractions. The fractions of which we have already spoken in section. the 1st, are usually denominated Vulgar Fractions, to distinguish them from another kind, hereafter to be mentioned, called decimal fractions. A fraction is an expression for a part of a unit, or integer, when it represents a whole of any kind. Thus, if a pound sterling be the unit, then a shilling will be the twentieth part of that unit, and four pence will be four-twelfths of that twentieth part. These represented according to the usual notation of Vulgar Fractions, will be and of respectively. The lower number of a fraction thus represented (denoting the number of parts into which the integer is supposed to be divided) is called the denominator; and the upper figure (which indicates the number of those parts expressed by the fraction) the numerator. Thus, in the fractions,, 7 and 15 are denominators, 5 and 8 numerators. Vulgar fractions are divided into proper, improper, mixed, simple, compound, and complex. Proper fractions have their numerators less than their denominators, as,, &c. Improper fractions have their numerators equal to, or greater than, their denominators, as 4, 12, &c. Mixed fractions, or numbers, are those compounded of whole numbers and fractions, as 7, 123, &c. Simple fractions are expressions for parts of given units, as , &c. Compound fractions are expressions for the parts of given. fractions, as of, 4 of 7, &c. Complex fractions have either one or both terms mixed 5 12 6 numbers as &c. 24' 14' 12' Any number which will divide two or more numbers without remainder is called their common measure. Reduction of Vulgar Fractions. This consists principally in changing them into a more commodious form for the operations of addition, subtraction, &c. Case 1.-To reduce fractions to their lowest terms: Divide the numerator and denominator of a fraction by any number that will divide them both, without a remainder; the quotient again, if possible, by any other number: and so on, till 1 is the greatest divisor. Thus, 1470 2203 spectively are the divisors. Or, 14793, by dividing at once by 735. 2205 = Note. This number 735 is called the greatest common measure of the terms of the fraction: it is found thus-Divide the greater of the two numbers by the less; the last divisor by the last remainder, and so on till nothing remains the last divisor is the greatest common measure required.* Case 2. To reduce an improper fraction to its equivalent whole or mixed number. Divide the numerator by the denominator, and the quotient will be the answer: as is evident from the nature of division. Ex.-Let 257 and 5480 be reduced to their equivalent whole or mixed numbers. * The following theorems are useful for abbreviating Vulgar Fractions: THEOREMS. 1. If any number terminates on the right hand with a cipher, or a digit divisible by 2, the whole is divisible by 2: for the one which remains in the second place is 10; but 2 measures 10; therefore the whole is divisible by 2. 2. If any number terminates on the right hand with a cipher or 5, the whole is divisible by 5; for every unit which remains in the second place is 10; but 5 measures every multiple of 10; therefore the whole is divisible by 5. 3. If the two right hand figures of any number are divisible by 4, the whole is divisible by 4: for every unit which remains in the third place is 100; but 4 measures every multiple of 100; therefore the whole is divisible by 4. 4. If the three right-hand figures of any number are divisible by 8, the whole is divisible by 8: for every unit which remains in the fourth place is 1000; but 8 measures every multiple of 1000; therefore the whole is divisible by 8. 5. If the sum of the digits constituting any number be divisible by 3 or 9, the whole is divisible by 3 or 9. 6. If the sum of the digits constituting any number be divisible by 6, and the right-hand digit by 2, the whole is divisible by 6: for by the data it is divisible both by 2 and 3. 7. If the sum of the 1st, 3d, 5th, &c. digits constituting any number be equal to that of the 2d, 4th, 6th, &c. that number is divisible by 11: for if a, b, c, d, e, Case 3. To reduce a mixed number to its equivalent improper fraction; or a whole number to an equivalent fraction having any assigned denominator. This is, evidently, the reverse of Case 2; therefore multiply the whole number by the denominator of the fraction, and add the numerator (if there be one) to obtain the numerator of the fraction required. Ex.-Reduce 221 to an improper fraction, and 20 to a fraction whose denominator shall be 274. (22 × 43) + 11 = 957 new numerator, and 257 the first fraction. 274 20 × 274 = 5480 new numerator, and 5480 the second fraction. Case 4.-To reduce a compound fraction to an equivalent simple one. Multiply all the numerators together for the numerator, and all the denominators together for the denominator, of the simple fraction required. If part of the compound fraction be a mixed or a whole number, reduce the former to an improper fraction, and make the latter a fraction by placing 1 under the numerator. When like factors are found in the numerators and denominators, cancel them both. Ex.-Reduce of 3 of 4 of 7 of to a simple fraction. = 1 X 5 X 8 1 X 5 X 4 20 = 99 2 x 3 x 5 X 7 X 8 2 X 5 X 8 3 X 4 X 7 X 9 X 11 4 X 9 X 11 Here the 3 and 7 common to numerator and denominator are first cancelled; then the fraction is divided by 2; and then by 2 again. Ex.-Reduce three farthings to the fraction of a pound sterling. A farthing is the fourth of a penny, a penny the twelfth of a shilling, and a shilling the twentieth of a pound. Therefore of of= 3 = 3 the answer. Here, reducing the mixed numbers to improper fractions, we have multiplying by 3, to get quit of the denominator of 8 24 m, n, be the digits, constituting any number, its digits, when multiplied by 11, will the upper fraction, we have: multiplying by 5, to get quit of the denominator of the lower fraction, we have 49; dividing both terms of this fraction by 8, there results for the simple fraction required. Case 5. To reduce fractions of different denominators to equivalent fractions having a common denominator. Multiply each numerator into all the denominators except its own, for new numerators; and all the denominators together for a common denominator. Ex.-Reduce, , and, to equivalent fractions having a common denominator. 5 x 3 x 7= 105 3 X 7 X 9 = 189, the common denominator. 162 42 54 63 63 Hence the fractions are 128, 183, 185, or 43, 4, 3, when abbreviated. Hence, also, it appears that exceed, and that exceed 4. 44 Ex.-Reduce of a penny, and of a shilling, each to the fraction of a pound; and then reduce the two to fractions having a common denominator. of a pound. 10 of a pound. of of 20 = = = 1200 = 300 of a penny = Hence of a shilling are 10 times as much as of a penny. Note.-Other methods of reduction will occur to the student after tolerable practice, and still more after the principles of algebra are acquired. Addition and Subtraction of Fractions. RULE. If the fractions have a common denominator, add or subtract the numerators, and place the sum or difference as a new numerator over the common denominator. If the fractions have not a common denominator, they must be reduced to that state before the operation is performed. In addition of mixed numbers, it is usually best to take the sum of the integers, and that of the fractions, separately; and then their sum, for the result required. |