DERIVATION OF FRACTIONS. A fraction is derived from division. If 15 dollars be equally divided between 4 men, how many dollars will each man receive ? 4) 1 5( 33 Each man will receive 3 dollars. 1 2 3 By taking 4 dollars from the whole number, we take away dollars enough to give each man one ; and by putting 1 in the quotient every time we take away 4 from the whole number of dollars, the quotient will show how many dollars each man will receive. By working the question, we find that 4 can be taken 3 times from the 15, and 3 remains. In 12 dollars, the number we have already subtracted, there are enough to give each man 3. As the number of dollars left is not equal to the number of men, there will not be another dollar for each man, but each will receive a part of another dollar. If we suppose the 3 dollars divided into quarters, the number would be 12, and each man would receive 3 quarters from the 12 quarters, in the same manner that he received 3 dollars from the 12 dollars. 4, the divisor, shows how many units of the dividend it requires that we may place an unit in the quotient; therefore, by making the divisor a denominator to a fraction, the 4 will show the number of units required to be taken from the dividend, to express one unit in the quotient. The numerator 3, which is the remainder, shows how many units we have left from the dividend, towards making another unit in the quotient. Again, let it be required to divide 19 by 12. 1 2)1 9 (111 1 2 In this example, it takes 12 in the dividend to make 1 in the quotient. We may suppose 1, the whole number in 7. the quotient, to be so much larger than the units in the dividend, that it requires 12 units in the dividend to make one unit as large as that in the quo. tient. By setting 12 as a denominator, it shows how many of the small units in the dividend are required to make one unit in the quotient. The units contained in 7 are just as large as those contained in 19, the dividend, and are considered just as large as those in 12, the divisor; therefore, if we place 7 for a numerator, it will show how many small units we have towards making another large unit in the quotient. But if it take 12 units in the dividend to make 1 in the quotient, an unit in the dividend is onetwelfth part of an unit in the quotient, and the 7 must be 7 twelfths of an unit in the quotient. Let the divisor be any number whatever, it will always require as many units in the dividend to equal one in the quotient, as there are ones in the divisor; therefore, the divisor will always be the proper number to show how many small units like those in the remainder, it will require to make an unit in the quo . tient; and the numerator being the remainder, will always denote the number of those small units we have. It may here be observed, that the units in the quotient are not always larger than those in the dividend, but as all operations in division are performed in the same manner as if they were, it is proper to consider them as such, in explaining the subject; and the scholar will understand them much better in this than in any othér. As we make use of the same figures in fractions as in whole numbers, it is difficult, many times, to explain the different operations in them, without using the word unit. For the sake of distinction, therefore, we shall call the parts of an unit denoted in a fraction, fractional units. li may not at first appear to the learner, how a fraction, standing by itself, arises from division. If it were required to divide one dollar equally between 3 men, the one dollar becomes the dividend, and the way, 3 men the divisor. The divisor being larger than the dividend, we cannot actually divide, but the quotient is properly expressed by writing the divisor under the dividend, thus, . Had the dividend been as large as the divisor, the quotient would have been 1, but as it is less, it is the same as a remainder, and by setting the divisór under the 1, the fraction shows what part of a dollar each man would receive. From what has been said we obtain the following, GENERAL PRINCIPLES OF FRACTIONS. 1. The denominator to a simple fraction is always equal to an unit in the whole number to which the fraction belongs. 2. The denominator and numerator are always of the same denomination; that is, a fractional unit in the denominator is always equal to a fractional unit in the numerator, 3. The value of a fraction is determined by dividing the numerator by the denominator. METHOD OF READING FRACTIONS. The numerator and denominator are read in the same manner as simple numbers, the numerator first. 17 is read, one hundred and twenty-seven, three hundred and fifty-fourths ; and His read twenty-five, three hundred and twenty-sixths. VULGAR FRACTIONS. DEFINITIONS. A Prime Numberis a number which cannot be divid. ed by any number without a remainder, except by itself or an unit, as 5, 11, 13, &c. A Multiple is a number which can be divided by some other number without a remainder; and is call ed a multiple, because it can be produced by multiplying two other numbers together. 9 is a multiple of 3. A Common Multiple of two or more numbers, is any number which can be divided by each of those numbers without a remainder. Ifit be the least number that can be divided by the numbers, it is called the least common multiple ; thus, 54 is a common multiple of 6, 9 and 18, but 18 is their least common multiple. A Common Divisor, or Common Measure, is a number that will divide two or more numbers without a remainder. If it be the greatest number that will divide without a remainder, it is called the greatest common divisor, or measure. PROBLEM IS To find the least common multiple of sederal nun bers. RULE. 1. Divide the given numbers by any number that will divide two or more of them without a remainder, and set the quotients and the numbers which have not been divided in a line below. 2. Divide the numbers brought down as before, and so continue to do until there are no two numbers that can be divided by one number without a remainder. Multiply all the divisors, the last quotients, and the last undivided numbers together, the last product will be the least common multiple of the given numbers. 1. What is the least number that can be divided by 8,12 or 5, without a remainder? Illustration. If one number can be divided by another without a remainder, and that number be multiplied by any simple number, the product can be divided by the same divisor without a remainder. 6 can be divided by three without a remainder. 6 be multiplied by 4, the product 24 can be divided by 3 without a remainder, for 6 is made up of two 3's, and every time we put 6 into the product, we put in a certain number of 3's. The product of a divisor multiplied by a quotient, can always be divided by the dividend, for the product will equal the dividend. 8, one of the given numbers in the last question, will divide the first product 8. 3 is tho quotient of 12 divided by 4, and 8 the first product is two 4's. Now if we multiply 8 by 3, the product is two 12's. 24, the second product, is made up of acertain number of 8's, or 12's ; 120 is made up of a certain number of 24's or 5's. But as 24 is made up af a certain number of 8's, or 12's, 120 must be; therefore 120 the continued product of the divisor, quotients, and undivided number, can be divided by 8, 12, or 5, without a remainder. 120 is the least number that can be divided by the three given numbers without a remainder, for if we divide one of the under numbers by itself, we must multiply by that divisor, which would produce the same product ; and if we could divide one of the under numbers by some other number, multiplying by |