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 2205 spectively are the divisors. 1470 Or, 178, by dividing at once by 735. 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. 274 * 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 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 2211 to an improper fraction, and 20 to a fraction whose denominator shall be 274. (22 × 43) + 11 fraction. = 957 new numerator, and 957 the first 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 2 of 4 of 7 of to a simple fraction. 2 x 3 x 5 X 7 X 8 2 X 5 X 8 = 1 X 5 X 8 1 X 5 X 4 20 1 x 9 x 11-99 3 X 4 X 7 X 9 X11 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 1 904 = 1 = 320 the answer. Ex.-Simplify the complex fraction 43 Here, reducing the mixed numbers to improper fractions, we 8 have multiplying by 3, to get quit of the denominator of m, n, be the digits, constituting any number, its digits, when multiplied by 11, will become 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. spectively are the divisors. 2, where 5, 3, 7, 7, re Or, 147, by dividing at once by 735. 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. 274 * 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 2211 to an improper fraction, and 20 to a fraction whose denominator shall be 274. (22 × 43) + 11 = 957 new numerator, and 957 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. 4 X 9 X 11 T 1 X 5 X 4 20 1 x 9 x 11-99 3 X 4 X 7 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 23/3 44 Ex.-Simplify the complex fraction 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 become The embarrassment and loss of time occasioned by the computation of quantities expressed in vulgar or ordinary fractions, have inspired the idea of fixing the denominator so as to know what it is without actually expressing it. Hence originate two dispositions of numbers, decimal fractions and complex numbers. Of the latter, such, for example, as when we express lineal measures in yards, in feet (or thirds of a yard), and inches (or twelfths of a foot), we shall treat after a few pages. We shall now treat of the former. Decimal fractions, or substantively, decimals, are fractions expressed as whole numbers, but whose values decrease from the place of units progressively to the right hand in the same decuple or tenfold proportion as the common scale of whole numbers increase to the left. They are usually separated from the integers by a dot placed between the upper part of the figures. Thus, 227 expressed according to the decimal notation |