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of the mixture. The rye was worth 96 cents per bushel, the oats 56 cents, and the mixture 72 cents. How many bushels did he use of each ?
Ans. 40 of rye and 60 of oats.
16. A person has two sorts of wine, one worth 40 cents a quart, and the other 24 cents. How much of each kind must he use to form a gallon worth 112 cents. Ans. 1 quart of the first, 3 quarts of the second.
17. A and B trade on a joint stock of 833 dollars, and clear 153 dollars. A's share of the gain is 45 dollars more than B's. What share of the capital did each possess ? Ans. A, $539; B, $294.
18. Two laborers, A and B, received 51 dollars. A had been employed 14 days, and B 15 days; A received for 6 days' labor 1 dollar more than B got for 4 days' labor. How many dollars did each receive per day? Ans. A, 1; B, 2.
19. In 80 pounds of an alloy of copper and tin, there are 7 lbs. of copper to 3 of tin. How much copper must be added to the alloy, that there may be 11 lbs. of copper to 4 of tin. Ans. 10 lbs.
20. In 3 battalions there are 1905 troops; the number in the first, together with the number in the second, is 60 less than the number in the third; the number in the third, together with the number in the first, is 165 less than the number of the second. How many are there in each battalion ?
Ans. 630, 675 and 600. 21. A grocer has three kinds of tea: 12 lbs. of the first, 13 of the second, and 14 of the third are together worth 25 dollars: 10 of the first, 17 of the
second, and 11 of the third are together worth 24 dol lars; 6 of the first, 12 of the second, and 6 of the third are together worth 15 dollars. What is the value of a pound of each?
Ans. 50 cents, 60 cents, and 80 cents.
22. A owes $1200, and B $2500; but neither has money enough to pay his debts. Says A to B, "Lend me of your fortune, and I can pay my debts;" says B to A, "Lend me of your fortune, and I can pay mine." What fortune had each ?
Ans. B, had $2400; and A, $900.
23. The united ages of a father and son are 80 years; and if the age of the son be doubled, it will exceed the father's age by 10 years. What is the age of each? Ans. 50 and 30.
24. A travels uniformly along a certain road, B starts an hour afterwards in pursuit, and after 4 hours finds by inquiry that he is travelling 1 miles per hour slower than A; he then doubles his rate of travel, and overtakes A, 6 hours from the time he started in pursuit. At what rate did A travel, and what was the rate that B travelled at first?
Ans. A's rate, 93 miles; B's, 81.
25. There are 32 gallons of wine in two casks. If from the first there be drawn into the second as much as there is in the second; then if there be drawn from the second into the first as much as remains in the first; and then if there be drawn from the first into the second as much as remains in the second, there will be 16 gallons in each cask. How many gallons were there originally in each? Ans. 22 and 10.
26. A cistern can be filled by 3 pipes. The first can fill it in 4 hours, the first and second together can fill it in 3 hours, and the third can fill it in 2 hours, How long will it take for them all to fill it together, and how long will it take the second alone to fill it? Ans. All in 1 h. 12 m. The second in 12 hours.
27. A cistern has two discharge cocks: they both run together for two hours when the first one is closed; the second one then empties it in 2 hours and 48 minutes. Had the second one been closed at the end of two hours, the first one would have emptied it in 4 hours and 40 minutes. In what time could each empty it alone?
Ans. The first in 10 hours; the second in 6 hours.
DISCUSSION OF PROBLEMS.
95. The DISCUSSION of a problem consists in making every possible supposition upon the arbitrary quantities which enter it, and interpreting the results.
An ARBITRARY QUANTITY is a quantity to which a value may be given at pleasure.
The discussion of the following problem indicates the general method of discussion, as well as illustrates the meaning of the plus and minus signs, as signs of interpretation, together with the ideas to be attached to the
symbols, 0, ∞, and
PROBLEM OF THE COURIERS.
96. Two couriers, A and B, travel along the same line, R' R, in the same direction, R' R, and at uniform rates; the courier A travels m miles per hour, and the
Whence, by solving,
Let the position of the rearmost courier, A, be taken as the origin of distances, and suppose all distances estimated towards B to be positive.
Denote the number of hours from the epoch to the time they are together by t. Denote the distance the courier B travels in the time t, by x; then will the distance that the courier A travels, in the same time, be denoted by a + x.
Then, since the distance travelled is equal to the number of hours multiplied by the rate per hour, we have the equations:
mt = a + x,
nt = x.
Now, supposing them to at any epoch, say 12
To discuss these values. The distance between the couriers may be assumed at pleasure; hence, a is arbitrary. The rates of travel may, also, be assumed at pleasure; hence, m and n are arbitrary. From the nature of the case, a can never be negative; hence, the only suppositions that can be made upon a, are that its positive, or a > 0; or that it is equal to zero, or a = 0. The only hypotheses that can be made upon m and n, are m>n, m<n, and m = n. these suppositions, we may obtain the six
By combining suppositions :
a > 0, and m>n.
a> 0, and m<n.
3. a > 0, and m = n.
We shall make each hypothesis separately.
α = 0, and m>n.
a = 0, and m<n.
6. α= 0, and m = n
1. a > 0, and m>n.
This hypothesis makes both terms of the fraction m
positive; hence, the value of t is positive.
We interpret this result as showing that the time required is after the epoch 12 o'clock.
This interpretation is in accordance with the supposition made; for, if m>n, the courier A, travels faster than the courier B; hence, after the epoch, the distance between them continually diminishes, and, consequently, at some time after the epoch they must come together.
The same hypothesis makes both x and x + a posi tive, showing, as before, that they are together some where to the right of the origin of distances.
2. a > 0, and m<n.
This hypothesis makes the numerator of the fraction positive, and the denominator negative; hence,
the value of t is negative.
We interpret this result as showing that the time when they are together is before the epoch 12 o'clock.
This interpretation is also in accordance with the supposition made; for, if m<n, the courier B travels faster than the courier A; hence, after the epoch, they will continually separate. Before the epoch, they must have been together at some time, previous to which B was approaching A, and after which the two were continually separating.