α

2

19. One number is a greater than another; and the sum of their second powers is b. What are the numbers?

2

20. Take the second power of each of these formulæ, square it, and add the two squares together; and, observe if the sum of their squares is equal to b; thus:

า

SECOND PART.

SECTION LXXXII.

Questions Solved, Using only One Letter to Represent the Unknown Number.

1. Said A to B, you have three times as much money as I have; and we have both together 555 dollars. How many dollars did each have?

2. One number is six times as many as another, and the sum of the numbers is 851. What are the numbers?

3. The difference of two numbers is 10; and twice one is equal to three times the other. What are the numbers? 4. The difference of two numbers is 30; and half of one is equal to one third of the other. What are the numbers? 5. The of a number exceeds its number?

by 20. What is the

6. The of a number exceeds its by 60. What is the number?

7. Three eights of a number exceeds of it by 11. What is the number?

8. A man, by spending 200 dollars more than of his income, finds himself at the end of 10 years, 1000 dollars in debt. What was his income?

9. A man who spends annually 50 dollars more than his income, finds by calculation, that if he can diminish his expenses, he will save annually, of his income, 50 dollars. is his income?

What

10. A is twice as old as B; if B were 10 years older, the difference of their ages would be 30 years. What are their ages?

11. There are two casks full of wine; one of which contains twice as much as the other. If 20 gallons were drawn

out of each, there would remain four times as much in the greater as in the less. How many gallons does each contain?

12. Two men had 100 dollars between them; one lost of his, the other of his; and, they then had 80 dollars between them. How many dollars had each at first?

13. The numerator of a fraction exceeds its denominator by 2; if the denominator were doubled, the value of the fraction would be. What is the fraction?

14. The value of a fraction is just ; but, if 4 be added to its numerator, and also to its denominator, the value will be 4, What are the numerator and denominator of the fraction?

15. A man, when he was married, was three times as old as his wife; after they had lived together some years, he was only twice as old as she was, and the sum of their ages was 90 years. How old were they when they were married; and, how long had they lived together when he was just twice as old as she was?

16. At what time, between 5 and 6 o'clock, will the hour and minute hands of a watch be together?

17. I once had 20 dollars; of which I gave away a certain sum, and had four times as much remaining as I gave away. How much did I give away?

18. A man lends a sum of money for 16 dollars per annum; of it at 6 per cent., and of it at 5 per cent. What was the sum ?

19. A man lends a sum of money for 44 dollars per annum; the sum at three per cent., at 4 per cent., and at 5 per What sum of money did he lend?

cent.

20. There are two numbers which make 11; and, the quotient of one by the other, gives 7. What are the numbers?

21. A man had 60 dollars, and lost a part of it; but, found he had three times as much left as he had lost. How much did he loose?

22. A man lost of his money; and then gained 20 dollars. He then lost of what he had, and afterward gained 10 dollars. Upon counting his money he found that he had just 40 dollars. What had he at first?

23. A man's annual expenses are just more than his income. If his income were 200 more than it is, he would

expend the same sum annually, and save 50 dollars. What is his annual income?

24. A man spent every year of his income; his income was afterward increased 500 dollars, without increasing his expenses, and he found that he saved just of it. What was his income?

25. A man has a certain sum of money given him; he spends of it the first year, the next year of what remains; upon making an estimate of his expenses for the two years, he found them to amount to 800 dollars. What was the sum given him?

26. A man being asked how many shillings he had; said, that of what he had added to of what he had, and 12 more, made 8 shillings more than the whole of what he had. How much had he?

27. A general, disposing of his army in a square form, finds that he has 284 men more than are necessary to complete the square. He increased the side of the square by one man, and found that he wanted 25 men to fill up the square. How many men had he?

28. A man had a number of workmen employed. He gave each as many shillings as there were workmen, and had 12 shillings left. If he had given them one shilling a-piece more, he would have had only 6 left. How many workmen were there?

29. A man with his wife and child dined together at an inn. The landlord charged one shilling for the child, and for the woman he charged as much as for the child and as much as for the man. How much did he charge for the man and his wife; supposing he charged as much for the man as he did for his wife and child both?

30. A commenced trading with B; A had 60 dollars and B 80. After several bargains, in which each lost and gained in turn, A found that he had six times as much as B had left. How much did B lose?

31. What number is that, which is such, that if you multiply it by 5, and add 6 to the product; then divide the sum by 2, and add 5 to the quotient, the result will be 40?

32. A certain number of men agreed to furnish a certain sum of money for a charitable purpose. They first contributed 10 dollars a-piece, and found that it wanted 10 dollars of