8 Vulgar Fractions are either proper, improper, single, compound or mixed. 1. A proper fraction is when the numerator is less than the denominator : as, 4, 5, 5, &c. which mean of 1, of 1, f of 1, &c. 2. An improper fraction is when the numerator exceeds the denominator : as, g, UI, &c. 3. A single fraction is a simple expression for any number of parts of the integer. 4. A compound fraction is the fraction of a fraction : as, 1 of 3, of , &c., which mean off of 1, of of &c. 5. A mixed number is composed of a whole number and a fraction; as, 8}, 1213, &c. 3 a Note.-Any number may be expressed like a fraction .-by writing 1 under it: Thus į means 6 ones or 6. A fraction having a fraction or mixed number for its numerator or denominator, or both, is called a complex fraction. A fraction denotes division, and its value is equal to the quotient obtained by dividing the numerator by the denominator : thus is equal to 3, and 20 equal to 4. Therefore, if the numerator be less than the denominator, the value of the fraction is less than 1. If the numerator be the same as the denominator, the fraction is just equal to 1. And if the numerator be greater than the denominator, the fraction is greater than 1. 6. The common measure of two or more numbers is that number which will divide each of them without a remainder ; and the greatest number that will do this, is called the greatest common measure. 7. A number which can be measured by two or more numbers, is called their common multiple ; and if it be the least number, which can be so measured, it is called their least common multiple. . more PROBLEM 1. numbers. Rule.-1. If there be two numbers only, divide the greater by the less, and this divisor by the remainder, and so on; always dividing the last divisor by the last remainder, until nothing remains; then will the last divisor be the greatest common measure required. 2. When there are more than two numbers, find the greatest common measure of two of them, as before ; and next find the greatest common measure of that common measure and one of the other numbers, and so on, through all the numbers to the last ; then will the greatest common measure last found be the answer. 3. If one happen to be the common measure, the given numbers are prime to each other, and found to be incommensurable. EXAMPLES 1. Required the greatest common measure of 918,1998 and 522. 918)1998(2 So 54 is the greatest common measure of 1998 and 918 162)91815 Hence 54)522(9 810 486 108)1621 103 36)54(1 36 54)108(2 108 19)36(2 36 Therefore 18 is the answer required. 2. What is the greatest common measure of 612 and 540? Ans. 36. 3. What is the greatest common measure of 117 and 91 ? Ans. 13. PROBLEM 2. To find the least common multiple of two or more numbers. RULE.-1. Divide by any number, that will divide two or more of the given nunbers without a remainder, and. set the quotient, together with the undivided numbers, in a line beneath. 2. Divide the second line as before, and so on, until there are no two numbers that can be divided ; then the continued product of the divisors and quotient will give the multiple required. EXAMPLES, 1. What is the least common multiple of 3, 5, 8, and 10? 2)3 5 8 10 5)3 5 4 5 3 14 1 Then 2x5x3x4=120 Ans. 2. What is the least common multiple of 9, 8, 15, 16 ! Ans. 720. 3. What is the least number that 3, 4, 8, and 12 will measure ? Ans. 24. 4. What is the least number that can be divided by the 9 digits without a remainder Ans. 2520. REDUCTION OF VULGAR FRACTIONS. REDUCTION OF VULGAR FRACTIONS is the bringing them out of one form into another, in order to prepare them for the operations of addition, subtraction, &c. CASE I. To abbreviate or reduce fractions to their lowest terms. RULE.-Divide the terms of the given fraction by any number that will divide them without a remainder, &c. as in Rule of Problem 1, page 76. Or divide both the terms of the fraction by their greatest common measure, and the quotients, will be the terms of the fraction required. If a fraction have ciphers on the right hand of both its terms, it may be reduced by cutting off an equal number froin both, EXAMPLES. 1. Reduce 344 to its lowest terms. (4) (3) (4) 1==i= the answer. Or thus, 144)24001 144 96)144(1 Therefore 48 is the greatest 48)96(2 common measure, and 96 48)244= Ans. Note.-1. Any number ending with an even number, or a cipher, is divisible by 2. 2. Any number ending with 5 or 0, is divisible by 5. 3. If the right hand place of any number be 0, the whole is divisible by 10. 4. If the two right hand figures, of any number are divisible by 4, the whole is divisible by 4. 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. All prime numbers, except 2 and 5, have 1, 3, 7, or 9, in the place of units; and consequently all other numbers are composite, and capable of being divided. 7. When numbers with the sign of addition or subtraction between them, are to be divided by any number, each of the numbers must be divided. Thus, 4+8+10 =2+4+5=11 2 8. But if the numbers have the sign of multiplication between them, only one of them must be divided. Thus, 3x8x10 X3 X4x10 1x4x10 1x2x10 =4=20 2x6 1x6 1 x2 1x1 9. If both the numerator and denominator of a fraction be multiplied or divided by the same number, the fraction will still retain its original value. Let and be two fractions proposed; then x-%; and 2-*=*. That is, if the numerator 4, and denominator 5, of the first fraction, be each multiplied by the same number 2, the produced fraction & is equal to the proposed one . For the numerator and denominator of the produced fraction, are in the same proportion as the numerator and denominator of the proposed one. Also, if the numerator 9, and the denominator 12, of the second fraction, be each divided by the same number 3, the fractions f and are equal for the same reason. To reduce a mixed number to its equivalent improper fraction. Rule.-Multiply the whole number by the denominator of the fraction, and add the numerator to the product; then that sum written above the denominator, will form the fraction required. EXAMPLES. 1. Reduce 273 to its equivalent improper fraction. 27 243 2 245 Or 27 x 9 +2 --=245 the answer. 9 9 2 Reduce 51415 to an improper fraction. Ans. 8439, 3. Reduce 121 to an improper fraction. Ans. 412. 4. Reduce 7915 to an improper fraction. Ans. 14f3. 5. Reduce 1001% to an improper fraction. Ans. 1992. 19 |