32. A tank containing 800 gallons has three pipes. The first lets in 8 gallons in 2 minutes, the second 10 gallons in 3 minutes, and the third 12 gallons in 5 minutes. In what time will the tank be filled by the three pipes all running together?

33. A can do a piece of work in 5 days, B in 7 days, and C in 10 days. How long will it take them all working together to do it?

34. A can build a boat in 18 days, but with the assistance of B he can do it in 12 days. How long would it take B working alone to do it?

35. A can do a piece of work in 5 days, B in 33 days, and C in 5 days. How long would it require them working together to do it?

36. A performs of a piece of work in 15 days; he then calls in B to help him, and the two together finish the work in 8 days. In how many days can each alone do the work?

37. A can do in 20 days a piece of work which B can do in 30 days. A begins the work, but after a time B takes his place and finishes it. B worked 10 days longer than A. How long did A work? Ans. 8 days.

38. A deer running at the rate of 40 rods a minute was first seen by a huntsman when 30 rods in advance of a hound pursuing at the rate of 50 rods a minute. The huntsman waited until 25 rods intervened between the hound and the deer, when he shot the deer. How long did he wait?

39. There are two places 154 miles apart, from which two persons start at the same time toward each other. One travels at the rate of 3 miles in 2 hours, and the other at the rate of 5 miles in 4 hours. Where will they meet?

40. A privateer sailing at the rate of 10 miles an hour discovers a ship 18 miles off sailing at the rate of 8 miles an hour. How many miles can the ship sail before it is overtaken ?

41. A freight train passes through a station at 20 miles an hour, and is followed at a distance of 2 miles by an express going 60 miles an hour. Where will the collision occur?

42. A courier sets out from a certain place and travels 17 miles in 4 hours, and 24 hours afterward a second courier, travelling 13 miles in 3 hours, is sent after him. How long and how far will the first go before being overtaken?

43. A person walked to the top of a mountain at the rate of 2 miles an hour, and down the same way at the rate of 31 miles an hour, and was out 5 hours. How far did he walk?

44. A hare takes 4 leaps to a greyhound's 3, but 2 of the greyhound's leaps are equivalent to 3 of the hare's. The hare has a start of 50 leaps. How many leaps must the greyhound take to catch the hare? Ans. 300.

45. A merchant bought two pieces of cloth, the first at the rate of $4 for 9 yards, and the second at that of $3 for 4 yards; the second piece contained as many times 3 yards as the first contained times 4 yards. He sold each piece at the rate of $5 for 9 yards, and lost

$5 by the bargain. How

many yards were there in each piece?

Ans. First, 144; second, 108.

46. At what time between 5 and 6 o'clock will the hands of

a watch be together?

47. At what time between 7 and 8 o'clock will the hands of a watch be opposite one another?

48. At what times between 6 and 7 o'clock will the hands of a clock be at right angles?

49. A certain fraction is equal to 3, and if its numerator is increased by 5 and its denominator by 9 it becomes §. Find the fraction.

50. Two laborers are employed at $3 and $5 a day each. The sum of the days they worked was 40. They each received the same sum. How many days was each employed?

CHAPTER XII.

EQUATIONS

OF THE FIRST DEGREE CONTAINING TWO OR MORE UNKNOWN NUMBERS.

176. Independent Equations are such as cannot be derived from one another, or reduced to the same form.

х y
+
2 2

independent equations, since any one of the three can be derived from any other one; or they can all be reduced to the form x + y = 10. 10 and 4x y are independent equations.

But x+y

=

177. Simultaneous Equations are those which are satisfied by the same values of the unknown numbers.

=

Thus, x+y= 10, and x + y 7, though independent, are not simultaneous equations, since no values of x and y will satisfy both equations. But x + y : 10 and 4xy are simultaneous, as well as independent equations.

=

178. Two independent simultaneous equations are necessary to determine the values of two unknown numbers.

x =

For from the equation x + y = 10 we cannot determine the value of either x or y in known terms. If y is transposed, we have - 10 — y; but since y is unknown, we have not determined the value of x. We may suppose y equal to any number whatever, and then x would equal the remainder obtained by subtracting y from 10. It is only required by the equation that the sum of two numbers shall equal 10; but there is an infinite number of pairs of numbers whose sum is equal to 10. But if we have also the equation 4 x = may put this value of y in the first equation, x + y = 10, and obtain x+4x= = 10, or x = 2; then 4x =8=y, and we have the value

of each of the unknown numbers.

=

y, we

But from the two equations

· 10 and x + y = 7 we can find no values of x and y that

x+y:
will satisfy both equations.

ELIMINATION.

179. Elimination is the method of deriving from the given equations a new equation, or equations, containing one (or more) less unknown number. The unknown number thus excluded is said to be eliminated.

There are three methods of elimination:

Transposing 3x in (1) and dividing by 7, we have (3), which gives an expression for the value of y. Substituting this value of y in (2), we have (4), which contains but one unknown number; that is, y has been eliminated. Reducing (4) we obtain (6), or x = 2. Substituting this value of x in (3), we obtain (7), or y = 3. the following

Rule.

Hence,

Find an expression for the value of one of the unknown numbers in one of the equations, and substitute this value for the same unknown number in the other equation.

NOTE. After eliminating, the resulting equation is reduced by the rule in Art. 163. The value of the unknown number thus found must be substituted in one of the equations containing the two unknown numbers, and this reduced by the rule in Art. 163.

Finding an expression for the value of x from both (1) and (2), we have (3) and (4). Placing these two values of x equal to each other (Art. 36, Ax. 8), we form (5), which contains but one unknown number. Reducing (5) we obtain (7), or y 3. Substituting this value of y in (4), we have (8), or x = 2. Hence the following

Rule.

=

Find an expression for the value of the same unknown number from each equation, and put these expressions equal to each other.

Solve the following equations by comparison: