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I. 1. Two men, A and B, trade in company, and gain 267 dollars, of which B-has twice as much as A. What is the share of each?
In this example the unknown quantities are the particular shares of A and B.
Let x represent the number of dollars in A's share, then 2 x will represent the number of dollars in B's share. Now these added together must make the number of dollars in both their shares, that is, 267 dollars.
x + 2x = 267 Putting all the x's together, 3x = 267
If 3 x are 267, 1 x is į of 267 in the same manner as if 3 oxen were worth $267, 1 ox would be worth į of it.
i = 89 = A's share.
2x = 178 = B's share. 2. Four men, A, B, C, and D, found a purse of money containing $325, but not agreeing about the division of it, each took as much as he could get; A got a certain sum, B got 5 times as much ; C, 7 times as much ; and D, as much as B and C both. How many dollars did each get?
Let x represent the number of dollars that A got; then B got 5x, C7x, and D (5x + 7x) = 12 x.
= 12 x. These, added together, must make $325, the whole number to be divided.
2 + 5x + 7x + 12 x= 325 Putting all the x's together, 25 x = 325
x = 13 = A's share. 5x =
65 = B's 7 * = 91 = C's
12 x = 156 = D's Note. All examples of this kind in algebra admit of proof. In this case the work is proved by adding together the several shares. If they are equal to the whole sum, 325, the work is right. As the answers are not given in this work, it will be well for the learner always to prove his results.
In the same manner perform the following examples.
3. Said A to B, my horse and saddle together are worth $130, but the horse is worth 9 times as much as the saddle. What is the value of each?
4. Three men, A, B, and C, trade in company, A puts in a certain sum, B puts in 3 times as much, and C puts in as much
as A and B both; they gain $656. What is each man's share of the gain?
5. A gentleman, meeting 4 poor persons, distributed 60 cents among them, giving the second twice, the third three times, and the fourth four times as much as the first. How many cents did he give to each?
6. A gentleman left 11000 crowns to be divided between his widow, two sons, and three daughters. He intended that the widow should receive twice the share of a son, and that each son should receive twice the share of a daughter. Required the share of each.
Let x represent the share of a daughter, then 2 x will represent the share of a son, &c.
7. Four gentlemen entered into a speculation, for which they subscribed $4755, of which В paid 3 times as much as A, and C paid as much as A and B, and D paid as much as B and c. What did each pay?
8. A man bought some oxen, some cows, and some sheep for $1400 ; there were an equal number of each sort. For the oxen he gave $42 apiece, for the cows $20, and for the sheep $8 apiece. How many were there of each sort ?
In this example the unknown quantity is the number of each sort, but the number of each sort being the same, one charac-, ter will express it.
Let x denote the number of each sort.
Then x oxen, at $42 apiece, will come to 42 x dolls., and x cows, at $20 apiece, will come to 20 x dolls., and a sheep, at $8 apiece, will come to 8x dolls. These added together must make the whole price.
42 x + 20x + 8x = 1400 Putting the a's together, 70x = 1400 Dividing by 70, .
X = 20
Ans. 20 of each sort. 9. A man sold some calves and some sheep for $374, the calves at $5, and the sheep at $7 apiece; there were three times as many calves as sheep. How many were there of each ?
Let x denote the number of sheep; then 3x will denote the number of calves.
Then x sheep, at $7 apiece, will come to 7 x dolls., and 3 x calves, at $5 apiece, will come to 5 times 3 x dolls., that is, 15 x dolls. These added together must make the whole price.
7x + 15x=374 Putting the x's together, 22 x = 374 Dividing by 22,
x = 17 =number of sheep. 30 51 =
The learner must have remarked by this time, that when a question is proposed, the first thing to be done, is to find, by means of the unknown quantity, an expression which shall be equal to a given quantity, and then from that, by arithmetical operations, to deduce the value of the unknown quantity.
This expression of equality between two quantities, is called an equation. In the last example, 7 x + 15x=374 is an equation.
The quantity or quantities on the left of the sign=are called the first member, those on the right, the second member of the equation. (7x + 15 x) is the first member of the above equation, and 374 is the second member.
Quantities comected by the signs +- and -- are called terms. 7 x and 15 x are terms in the above equation.
The figure written before a letter showing how many times the letter is to be taken, is called the coefficient of that letter. In the quantities 7 x, 15 x, 22 x;7, 15, 22, are coefficients of x.
The process of forming an equation by the conditions of a question, is called putting the question into an equation.
The process by which the value of the unknown quantity is found, after the question is put into an equation, is called solving or reducing the equation.
No rules can be given for putting questions into equations ; this must be learned by practice; but rules may be found for solving most of the equations that ever occur.
After the preceding questions were put into equation, the first thing was to reduce all the terms containing the unknown quantity to one term, which was done by adding the coefficients. As 7x + 15 x are 22 x. Then, since 22 x = 374, 1 x must be equal to 1 of 374. That is,
When the unknown quantity in one member is reduced to one term, and stands equal to a known quantity in the other, its value is
found by dividing the known quantity by the coefficient of the unknown quantity.
10. A man bought some oranges, some lemons, and some pears, for 156 cents; the oranges at 6 cents each, the lemons at 4 cents, and the pears at 3 cents; there was an equal number of each sort. Required the number of each.
11. In-fencing the side of a field, the length of which was 450 yards, two workmen were employed; one fenced 9 yards, and the other 6 yards per day. How many days did they work ?
12. Three men built 780 rods of fence; the first built 9 rods per day, the second 7, and the third 5; the second worked three times as many days as the first, and the third, twice as many days as the second. How many days did each work?
13. A man bought some oxen, some cows, and some calves for $348 ; the oxen at $38 each, the cows at $18, and the calves at $4. There were three times as many cows as oxen, and twice as many calves as cows. How many were there of each sort ?
14. A merchant bought a quantity of flour for $132; for one half of it he gave $5 per barrel, and for the other half $7. How
many barrels were there in the whole ? Let x denote one half the number of barrels.
15. From two towns, which are 187 miles apart, two travellers set out at the same time with an intention of meeting ; one of them travels at the rate of 8, the other of 9 miles each day. In how many days will they meet ?
II. 1. A cask of wine was sold for $45, which was only of what it cost. Required the cost. Let x denote the cost.
3x x x
The latter is
= 45 4
1 x If of x comes to 45, then must come to f of 45, or
4 15, and x will be 4 times 15, or 60.
A better method.
x = 60
Observe, that is the same as of 3x. Now if i of 3 x
4 is 45, 3 x itself must be 4 times. 45, or 180; 3 x being 180, z must be of 180, which is 60.
2. A man, being asked his age, answered, that if its half and its third were added to it the sum would be 88. What was his age ? Let x denote his age; then,
3 Reducing the terms to a com 6 x 3 x
+ + =88 mon denominator,
6 6 6
Adding them together,
X = 48
years. 3. If of a hogshead of wine cost $65; what will a hogshead cost at that rate ?
4. There is a pole and under water, and 5 feet out of water ; what is the length of the pole?
Let x denote the whole length. Then + + 5 must be equal to the whole length. Hence,
ta +5 Reducing to a common denominator,
+ 265 6