EXAMPLES X. b. 1. Find a number such that the sum of its sixth and ninth parts may be equal to 15. 2. What is the number whose eighth, sixth, and fourth parts together make up 13? 3. There is a number whose fifth part is less than its fourth part by 3: find it. 4. Find a number such that six-sevenths of it shall exceed fourfifths of it by 2. 5. The fifth, fifteenth, and twenty-fifth parts of a number together make up 23: find the number. 6. Two consecutive numbers are such that one-fourth of the less exceeds one-fifth of the greater by 1: find the numbers. 7. Two numbers differ by 28, and one is eight-ninths of the other; find them. 8. There are two consecutive numbers such that one-fifth of the greater exceeds one-seventh of the less by 3: find them. 9. Find three consecutive numbers such that if they be divided by 10, 17, and 26 respectively, the sum of the quotients will be 10. A and B begin to play with equal sums, and when B has lost five-elevenths of what he had to begin with, A has gained £6 more than half of what B has left: what had they at first? 10. 11. From a certain number 3 is taken, and the remainder is divided by 4; the quotient is then increased by 4 and divided by 5 and the result is 2: find the number. 12. In a cellar one-fifth of the wine is port and one-third claret: besides this it contains 15 dozen of sherry and 30 bottles of spirits. How much port and claret does it contain? 13. Two-fifths of A's money is equal to B's, and seven-ninths of B's is equal to C's: in all they have £770, what have they each? 14. A, B, and C have £1285 between them: A's share is greater than five-sixths of B's by £25, and C's is four-fifteenths of B's: find the share of each. 15. A man sold a horse for £35 and half as much as he gave for it, and gained thereby ten guineas: what did he pay for the horse? 16. The width of a room is two-thirds of its length. If the width had been 3 feet more, and the length 3 feet less, the room would have been square; find its dimensions. 17. What is the property of a person whose income is £430, when he has two-thirds of it invested at 4 per cent., one-fourth at 3 per cent., and the remainder at 2 per cent.? 18. I bought a certain number of apples at three a penny, and five-sixths of that number at four a penny; by selling them at sixteen for sixpence I gained 3d.; how many apples did I buy? CHAPTER XI. HIGHEST COMMON FACTOR. LOWEST COMMON MULTIPLE. SIMPLE EXPRESSIONS. 88. DEFINITION. The highest common factor of two or more algebraical expressions is the expression of highest dimensions [Art. 10] which divides each of them without remainder. The abbreviation H. C. F. is sometimes used instead of the words highest common factor. 89. In the case of simple expressions the highest common factor can be written down by inspection. Example 1. The highest common factor of a1, a3, a2, a6 is a2. Example 2. The highest common factor of a3b4, ab5c2, a2bc is ab1; for a is the highest power of a that will divide a3, a, a2; b4 is the highest power of b that will divide b1, b5, b7; and c is not a common factor. 90. If the expressions have numerical coefficients, find by Arithmetic their greatest common measure, and prefix it as a coefficient to the algebraical highest common factor. Example. The highest common factor of 21a4x3y, 35a2x1y, 28a3xy1 is 7a2xy; for it consists of the product of (1) the numerical greatest common measure of the coefficients; (2) the highest power of each letter which divides every one of the given expressions. 91. DEFINITION. The lowest common multiple of two or more algebraical expressions is the expression of lowest dimensions which is divisible by each of them without remainder. The abbreviation L. C. M. is sometimes used instead of the words lowest common multiple. 92. In the case of simple expressions the lowest common multiple can be written down by inspection. Example 1. The lowest common multiple of aa, a3, a2, a¤ is ao. Example 2. The lowest common multiple of a3b4, ab5, a2b7 is a3b7; for a3 is the lowest power of a that is divisible by each of the quantities a3, a, a2; and b7 is the lowest power of b that is divisible by each of the quantities b1, b5, b7. 93. If the expressions have numerical coefficients, find by Arithmetic their least common multiple, and prefix it as a coefficient to the algebraical lowest common multiple. Example. The lowest common multiple of 21a4x3y, 35a2x1y, 28a3xy1 is 420a1x4y1; for it consists of the product of (1) the numerical least common multiple of the coefficients; (2) the lowest power of each letter which is divisible by every power of that letter occurring in the given expressions. EXAMPLES XI. c. Find both the highest common factor and the lowest common 94. IN this Chapter we propose to deal only with the easier kinds of fractions, where the numerator and denominator are simple expressions. Their reduction and simplification will be performed by the usual arithmetical rules. The proofs of these rules we reserve for a later chapter where the subject of fractions will be treated more fully. 95. RULE. To reduce a fraction to its lowest terms: divide numerator and denominator by every factor which is common to them both, that is by their highest common factor. Dividing numerator and denominator of a fraction by a common factor is called cancelling that factor. MULTIPLICATION AND DIVISION OF FRACTIONS. 96. RULE. To multiply algebraical fractions: as in Arithmetic, multiply together all the numerators for a new numerator, and all the denominators for a new denominator. Example 1. 2a 5x2 3b2 2a × 5x2 × 3b2 5х X 3b 2a2b 2x 3b × 2a2b × 2x 2a' by cancelling like factors in numerator and denominator. |