RULE. Write the numbers perpendicularly under each other, in order, with the least at the top. Divide each denomination by that number which it takes to make one of the next greater, and place the quotient at the right hand of the denomination next greater than that divided, beginning with the least and proceeding in order to the greatest. The last quotient will be the required decimal. III. Find the value of a decimal in the terms of an integer or whole number. What is the value of .264583 of a pound? 264583 20 5.291660 12 3.499920 4 1.999688 This is the reverse of the preceding case. In dividing 3.5 by 12 we obtained the decimal .29166; but there was still a remainder, 8, by annexing ciphers to which and then dividing by 12, would always leave the same remainder. Such a decimal is called a running decimal. But the value of it, after the decimal has been carried to 5 or 6 figures, is so small, that it may be neglected. Hence the answer in this example is less than the given sum in the preceding, by a small part of a farthing. Ans. 5s. 3 d. 1.99968 qrs. RULE. Multiply the decimal by that number which it takes of the next less denomination to that in which the decimal is given. From the product cut off as many figures as there are in the given decimal for a remainder. Multiply the remainder and cut off as before through all the denominations. The figures cut off at the left hand are the answer, as seen in the preceding example. SECTION XVI. VULGAR FRACTIONS. Though Fractions properly denote parts of a unit, sometimes the term is used to denote numbers larger than unity. Vulgar Fractions, as before stated, are used to denote various irregular divisions of a unit. The lower number is called the denominator, because it shows into how many equal parts the unit is divided; and the upper number is called the numerator because it shows how many of those parts are taken, or denoted by the fraction. Let a dollar be divided into 8 equal parts; and one of these be given to A, 3 to B, and 4 to C; then A has, B , and C of a dollar. 8)239 (297 In performing division we often find a remainder, the value of which is fractional. For instance, in dividing 239 by 8,7 remain undivided. In this example it will be perceived that 29 times 8 have been taken from 239, and 7 remain undivided; and as we cannot divide it by 8, we may express its value by placing it for the numerator of a fraction, the denominator of which is 8, the divisor; thus, i. e. 79 7 seven eighths of another 8, besides the 29 times 8; for 8 units of the dividend are just equal to one of the quotient; then 7 of the dividend are equal to 7 eighths of 1 in the quotient; and the quotient in this case therefore, is 293. is a fraction, the denominator of which, 8, represents the number of parts into which each of the quotient figures must be divided in order to produce the dividend, and the numerator, 7, expresses the number of these parts which are contained in the remainder, after the 29 times 8 have been taken from the dividend. Let a dollar be divided into 10 equal parts; they will of course be dimes. of a dollar then is 10 cents, and 30 cents, &c. Hence, 20 cents; If the numerator of a fraction be increased, while the denominator remains the same, the value of the fraction is proportionally increased; and the reverse. Again let the fraction of a dollar be expressed thus ; this is only equal to 5 cents; and to only 4. Hence If the denominator of a fraction be increased, the value of the fraction is decreased in the same proportion. Most of the operations in fractions depend on these two principles. = Again take of a dollar 10 cents. numerator and denominator by the same Multiply both the number, 2 for in stance: 2. Here we have 2 twentieths of a dollar, which are equal to 1 tenth, for a twentieth is one half of one tenth. 50 Hence if both the numerator and denominator be multiplied, or divided by the same number, the value of the fraction will not be altered. Fractions of the same value may therefore be expressed in an infinite number of ways: 1=4=8=18=%, &c. In each of these the numerator is just half of the denominator. The most simple of the above fractions is, and the others may be reduced to it by dividing both terms by any number that will divide both without a remainder. In reducing fractions to their most simple terms, it is very convenient to know the largest number which will divide both terms without a remainder. What is the greatest number that will divide both terms of the fraction,, without a remainder ? 143)637(4 572 65 It is evident that any number can be divided by itself; but by no larger number; and the quotient in such cases is always 1. If then 637 the denominator can be divided by 143 the numerator, this is the greatest common measure of the fraction. But 637 divided by 143 leaves65 remainder. Every common measure of 143 and 637 will also measure 65; for 637 = 143 × 4 + 65. As 65 is the greatest measure of itself, the common measure of 65 and 143 cannot be greater than 65. But 14365 leaves 13 remainder; hence 65 is not the common measure sought. Every common measure of 65 and 143 must also measure 65 and 13, for 143=65 × 2 +13. The common measure of 65 and 13 cannot be greater than 13. We therefore ascertain whether 13 will measure 65, and find it will. 13, therefore, is the greatest common measure sought. The whole operation may be shown together, thus: RULE. To find the greatest common measure of any two numbers, divide the greater by the less, and the less by the remainder of the first division; then this remainder by the remainder of the second division, then this second remainder by the third, and so on till nothing remains. The last remainder used as a divisor will be the common measure required. If the greatest common measure of more than two numbers be required, find the greatest common measure of two of them as before; then of that common measure and one of the other numbers, and so on through the whole. The greatest common measure last found will be the one sought. It is very convenient also, in many instances, to know the least common multiple of two or more numbers, that is, the least number that can be divided by each of those numbers without a remainder. What is the least common multiple of 4, 6, 8, and 10? 4) 4 6 8 10 2) 1 6 2 10 1 3 1 5 As 4 X 6 X 8 X 10 = 1920, this sum is evidently a common multiple of each 4X2X1X3X1X5=120, Ans. of these numbers, but 8 is a multiple of 4 (the first divisor in the above examples ;) therefore we may take the quotient of 8 divided by 4=2; and 4 × 6 × 2 × 10=480, which is also a multiple of 4, 6, 8, and 10. Also 10 is a multiple of 2, (the second divisor,) and 6 of 2 likewise; therefore 3 × 5 × 2 × 4 = 120, can be measured by 4, 6, 8, and 10, and is the least number that can be measured by them. Hence we derive the following RULE. To find the least common multiple of two or more numbers, divide them separately by any number that will divide two or more of them without a remainder, and set the quotients and the undivided numbers also in a line below. Again divide as before, and so on till no two numbers can be divided by any common divisor. Then multiply all the undivided numbers by all the divisors, and the product will be the required least common multiple, as seen in the example. REDUCTION OF Vulgar FRACTIONS. Reduction of Vulgar Fractions is the bringing of them from one form or denomination into another, to prepare them for the operations of addition, subtraction, &c. I. Reduce 288 to its lowest terms. As dividing the two terms of a fraction by the same number does not alter its value, we may here divide the 288 and 480 by any number that will divide both without a remainder. 12 Thus 8)28888=3, Ans. We first divide by 8, and the quotients by 12 and obtain. No number will divide 3 and 5 without a remainder; hence are the lowest terms of the fraction Or we may find the greatest common measure of the numbers, thus: To reduce fractions to their lowest terms, divide both terms by any number that will divide them without a remainder; divide the quotients in the same manner and so on till no number will divide them without a remainder. The last quotients will be the terms sought. In the first instance we divided by 8 and 12, which gave the answer; in the second, after finding the greatest common measure 96, (in the manner before explained,) we divided both terms by it, which gave the same result, as must necessarily be the case, for 8 × 12=96. Every number which terminates in 0, 2, 4, 6, or 8, can be successively divided by 2. Every number terminating in 0, or 5, is divisible by 5. |