BOOK II, CHAP. I. OF FRACTIONS. 1. 1A' DEFINITIONS. FRACTION is a part or parts of Unity (representing any whole which may be divided.) 2. A Fraction is expressed by two numbers placed one above the other, with a line drawn between them, as, ; the number above the line is called the Numerator, and the number below the line the Denominator. c.g. 1 3 11 Numerator, 2 5 12 Denominator, one-half, three-fifths, eleven-twelfths. which are read thus, 3. The Numerator denotes how many parts of the Whole the Fraction consists of. 4. The Denominator denotes how many parts the Whole is divided into. 5. A proper Fraction is that whose Numerator is less than its Denominator, as, 2. 3 6. An improper Fraction is that whose Numerator is equal to, or greater than its Denominator, as,, 4: 7. A compound Fraction is a Fraction of a Fraction, as of 2, or of 1% of 2'0. 8. A whole number with a Fraction annexed, is called a mixed number, as 23, 14. 9. A Fraction is said to be in its least terms, when it is expressed by the least numbers possible. Quest. What is a Fraction? A. A part or parts of one whole Thing. 2. How is a Fraction expressed? A. By A. By two numbers placed one above the other, with a line drawn between them. 2. What are these numbers called? A. That above the line is called the Numerator, and that below the Denominator. 2. What is a proper Fraction? A. That whose Numerator is less than the Denominator. 2. What is an improper Fraction? A. That whose Numerator is equal to, or greater than the Denominator. 2. What is a compound Fraction? A. A Fraction of a Fraction, as of 2. 2. What is a mixt Number? A. A whole number with a Fraction annexed, as 23. 1. What is the greatest common measure of 112 and 120? Answ. 8. 2. What is the greatest common measure of 26 and 62? Answ. 2. 3. What is the greatest common measure of 279 and 403? Answ 31. 2nd Preparatory Problem. To find the least common multiple of any given numbers. Their product, if each prime to th' rest, Fxamples. What is the least multiple of 3, 5, 8 and 10? 5)-3, 5, 8, 10 I survey my given numbers and discover 5 will divide two of them; viz. 5 and 10 which I divide by 5, bringing into a line with the quotients the numbers, which 5 will not measure, Again, I view the numbers in the second line, and find 2 will measure 8 and 2, and these I divide by 2, and in the third line get 3, 1, 4, 1, all prime, I multiply the numbers in the said line toge ther with the divisors continually into each other for the number sought, and find it 120. 4. What is the least number which 3, 4, 8 and 12 will measure? Answ. 24. 5. What number is the least that 7, 8, 16 and 28 will measure? Answ. 112. 6. What is the least number which 5, 6, 12 and 16 will measure? Answ. 240. Problem I. To reduce a fraction to its least terms. Rule. Find the greatest common measure of the numerator and denominator of the given fraction, and divide them there by, by, the quotients will be the least term required, viz. the quotient found by dividing the numerator will be the numerator, and the quotient of the denominator the denominator of the fraction required. Take any common measure of the numerator and deno minator of the given fraction, and divide them thereby, making the quotients respectively the numerator and denominator of the new fraction, which divide in like manner, and so proceed till the terms be prime to each other, and the thing proposed is manifestly effected. Note. If the numerator and denominator be both even, 2 is a common measure to them. If one be odd (if they have any) their common measure is some odd number, as 3, 7, 11, &c. If both have for their lowest figure 5, or one 5, and the other a cypher, 5 is a common measure; and if both have cyphers in units place, &c. cut off the cyphers, 10, 100, &c. being a common measure. Bring the following fractions to their lowest terms, viz. Rule II. To change or reduce a given fraction to another which shall be equal thereto, and have a given denominator. Rule. Multiply the given denominator by the numerator `of the given fraction, and divide the product by the denominator thereof, and the quotient will be the numerator of the fraction sought. Otherwise thus: Divide the given denominator by the denominator of the given fraction, and multiply the quotient by the numerator. Examples. 21. Reduce to a fraction whose denominator shall be 15? Answ 22. Bring to a fraction whose denominator shall be 15? Answ.. 23. Reduce both tors shall be 112? 24. Reduce to a fraction whose denominator shall be 100? Answ. 75 Töö 25. Reduce to a fraction whose denominator shall be 100J? Answ. 625 Problem III. To reduce any given fractions to others, which shall have one common denominator. Rule. 1. Find the least number which all the denominators of the given fractions will measure, for a common denominator. 2 Reduce each fraction to another whose denominator shall be the said common denominator, Examples |