5. A grocer sold to one person 5 pounds of coffee and 3 pounds of sugar, for 79 cents; and to another, at the same prices, 3 pounds of coffee and 5 pounds of sugar, for 73 cents; what was the price of a pound of each? Ans. Coffee 11 cts., sugar 8 cts.

6. A farmer sold to one person 9 horses and 7 cows, for 300 dollars; and to another, at the same prices, 6 horses and 13 cows, for the same sum; what was the price of each?

Ans. Horses $24, and cows $12 each.

7. A vintner sold at one time, 20 dozen of port wine and 30 of sherry, and for the whole received 120 dollars; and, at another, 30 dozen of port and 25 of sherry, at the same prices as before, for 140 dollars; what was the price of a dozen of each sort of wine? Ans. Port $3, and sherry $2 per doz. 8. It is required to find two numbers, such that of the first and of the second shall be 22, and of the first and of the second shall be 12.

9. If the greater of two numbers be added to will be 37; but if the less be diminished by difference will be 20; what are the numbers?

Ans. 24 and 30. of the less, the sum of the greater, the Ans. 28 and 27. of the first dimin

10. What two numbers are those, such that ished by of the second, shall be 5, and of the first diminished by of the second, shall be 2?

Ans. 20 and 15.

11. A farmer has 2 horses, and a saddle worth 25 dollars; now, if the saddle be put on the first horse, his value will be double that of the second; but, if the saddle be put on the second horse, his value will be three times that of the first. Required the valuc of each horse. Ans. First $15, second $20.

12. A and B are in trade together with different sums; if 50 dollars be added to A's property, and 20 dollars taken from B's, they will have the same sum; and if A's property was 3 times, and B's 5 times as great as each really is, they would together have 2350 dollars; how much has each? Ans. A $250, B $320. 13. A has two vessels containing wine, and finds, that of the first contains 96 gallons less than of the second; and that of the second contains as much as of the first; how much does each vessel hold? Ans. 720 and 512 galls. 14. There is a number consisting of two digits, which, divided by their sum, gives a quotient, 7; but if the digits be written in an inverse order, and the number so arising, be divided by their sum increased by 4, the quotient will be 3. Required the number. Ans. 84.

15. If we add 8 to the numerator of a certain fraction, its value becomes 2; and if we subtract 5 from the denominator, its value becomes 3; required the fraction. Ans..

16. If to the ages of A and B 18 be added, the result will be double the age of A; but, if from their difference 6 be subtracted, the result will be the age of B; required their ages.

Ans. A 30, B 12 yrs.

17. There are two numbers whose sum is 37, and if 3 times the less be subtracted from 4 times the greater, and the difference divided by 6, the quotient will be 6; what are the numbers? Ans. 16 and 21. 18. It is required to find a fraction, such that if 3 be subtracted from the numerator and denominator, the value will be; and if 5 be added to the numerator and denominator, the value will be

Ans. 19.

19. A father gave his two sons, A and B, together 2400 dollars, to engage in trade; at the close of the year, A has lost 4 of his capital, while B, having gained a sum equal to of his capital, finds that his money is just equal to that of his brother; what was the sum given by the father to each? Ans. A $1500, B $900.

20. If from the greater of two numbers 1 be subtracted, the remainder will be equal to 4 times the less; but, if to the less 3 be added, the sum will be of the greater; required the numbers. Ans. 8 and 33.

21. A said to B, "Give me 100 dollars, and then I shall have as much as you." B said to A, "Give me 100 dollars, and then I shall have twice as much as you." How many dollars had each? Ans. A $500, B $700.

22. If the greater of two numbers be multiplied by 5, and the less by 7, the sum of their products is 198; but if the greater be divided by 5, and the less by 7, the sum of their quotients is 6; what are the numbers? Ans. 20 and 14.

23 Seven years ago the age of A was just three times that of B; and seven years hence, A's age will be just double that of B; what are their ages? Ans. A's 49, B's 21 yrs.

24. There is a certain number consisting of two places of figures, which being divided by the sum of its digits, the quotient is 4, and if 27 be added to it, the digits will be inverted; required the number.

Ans. 36.

25. A grocer has two kinds of sugar, of such quality that one pound of each are together worth 20 cents; but if 3 pounds of the first, and 5 pounds of the second kind be mixed, a pound of the mixture will be worth 11 cents; what is the value of a pound of each sort? Ans. 6 cts., and 14 cts.

26. A boy lays out 84 cents for lemons and oranges, giving 3 cents a piece for the lemons, and 5 cents a piece for the oranges; he afterward sold of the lemons and of the oranges, for 40

cents, and by so doing cleared 8 cents on what he sold; what number of each did he purchase?

Ans. 8 lemons and 12 oranges.

27. A person spends 30 cents for peaches and apples, buying his peaches at 4, and his apples at 5 for a cent; he afterward sells of his peaches, and of his apples, at the same rate he bought them, for 13 cents; how many of each did he buy?

Ans. 72 peaches and 60 apples.

28. A owes 500 dollars, and Bowes 600 dollars, but neither has sufficient money to pay his debts. A said to B, "Lend me of your money, and I shall have enough to discharge my debts." B said to A," Lend me of your money, and I can pay mine." How much money has each? Ans. A $400, B $500.

29. A merchant bought two pieces of cloth for 236 dollars, the first piece at 4, and the second at 7 dollars per yard; but the cloth getting damaged, he sold of the first piece, and 3 of the second, for 160 dollars, by which he lost 8 dollars on what he sold; what was the number of yards in each piece?

Ans. 24 yards in the first, and 20 yards in the second. 30. A son said to his father, "How old are we?" The father replied, "Six years ago my age was 33 times yours, but 3 years hence, my age will be only 2 times yours." Required the age of each. Ans. Father's age 36, son's 15 yrs.

31. A person has two horses, and two saddles, one of which cost 50, and the other 2 dollars. If he places the best saddle upon the first horse, and the other on the second, then the latter is worth 8 dollars less than the former; but if he puts the worst saddle upon the first, and the best upon the second horse, then the value of the latter is to that of the former as 15 to 4. Required the value of each horse. Ans. First $30, second $70.

32. A farmer having mixed a certain number of bushels of oats and rye, found, that if he had mixed 6 bushels more of each, he would have mixed 7 bushels of oats for every 6 of rye; but if he had mixed 6 bushels less of each, he would have put in 6 bushels of oats for every 5 of rye. How many bushels of each did he mix? Ans. Oats 78, rye 66 bu.

33. A person having laid out a rectangular yard, observed, that if each side had been 4 yards longer, the length would have been to the breadth, as 5 to 4; but, if each had been 4 yards shorter, the length would have been to the breadth, as 4 to 3; required the length of the sides. Ans. Length, 36, breadth 28 yards.

34. A farmer rents a farm for 245 dollars per annum; the tillable land being valued at 2 dollars an acre, and the pasture at 1 dollar and 40 cents an acre; now the number of acres tillable, is

to the excess of the tillable above the pasture, as 14 to 9; how many were there of each? A. Tillable 98, pasture 35 acres.

35. Two shepherds, A and B, are intrusted with the charge of two flocks of sheep; at the end of the first year, it is found, that A's flock has increased 10, and B's diminished 20, when their numbers are to each other, as 4 to 3; during the second year, A's flock loses 20, and B's gains 10, when their numbers are to each other as 6 to 7. Required the number in each flock at first.

Ans. A's had 70, and B's 80 sheep.

36. After drawing 15 gallons from each of 2 casks of wine, the quantity remaining in the first, is of that in the second; after drawing 25 gallons more from each, the quantity left in the first, is only half that in the second. Required the number of gallons in each before the first drawing. Ans. 65 and 90 galls.

37. There is a fraction, such that if 1 be added to the numerator, and the numerator to the denominator, its value will be but if the denominator be increased by unity, and the numerator by the denominator, its value will be ; find it. Ans.

38. Find two numbers in the ratio of 5 to 7, to which two other required numbers, in the ratio of 3 to 5, being respectively added, the sums shall be in the ratio of 9 to 13, and the difference of their sums equal to 16. Ans. 30 and 42, 6 and 10. Let the first two numbers be represented by 5x and 7x, and the other two by 3y and 5y.

39. A farmer, with 28 bushels of barley, worth 28 cents per bushel, would mix rye at 36 cents, and wheat at 48 cents per bushel, so that the whole mixture may consist of 100 bushels, and be worth 40 cents a bushel; how many bushels of rye, and how many of wheat must be mixed with the barley?

Ans. Rye 20, and wheat 52 bu. 40. Two loaded wagons were weighed, and their weights were found to be in the ratio of 4 to 5; part of their loads, which were in the ratio of 6 to 7, being taken out, their weights were then found to be in the ratio of 2 to 3, and the sum of their weights was then 10 tons; what were their weights at first?

Ans. 16 and 20 tons.

41. A person had two casks and a certain quantity of wine in each; in order to have the same quantity in each cask, he poured as much out of the first cask into the second as it already contained; he next poured as much out of the second, into the first, as it then contained; and lastly, he poured out as much from the first into the second, as there was remaining in it; after this, he had 16 gallons in each cask; how many gallons did each contain at first? Ans. First 22, and second 10 galls.

SIMPLE EQUATIONS, CONTAINING THREE OR MORE UNKNOWN

QUANTITIES.

ART. 162.-Equations involving three or more unknown quantities may be solved, by either of the three methods of elimination explained in the preceding Article, as we shall now proceed to show, by solving an example by each of these methods.

Suppose we have the three following equations, in which it is required to find the values of x, y, and z.

x+2y+ z=2
-20 (1.)

2x+y+3z=31 (2.)

3x+4y+22=44 (3.)

Solution by substitution.

From equation (1), x=20-2y-z.

Substituting this in equation (2), we have

2(20-2y-z)+y+3z=31.

or, 40-4y-2z+y+3z=31.
3y-2-9 (4.)

Substituting the same value of x in equation (3), we have
3(20-2y-z)+4y+2z=44.

or, 60-6y-3z+4y+2z=44.
2y+2=16 (5.)
3y-z=9 (4.)

Here the values of y and z are readily found by the rule, Art. 158, to be 5 and 6; then substituting these values in equation (1), we find x=4.

Comparing the first and second values of x, we have

31-y-3z 20—2y—z= 2

or, 40-4y-2z=31-y-3z

or, 3y-z-9 (4.)

Comparing the first and third values of x, we have

44-4y-2z 20-2y-z= 3

or, 60-6y-3z-44-4y-2z

2y+z=16 (5.)

From equations (4) and (5), the values of y and z, and then x,

may be found by the rule, Art. 159.