ber of ounces of silver taken from the second and third, to form the fourth; but from the enunciation, one pound of this fourth ingot contains 8 ounces of silver. We have, then, for the first equation,

or, making the denominators disappear,

7x+12y+4z=128.

As respects the copper, we should find

3x+3y+72=60,

and with reference to the pewter

6x+y+5z=68.

As the coefficients of y in these three equations, are the most simple, it is most convenient to eliminate this unknown quantity first.

Multiplying the second uation by 4, and subtracting the first, we have

5x+24z=112.

Multiplying the third equation by 3, and subtracting the second from the product,

15x+8z 144.

Multiplying this last equation by 3, and subtracting the preceding one from the product, we obtain

Lastly, the two values x8, z=3, being substituted in the equation

Therefore, in order to form a pound of the fourth ingot, we must take 8 ounces of the first, 5 ounces of the second, and 3 of the third.

Verification.

If there be 7 ounces of silver in 16 ounces of the first ingot, in 8 ounces of it, there should be a number of ounces of silver expressed by

will express the quantity of silver contained in 5 ounces of

the second ingot, and 3 ounces of the third.

therefore, a pound of the fourth ingot contains 8 ounces of silver, as required by the enunciation. The same conditions may be verified relative to the copper and pewter.

6. A's age is double B's, and B's is triple of C's, and the

sum of all their ages is 140.

What is the age of each?

84, B's 42, and C's=14.

Ans. A's

7. A person bought a chaise, horse, and harness, for £60; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness. What did he give for each?

£13 6s. 8d.

Ans.

£ 6 13s. 4d.

for the horse. for the harness.

£40

8. To divide the number 36 into three

of the first, of the second, and all equal to each other.

for the chaise.

such parts that

of the third, may be

Ans. 8, 12, and 16.

9. If A and B together can do a piece of work in 8 days, A and C together in 9 days, and B and C in ten days; how many days would it take each to perform the same work alone? Ans. A 143, B 1721, C 233 10. Three persons, A, B, and C, begin to play together, having among them all $600. At the end of the first game A has won one-half of B's money, which, added to his own, makes double the amount B had at first. In the second game, A loses and B winst as much as C had at the beginning, when A leaves with exactly what he had at first. How much had each at the beginning?

Ans. A $300, B $200, C $100. 11. Three persons, A, B, and C, together possess $3640. If B gives A $400 of his money, then A will have $320 more than B; but if B takes $140 of C's money, then B and C will have equal sums. How much has each ?

Ans. A $800, B $1280, C $1560. 12. Three persons have a bill to pay, which neither alone is able to discharge. A says to B, "Give me the 4th of your money, and then I can pay the bill." B says to C, "Give me the 8th of yours, and I can pay it.

Bat

C says to A, "You must give me the half of yours before I can pay it, as I have but $8." What was the amount of their bill, and how much money had A and B ?

13. A person possessed a certain capital, which he placed out at a certain interest. Another person, who possessed 10000 dollars more than the first, and who put out his capital 1 per cent. more advantageously, had an income greater by 800 dollars. A third person, who possessed 15000 dollars more than the first, putting out his capital 2 per cent. more advantageously, had an income greater by 1500 dollars. Required the capitals of the three persons, and the rates of interest.

Ans. {

Sums at interest,
Rates of interest,

$30000, 40000, 45000.

4

5

6 pr. ct. 14. A widow receives an estate of $15000 from her deceased husband, with directions to divide it among two sons and three daughters, so that each son may receive twice as much as each daughter, and she herself to receive $1000 more than all the children together. What was her share, and what the share of each child?

15. A certain sum of money is to be divided between three persons, A, B, and C. A is to receive $3000 less than half of it, B $1000 less than one third part, and C to receive $800 more than the fourth part of the whole. What is the sum to be divided, and what does each receive?

CHAPTER IV.

Of Powers.

79. If a quantity be multiplied several times by itself, the product is called the power of the quantity. Thus,

a=a is the root, or first power of a.

axaa2 is the square, or second power of a. axaxa=a3 is the cube, or third power of a. a×a×a×a=at is the fourth power of a.

αχαχαχαχαπαδ

is the fifth power

of a.

In every power there are three things to be considered · 1st. The quantity which is multiplied by itself, and which is called the root or the first power.

2nd. The small figure which is placed at the right, and a little above the letter. This figure is called the exponent of the power, and shows how many times the letter enters as a factor.

3rd. The power itself, which is the final product, or result of the multiplications.

QUEST.-79. If a quantity be continually multiplied by itself, what is the product called? How many things are to be considered in every power? What are they?