REDUCTION OF FRACTIONS TO THEIR MOST SIMPLE EXPRESSION. 92. A fraction is simplified by expressing, in a smaller number of figures, the value of its two terms. This simplification takes place when the numerator and denominator are divisible by the same number. Their value is not changed by this operation (89), which is conducted in the following manner. 93. Divide the numerator and denominator, each by 2, and repeat the division till it cannot be performed without a remainder. Then divide both the terms by 3, until they cannot be divided without a remainder. Continue the same operation by dividing both terms of the fraction successively by the numbers, 5, 7, 11, 13, 17, &c.; that is, by numbers which have no divisor but themselves. These numbers are called prime numbers. The only difficulty in these operations is to know when the terms of the fraction are divisible by 2, 3, 5, &c. But the following observations may be useful in removing it. 94. Every number terminated by a figure which expresses an even number of units, will be divisible by 2. Any number, the sum of whose figures added together as simple units, make 3 or a multiple of 3, that is, an exact number of times 3, will be divisible by 3. For example, 54231 is divisible by 3, because the sum of its figures 5,4, 2, 3, 1, is 15, which is 5 times 3. The same property belongs to the number 9, or to a multiple of 9. The reason of this rule is founded on a principle similar to that demonstrated in article (75). Every number, terminated by 5, or a cipher, is divisible by 5. Similar rules might be found for 7, and the numbers following. But as they require a long operation, it is better to use division. 2898* It is required to reduce the fraction 2018. We divide both the terms by 2, because the last figure in each term is an even number, and have 1008. We divide again by 2 and have 504 We next divide by 3, according to the rule given above, and have 168. We divide again by 3 and have 56 We now attempt to divide by 79 the division succeeds, and we have 1449 161 483 The reason why we have directed, that division should be attempted only by the prime numbers 2, 3, 5, 7, &c.," is, that having exhausted the division by 2, for instance, it would be useless to attempt it by 4, because 4 is a multiple of 2. 95. But the readiest method for reducing a fraction to its most simple expression, is to divide both its terms by their greatest common divisor. The rule for finding the greatest common divisor is as follows. Divide the greater of the two terms by the less, and if there is no remainder, the less term is the greatest common divisor. But if there be a remainder, divide the less term by it ; if the division is exact, the first remainder is the greatest common divisor. If the second division leave a remainder, divide the first remainder by the second; and continue thus to divide the preceding remainder by the last, until there is an exact division. Then the last divisor will be the greatest common divisor of the two terms of the fraction. When the last divisor is unity, it is a proof that the fraction cannot be reduced. For illustrating this rule take the fraction 37. We divide 9024 by 3760 and have 2 for quotient and 1504 for remainder; then 3760 by 1504 and 2 is the quotient and 752 the remainder The first remainder 1504 is then divided by the second remainder 752; te division is exact. We conclude therefore, that 752, is the greatest common divisor of the fraction 3760 Ou dividing both terms by it, the operation gives 2, the simplest expression of the fraction. 9024 5 And As 752 divides 1504, it ought also to divide $760, which is composed of twice 1504 added to 752. as 9024 is composed of twice 5760 added to 1504, it ought to be divisible by 752. It is also plain, that 752 is the greatest common divisor, which 3760 and 9024 can have; because there can be no divisor common to 9024 and 3760, which is not common to 3760 and 1504; and to these last there can be no common divisor, which is not common to 1504 and 752. And to the last there can be no greater divisor than 752. EXAMPLES FOR PRACTICE. 1. Reduce 144 to its simplest expression. 2. Reduce 4 to its simplest expression. 272 Ans. 43105 6. Reduce 6896800 as much as possible. 36700160 DIFFERENT WAYS OF CONSIDERING A FRACTION AND THE CONSEQUENCES WHICH FOLLOW. 96. The idea we have given of a fraction is, that the denominator shows the number of parts of which the unit is composed; and the numerator, the number of parts in the quantity, which the fraction expresses. 97. The numerator of a fraction may also be considere as a dividend, and the denominator as a divisor. Hence the reason why the remainders of division were put under the form given them in article (60). 98. From these considerations it follows, 1st. that a whole number may be put under the form of a fraction, by making the whole number the numerator, and giving to it unity for a denominator. Thus 8 or are the same thing; and so are 5 or, 10 or 10, &c. 99. 2d. Any fraction may be changed into decimals by considering the numerator as the remainder of a division in which the denominator was divisor, and operating according to the direction given in (page 59)-observing to put a cipher in place of units in the quotient. Thus, we find that 0,6 is the decimal expression for 3, 0,555 for 5, 0,84 for; and so on. On this principle is founded a rule for reducing complex numbers to decimals. For example, if it is required to reduce 3 fathoms, 5 feet, 8 inches, 7 lines, to the decimal of the fathom expressed in half-lines; we observe, that the fathom contains 864 lines, and cousequently H 1728 half-lines; therefore, in order to express the decimal value in half-lines, we must continue the operation to the fourth place of decimals. The 5ft. 8in. 71. are now reduced to lines; and we have 823 lines or 23 of a fathom. After reducing this fraction to decimals in the manner directed, we have 0,9525; and consequently 3,9525fa. for the proposed number. The following rule, however, is in more common use. "To reduce a number from a lower denomination to the decimal of a higher, we first change it, or suppose it to be changed, into a fraction having 10, or some multiple of 10, for its denominator, and divide the numerator by as many as make one of this higher denomination, and the quotient is the required decimal; which, together with the whole number of this denomination, may again be converted into a fraction, having 10 or a multiple of 10 for its denominator, and thus by division be reduced to a still higher name, and so on. EXAMPLES FOR PRACTICE. 1. Reduce 19s. 3d. 3qr. to the decimal of a pound. The operation is as follows. In this example, 3, the number of farthings, is considered as 30qr. which are reduced to hundredths of a penny by dividing by 4, the figure on the left, and the quotient 75 is placed as a decimal on the right of the pence; we then take this sum, considered as 375d. or 3758d. that is, annex as many ciphers as are necessary, and divide it by 12, which brings it into decimals of a shilling. Lastly, the shillings, and parts of a shilling, 19,3125s. considered as s. are reduced to decimals of a pound by dividing by 20, which gives the result above found. 19312500 2. Reduce 17pls. 1ft. 6in. to the decimal of a mile. 3. Reduce 9s. to the decimal of a pound. Ans.,45. 4. Reduce 19s. 5d. 2qr. to the decimal of a pound. Ans. ,972916. 5. Reduce 10oz. 18dwt. 16gr. to the decimal of a pound Troy. Ans. ,911111, &c. 6. Reduce 2qrs. 14. to the decimal of a cwt. Ans. ,625, &c. 7. Reduce 17yd. 1ft. 6in. to the decimal of a mile. Ans. ,00994318, &c. 8. Reduce Sqrs. 2nls. to the decimal of a yard. Ans. ,875. 9. Reduce 1gal. of wine to the decimal of a hogshead. Ans. ,015873. 10. Reduce Sbus. 1pk. to the decimal of a quarter. Ans. ,40625. 11. Reduce 10w. 2d. to the decimal of a year. Ans. ,1972602, &c. OPERATIONS OF ARITHMETIC UPON FRACTIONS. 100. The same operations are performed upon fractions as upon whole numbers. Addition and Subtraction often require a preparatory operation; Multiplication and Division require very little preparation. ADDITION OF FRACTIONS. 101. If the fractions have the same denominator, add all the numerators, and affix to their sum the common denominator. Thus, to add,, and, we add together 2, 3, and 5, the numerators, and have 10, which is reduced to 13, (85.) 102. If the fractions have not the same denominator, they must be reduced to a common denominator by following the rule given in articles (90) and (91.) Then add |