L and carriage, as the 12th part of the purchase money amounted to. For how much did he buy it? Ans. $300. 10. Divide the number 60 into two such parts that their Ans. 44 and 16. produce shall be 704. 11. A merchant sold a piece of cloth for $39, and gained as much per cent. as it cost him. What did he pay for it? Ans. 30. 4 12. A and B distributed 1200 dollars each, among a certain number of persons. A relieved 40 persons more than B, and B gave to each individual 5 dollars more than A. How many were relieved by A and B? Ans. 120 by A, and 80 by B. This problem can be brought into a pure equation, in like manner as (Problem 1.) 13. A vintner sold 7 dozen of sherry and 12 dozen of claret for £50, and finds that he has sold 3 dozen more of sherry for £10 than he has of claret for £6. Required the price of each? Ans. Sherry, £2 per dozen; claret, £3. 14. A set out from C towards D, and travelled 7 miles a day. After he had gone 32 miles, B set out from D towards C, and went every day of the whole journey; and after he had travelled as many days as he went miles in a day, he met A. Required the distance from C to D. Ans. 76 or 152 miles; both numbers will answer the condition. 15. A farmer received $24 for a certain quantity of wheat, and an equal sum at a price 25 cents less by the bushel for a quantity of barley, which exceeded the quantity of wheat by 16 bushels. How many bushels were there of each? Ans. 32 bushels of wheat, and 48 of barley. 16. A and B hired a pasture, into which A put 4 horses, and B as many as cost him 18 shillings a week; afterwards B put in two additional horses, and found that he must pay 20 shillings a week. At what rate was the pasture hired? Ans. B had 6 horses in the pasture at first, and the price of the whole pasture was 50 shillings per week. 17. A mercer bought a piece of silk for £16 4s., and the number of shillings he paid per yard, was to the number of yards as 4 to 9. How many yards did he buy, and what was the price per yard? Ans. 27 yards, at 12 shillings per yard. 18. If a certain number be divided by the product of its two digits, the quotient will be 2, and if 27 be added to the nuinber, the digits will be inverted. What is the number? Ans. 36. 19. It is required to find three numbers, whose sum is 33, such that the difference of the first and second shall exceed the difference of the second and third by 6, and the sum of whose squares is 441. Ans. 4, 13, and 16 20. Find those two numeral quantities whose sum, product, and sum of their squares are all equal to each other. Ans. No such numeral quantities exist. In a strict alge braic sense, the quantities are 21. What two numbers are those, whose product is 24, and whose sum added to the sum of their squares is 62? Ans. 4 and 6. 22. It is required to find two numbers, such that if their product be added to their sum it shall make 47, and if their sum be taken from the sum of their squares, the remainder shall be 12? Ans. 7 and 5. 23. The sum of two numbers is 27, and the sum of their cubes 5103. What are their numbers? Ans. 12 and 15. 24. The sum of two numbers is 9, and the sum of their fourth powers 2417. What are the numbers? Ans. 7 and 2. 25. The product of two numbers multiplied by the sum of their squares, is 1248, and the difference of their squares is 20. What are the numbers? Ans. 6 and 4. Let x+y=the greater, and x-y = the less. 26. Two men are employed to do a piece of work, which they can finish in 12 days. In how many days could each do the work alone, provided it would take one 10 days longer than the other? Ans. 20 and 30 days. For brief and concise solutions of Problems 8, 9, 12, 13, 14, 22, and 25, see Universal Key to Science of Algebra. SECTION V. CHAPTER I. ARITHMETICAL PROGRESSION. A series of numbers or quantities, increasing or decreas ing by the same difference, from term to term, is called arithmetical progression. Thus, 2, 4, 6, 8, 10, 12, &c., is an increasing or ascending arithmetical series, having a common difference of 2; and 20, 17, 14, 11, 8, &c., is a decreasing series, whose common difference is 3. (Art. 115.) We can more readily investigate the proper ties of an arithmetical series from literal than from numeral terms. Thus let a represent the first term of a series, and d the common difference. Then a (a+d) (a+2d) (a+3d) (a+4d), &c. represents an ascending series; and a (a-d) (a-2d) (a-3d) (a-4d), &c., represents a descending series. Observe that the coefficient of d, in any term, is equal to he number of terms before it. The first term exists without the common difference. All ther terms consist of the first term and the common differnce multiplied by one less than the number of terms from he first. Wherever the series is supposed to terminate, is the last erm, and if designated by L, and the number of terms by 1, the last term must be a+(n-1)d, or a-(n-1)d, according as the series may be ascending or descending, which we draw from inspection. (Art. 116.) It is manifest that the sum of the terms will be the same, in whatever order they are written. Take, for instance, the series 3, 5, 7, 9, 11, 11, 9, 7, 5, 3. : The sums of the terms will be 14, 14, 14, 14, 14. Take the series a, a+ d, a+2d, a+3d, a+4d, a+4d, a+3d, a+2d, a+ d, a 2a+4d, 2a+1d, 2a+1d, 2a-4d, 2a+4d. Inverted, Sums will be Here we discover the important property, that, in an arithmetical progression, the sum of the extremes is equal to the sum of any other two terms equally distant from the extremes. Also, that twice the sum of any series is equal to the extremes, or first and last term repeated as many times as the series contains terms. : Hence, if s represents the sum of a series, and n the number of terms, a the first term, and L the last term, we shall have Or 2s=n(a+L) The two equations (A) and (B) contain five quantities, a, d, L, n, and s; any three of them being given, the other two can be determined. These two equations are sufficient for problems relating to arithmetical series, and we may use them without modification by putting in the given values just as they stand, and afterwards reducing them as numeral equations. EXAMPLES. 1. The sum of an arithmetical series is 1455, the first term 5, and the number of terms 30. What is the common difference? Ans. 3. Land dare sought. Reduced L=92 Reduced d=3, ans Here s=1455, a=5, n=30. Equation (B) 1455=(5+L)15. 92=5+29d. g 2. The sum of an arithmetical series is 567, the first term 7, and the common difference 2. What is the number of Ans. 21, terms? Here s=567, a=7, d=2. Land n are sought. L=7+2n-2=5+2n Equation (B) 567=(7+5+2n)=6n+n2 Or 2 n2+6n+9=576 n+3=24, or n=21, ans. 3. Find seven arithmetical means between 1 and 49. Observe that the series must consist of 9 terms. Hence, a=1, L=49, n=9. Ans. 7, 13, 19, 25, 31, 37, 43. 4. The first term of an arithmetical series is 1, the sum of the terms 280, the number of terms 32. What is the common difference, and the last term? Ans.d=, L=161. 5. Insert three arithmetical means between + and. Ans. The means are, 129 24 |