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RULE.-Square the sum of the numbers, and from that square, subtract 4 times the product, the square root of that difference will give the difference of the numbers, then agreeably to Theorem 1, by having the sum and difference we can easily find the numbers.
THEOREM 7. The sum, and sum of the squares of two numbers being given to find the numbers.
RULE.-Subtract the sum of the squares of the numbers, from the square of their sum, and half the remainder will be their product. When the sum and product are given, we can find the numbers agreeably to Theorem 6.
QUESTIONS FOR EXERCISE.
And both were very fair,
One round, the other square.
Each piece its value had;
For it exactly paid.
As it is very near,
She that had the round or square?
25090560 acres. Area of the circle 19706125.8240 acres. A. A bullet is dropped from the top of a building, and found to reach the ground in 2 seconds. Required its height? Ans. 2 X4 = 8 and 8 x 8
= 64 feet. B. Admitting I let fall a bullet from the top of a building, and found it reached the ground in 13 seconds. Required its height?
Ans. 36 ft. 2. If a roll of butter weighs in one scale 21 lbs.,
and being changed into the other weighs 4 lbs.; what is the true weight of the butter?
Ans. 3 lbs,
3. A farmer hired two men, Reuben and Richard, for $30, (being the wages of both for 1 month,) to Reuben he gave $4 per month more than to Richard. How much did each get per month?
Ans. Reuben got $17 and Richard $13. 4. There is a certain number which being divided by 11, the quotient resulting multiplied by 5, that product divided by 4, from the quotient subtract 75, to the remainder add 40, and half the sum shall make 45. Required the number?
Ans. 1100. 5. What is the discount of $400 for 2 years, 8 months, 24 days, at 6 per cent per annum? Ans. $56.36.
6. What is the interest of $400 for 2 years, 8 months, 24 days, at 6 per cent.?
Ans. $65.60. 7. A merchant in Baltimore received from New Orleans a bill at 30 days sight; he allowed 1 per cent. discount for present payment, and received $2530.44; what sun was the bill drawn for; and what was the discount?
Ans $2555.74 + disct. $25.30. 8. Bunker Hill monument is 30 feet square at its base, and is to be 15 feet square at its top, its height is to be 220, from the bottom to the top through its centre, is a cylindrical avenue 15 feet in diameter at the bottom, and about 11 feet at the top. How many cubic feet will there be in the monument?
Ans. 86068.518 + ft. 9. Divide 40 into 4 such parts, that if to the first you add 4, from the second subtract 4, multiply the third by 4, and divide the fourth by 4; the sum, difference, product and quotient will be equal? Ans. ,5, and it.
10. Find two numbers in a given ratio as 6 to:9, so that their sum and product may be equal? 24 and 15.
11. Two drovers, A and B, (from the state of Ohio) stopped at the “Three Tun Tavern," in the city of Baltimore, each having a drove of cattle; A had 21 head, in his drove, and B 19; they agreed to join both together, and sell them for $27 per head; and, that A should receive
$3 a head more than B; how much did each get for his cattle?
Ans. A got $596.921; B $483.071. 12. What number is it to which į of of j be added, the sum will be 1?
Ans. . 13. What number is it whose , ) and } added together make 78?
Ans. 72. 14. To find a number, which being multiplied by 3, subtracting 5 from the product and the remainder divided by 2, if the number sought be added to the quotient, the sum will be 40?
Ans. 17. 15. What number is it which being added to 4, and also multiplied by 4, the product shall be treble the sum?
Ans. 12. 16. To find a number, to which if 11 be added and 7 subtracted from the same number, the sum of the addition will be double the remainder:
Ans. 25. 17. To find a number which being added to itself and the sum being multiplied by the same, and the same number still subtracted from the product; and lastly, the remainder divided by the same, the quotient may be 13?
Ans. 7. 18. To find three numbers such, that the sum of the first and second shall be 15, the sum of the first and third 16, and the sum of the second and third 17?
Ans. 7, 8 and 9. 19. Three persons, A, B and C, owe certain sums of money, so that A and B owe $210, B and C 290, and C and A 400; what did each of them owe?.
Ans. A 160, B 50, and C 240. 20. A man having a certain number of dollars in his hand, being asked how many he had, replied if the number be divided by 5 and 19 added to the quotient, I shall then have $23; how many dollars had he? Ans. 20.
21. A person said he had 20 children, and that it happened there was a year and a half between each of their ages,
his eldest was born when he was 24 years old, and
Ans. 731 years.
the age of the youngest is now 21; what was the fathers age?
22. When I sold cloth at $7 per yard, I gained 56 cents; what would I gain by selling 3 pieces, which cost me $400?
Ans. $32.144. 23. A and B dissolve partnership, and equally divide their gain; A's share which was $332.50 lay for 21 months; B's for 9 months only; the adventure of B is required?
Ans. $775.833. 24. I bought 25 yards of lace at 30 cents for the first yard, and for the last yard 96; the price of each yard increasing in arithmetical progression; what did the whole amount to?
Ans. $15.75. 25. A case of dry goods amounting to £230 5s. sterling is sold in Philadelphia, at 20 per cent. advance; what is the amount in American currency? Ans. $1228.
26. What is the difference between the interest of $1000 at 6 per cent. for 8 years, and the discount of the same sum at the same rate, and for the same time?
Ans. the interest exceeds the discount by $155.671.
27. A gentleman sells a piece of cloth at $56.25, and thereby loses 71 per cent.; what was the first cost?
Ans. $60.47. 28. A has 28 cwt. of cotton which cost him $264; how must he rate it per lb., to D, so that by taking his note payable at 9 months he may clear $30, allowing interest at 6
Ans. 9 c. 29. In a series of proportional numbers the first is 5, the second is 8, the product of the second and third is 78.4, what is the difference of the first and fourth?
Ans. 10.68. 30. If 3481 soldiers are to be placed in square file, how many are to be set in each rank? Ans, 59.
31. A can do a piece of work in 10 days, B in 13, set them both at it, in what time will it be finished?
32. A stationer sold quills at $10 per thousand, by which he gained of the money, but growing scarce raised them to $12,50 per thousand, what is the gain per cent. by the latter price?
Ans. 100 33. Richard - Cotter, of Baltimore, remits to William Denman, of New York, a bill of exchange on London, the avails of which he wishes to be invested in goods on his account. Denman disposes of the bill at 74 per cent. advance, and received $9675.00, having reserved for himself į per cent. on the sale of the bill, and 2 per cent. for commission, what will remain for investment, and how much was the bill drawn for?
Ans. for investment $9457.311. The bill was drawn for £2025 sterling.
SOUND. Sound if not interrupted, will move at the rate of about
1150 feet in a second of time. 34. I have seen the flash of a cannon fired from Fort McHenry, and heard the report 47 seconds afterwards, what distance was the fort from where I stood?
Ans. 54050 feet. 35. Bought a quantity of cloth for $750, \ of which I found to be inferior, which I had to sell at $1.25 per yard, and by this I lost $100, what must I sell the rest at per yard, that I shall lose nothing by the whole?
Ans. $3.154i. 36. A circular fish pond is to be dug in a garden that shall take up just half an acre, what must be the radius of the circle?
Ans. 27.75 yds. 37. Bought a horse which was worth 30 per cent. more than I gave for him, but having been injured I sold him for 25 per cent. less than what he cost, and thereby lost $55 on his real value, what was received for the horse?
Ans. $75. 38. A son having asked his father's age the father thus replied: "your age is 8 years, to which if five-eights of