55. Solution of problems. The essential step in solving a problem by algebra is the expression of the conditions of the problem by algebraic symbols. This is, in fact, nothing else than a translation of the problem from the English language into the language of algebra. The translation should be made as close as possible, clause by clause in most cases. In general the result sought should be represented by the variable, which for that reason is often called the unknown quantity. EXAMPLE. What number is it whose third part exceeds its fourth part by sixteen? Solution: "What number is it " is translated by x. Thus we let x represent the number sought. "Whose third part" is translated by "Exceeds its 3 fourth part" is translated by х i.e. the third part less the fourth part 9 3 leaves something. 66 By sixteen" gives us the amount of the remainder. Thus the translation of the problem into algebraic language is Let x represent the number sought. This equation should be solved and checked by the methods already given. PROBLEMS 1. What number is it whose third and fifth parts together make 88? 2. What number increased by 3 times itself and 5 times itself gives 99? 3. What is the number whose third, fourth, sixth, and eighth parts together are 3 less than the number itself? 4. What number is it whose double is 7 more than its fourth part? 5. In 10 years a young man will be 3 times as old as his brother is now. The brother is 7 years old. How old is the young man? 6. A father who is 53 years old is 3 years more than 12 times as old as his son. How old is the son ? 7. If you can tell how many apples I have in my basket, you may have 4 more than, or, what is the same thing, 4 less than of them. How many have I? 8. If Mr. A received more salary than at present, he would receive $2100. How much does he receive? 9. A boy spends of his money in one store and of what remains in another, and has 24 cents left. How much had he? 10. A man who is 3 months past his fifty-fifth birthday is 4 times as old as his son. How old is the son? 11. In a school are four classes. In the first is of all the pupils; in the second,; in the third, ; in the fourth, 37. How many pupils are in the school? 12. A merchant sold to successive customers, 1, and of the original length of a piece of cloth. He had left 2 yards less than half. How long was the piece? 13. How may one divide 77 into two parts of which one is 21⁄2 times as great as the other? 14. The sum of two numbers is 73 and their difference is 15. What are the numbers? 15. A father is 4 times as old as his son. Father and son together are 27 years younger than the grandfather, who is 71 years old. How old are father and son? 16. The sum of two numbers is 999. If one divides the first by 9 and the second by 6, the sum of these quotients is 138. What are the numbers ? 17. The first of two numbers whose sum is a is b times the second. What are the numbers ? 18. If the city of A had 14,400 more inhabitants, it would have 3 times as many as the city of B. Both A and B have together 12,800 more than the city of C, where there are 172,800 inhabitants. How many are in A and B ? 19. Two men who are 25 miles apart walk toward each other at the rates of 3 and 4 miles an hour respectively. After how long do they meet ? 20. A courier leaves a town riding at the rate of 6 miles an hour. Seven hours later a second courier follows him at the rate of 10 miles an hour. How soon is the first overtaken ? 21. A can copy 14 sheets of manuscript a day. When he had been working 6 days, B began, copying 18 sheets daily. How many sheets had each written when B had finished as many as A? 22. The pendulum of a clock swings 387 times in 5 minutes, while that of a second clock swings 341 times in 3 minutes. After how long will the second have swung 1632 times more than the first? 23. The difference in the squares of two numbers is 221. Their sum is 17. What are the numbers? 24. If a book had 236 more pages it would have as many over 400 pages as it now lacks of that number. How many pages has the book? 25. A man is now 63 years old and his son 21. When was the father 19 times as old as his son? 26. If 7 oranges cost as much less than 50 cents as 13 do more than 50 cents, how much do they cost apiece? 27. The numerator of a fraction is 6 more than the denominator. Diminish both numerator and denominator by 1 and the fraction equals. Find the fraction. 28. The sum of three numbers is 100. The first and second are respectively 9 and 7 greater than the third. What are the numbers? 29. Out of 19 people there were as many children as women, and 11⁄2 times as many men as women. How many were there of each ? 30. A boy has twice as many brothers as sisters. His sister has 5 times as many brothers as sisters. How many sons and daughters were there? 31. A dealer has 5000 gallons of alcohol which is 85% pure. He wishes to add water so that it will be 75% pure. How much water must he add? 32. How much water must be added to 5 quarts of acid which is 10% full strength to make the mixture 81% full strength? 33. A merchant estimated that his supply of coffee would last 12 weeks. He sold on the average 18 pounds a week more than he expected, and it lasted him 10 weeks. How much did he have? 34. At what time between 3 and 4 o'clock are the hands of a clock pointing in the same direction? 35. At what time between 11 and 12 o'clock is the minute hand at right angles to the hour hand? 36. A merchant bought cloth for $2 a yard, which he was obliged to sell for $1.75 a yard. Since the piece contained 3 yards more than he expected, How many yards actually in the piece? he lost only 2%. 37. A man has three casks. If he fills the second out of the first, the latter is still full. If he fills the third out of the second, the latter is still full. The second and third together hold 100 quarts less than the first. How much does each hold? 38. A crew that can cover 4 miles in 20 minutes if the water is still, can row a mile downstream in the time that it can row the mile upstream. How rapid is the stream? 39. A cask is emptied by three taps, the first of which could empty it in 20 minutes, the second in 30 minutes, the third in 35 minutes. How long is required for all three to empty the cask? 40. A can dig a trench in the time that B can; B can dig it in the time that C can; and A and C can dig it in 8 days. How long is required by all working together? |