1 - 25. (+4) - (20 - 2) = (5 - 1) - Da - 13) + 8. ¦ x 18 37. Find the value of x which makes the two expressions (3x-1)(4x-11) and 6(2x-1)(x-3) equal. 38. What value of x will make the expression 77x-3(2x-1)(4x − 2) equal to 337-8(3x − 1)(x + 1) ? CHAPTER X. SYMBOLICAL EXPRESSION. 82. IN solving algebraical problems the chief difficulty of the beginner is to express the conditions of the question by means of symbols. A question proposed in algebraical symbols will frequently be found puzzling, when a similar arithmetical question would present no difficulty. Thus, the answer to the question "find a number greater than x by a" may not be selfevident to the beginner, who would of course readily answer an analogous arithmetical question, "find a number greater than 50 by 6." The process of addition which gives the answer in the second case supplies the necessary hint; and, just as the number which is greater than 50 by 6 is 50+6, so the number which is greater than x by a is x+a. 83. The following examples will perhaps be the best introduction to the subject of this chapter. After the first we leave to the student the choice of arithmetical instances, should he find them necessary. Example 1. By how much does x exceed 17? Take a numerical instance; "by how much does 27 exceed 17?" - Similarly the defect of x from 17 is 17 - x. Example 2. If x is one part of 45 the other part is 45 - x. Example 3. If x is one factor of 45 the other factor is How far can a man walk in a hours at the rate of Example 4. 4 miles an hour? In 1 hour he walks 4 miles. In a hours he walks a times as far, that is, 4a miles. 20 Example 5. If $20 is divided equally among y persons, the share of each is the total sum divided by the number of persons, or $ Example 6. Out of a purse containing $x and man spends z quarters; express in cents the sum left. and $x=4x quarters, y half-dollars=2y quarters; .. the sum left=(4x+2y-z) quarters, =25(4x+2y-z) cents. y half-dollars a 5. By what must 5 be multiplied to make a? 6. What is the quotient when 3 is divided by a? 7. By what must 6x be divided to get 2? 8. By how much does 6x exceed 2x ? 9. The sum of two numbers is x and one of the numbers is 10; what is the other? 10. The sum of three numbers is 100; if one of them is 25 and another is x, what is the third ? 11. The product of two factors is 4x; if one of the factors is 4, what is the other? 12. The product of two numbers is p, and one of them is m; what is the other? 13. How many times is x contained in 2y? 14. The difference of two numbers is 8, and the greater of them is a; what is the other? 15. The difference of two numbers is x, and the less of them is 6; what is the other? 16. What number is less than 30 by y? 17. The sum of 12 equal numbers is 48x; what is the value of each number? 18. How many numbers each equal to y must be taken to make 15xy? 19. If there are a numbers each equal to 2a, what is their sum? 20. If there are 5 numbers each equal to x, what is their product? 21. 22. cost? If there are x numbers each equal to p, what is their product? 23. If n books of equal value cost x dollars, what does each cost? 24. How many books each worth two dollars can be bought for y dollars? 25. If apples are sold at x for a dime, what will be the cost in cents of y apples? 26. What is the price in cents of n oranges at six cents a score? 27. If I spend n dimes out of a sum of $5, how many dimes have I left? 28. What is the daily wage in dimes of a man who earns $12 in p weeks, working 6 days a week? 29. How many days must a man work in order to earn $6 at the rate of y dimes a day? 30. If x persons combine to pay a bill of $y, what is the share of each in dimes? 31. How many dimes must a man pay out of a sum of $p so as to have left 30x cents? 32. How many persons must contribute equally to a fund consisting of $x, so that the subscription of each may equal y quarters? 33. How many hours will it take to travel x miles at 10 miles an hour? 34. How far can I walk in p hours at the rate of 9 miles an hour? 35. If I can walk m miles in n days, what is my rate per day? 36. How many days will it take to travel y miles at x miles a day? 84. We subjoin a few harder examples worked out in full. Example 1. What is (1) the sum, (2) the product of three consecutive numbers of which the least is n? The two numbers consecutive to n are n+1 and n + 2; And the product .. the sum = n + (n + 1) + (n+2) =3n+3. = n(n+1)(n+2). Example 2. A boy is x years old, and five years hence his age will be half that of his father: how old is the father now? In five years the boy will be x +5 years old; therefore his father will then be 2(x+5), or 2x + 10 years old; his present age must therefore be 2x+10-5 or 2x+5 years. Example 3. A and B are playing for money; A begins with $p and B with q dimes. B wins $x; express by an equation the fact that A has now 3 times as much as B. What B has won A has lost; ... A has p-x dollars, that is 10(p-x) dimes, B has q dimes +x dollars, that is q+10x dimes. Thus the required equation is 10(p-x)=3(q+10x). Example 4. A man travels a miles by coach and b miles by train; if the coach goes at the rate of 7 miles an hour, and the train at the rate of 25 miles per hour, how long does the journey take? The coach travels 7 miles in 1 hour; Example 5. How many men will be required to do in p hours what q men do in mp hours? mp hours is the time occupied by q men ; 1. Write down three consecutive numbers of which a is the least. 2. Write down four consecutive numbers of which b is the greatest. 3. Write down five consecutive numbers of which c is the middle one. 4. What is the next odd number after 2n − 1 ? 5. What is the even number next before 2n? 6. Write down the product of three odd numbers of which the middle one is 2x+1. 7. How old is a man who will be x years old in 15 years? 8. How old was a man x years ago if his present age is n years? 9. In 2x years a man will be y years old, what is his present age? |