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Prob. 13. A metallic compound consists of 4 parts copper and 3 parts silver. How much copper must be added to 943 pounds of the compound, in order that the proportions may be 7 parts of copper to 2 parts of silver? Ans. 873 pounds.

Prob. 14. In 255 pounds of spirit of wine, water and pure al. cohol are combined by weight in the ratio of 2 to 3. How much water must be extracted by distillation, in order that the ratio of the water to the alcohol may be 3 to 17 by weight?

Ans. 75 pounds. Prob. 15. It is required to diminish each of the factors of the two unequal products, 52 x 45 and 66 x 37, by the same number, so that the new products may be equal to each other. What is that number?

Ans. 17. Prob. 16. The square of a certain number is 1188 greater than the square of a number smaller by 6 than the former. What is that number?

Ans. 102. Prob. 17. I have a certain number of dollars in my possession, which I undertook to arrange in the form of a square, and found that I wanted 25 dollars to complete the square; but if I diminish each side of the square by 2, there remain 31 dollars over. How many dollars have I?

Ans. 200. Prob. 18. A vine-tiller has a rectangular garden, whose length is to its breadth as 7 to 5, which he wishes to plant with vines. If he sets the plants at a certain uniform distance from each other, he finds that he has 2832 plants remaining. But if he places them nearer together, so as to make 14 more on each longer side, and 10 more on each shorter side, he bas only 172 plants remaining. How many plants has he?

Ans. 14,172. Prob. 19. In the composition of a certain quantity of gunpowder, the nitre was ten pounds more than two thirds of the whole; the sulphur. was four and a half pounds less than one sixth of the whole; and the charcoal was two pounds less than one seventh of the nitre. How many pounds of gunpowder were there?

Ans. 69 pounds. Prob. 20. There are three numbers in the ratio of 3, 4, and 5. Five times the first number, together with four times the

second number, and three times the third number, make 690. What are the three numbers ?

Ans. 45, 60, and 75. Prob. 21. Divide the number 165 into five such parts that the first increased by one, the second increased by two, the third diminished by three, the fourth multiplied by four, and the fifth divided by five, may all be equal.

Ans. 19, 18, 23, 5, and 100. Prob. 22. A criminal, having escaped from prison, traveled ten hours before his escape was known. He was then pursued, so as to be gained upon three miles an hour. After his pursuers had traveled eight hours, they met an express going at the same rate as themselves, who met the criminal two hours and twenty-four minutes before. In what time from the commencement of the pursuit will they overtake him?

Ans. 20 boars. Prob. 23. There is a wagon with a mechanical contrivance by which the difference of the number of revolutions of the wheels on a journey is noted. The circumference of the fore wheel is a feet, and of the hind wheel 6 feet. What is the distance gone over when the fore wheel has made n revolutions more than the hind wheel?

abn Ans.

b Prob. 24. A cistern can be filled by four pipes; by the first in a hours, by the second in 6 hours, by the third in c hours, and by the fourth in d hours. In what time will the cistern be filled when the four pipes are opened at once ?

abcd · Ans.

hours. abc+abd+acd+bcd

feet.

-a

EQUATIONS OF THE FIRST DEGREE WITH SEVERAL UNKNOWN

QUANTITIES.

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Ex. 8.

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Ex. 9.

Ans.

y

1

a+b
y

1
+
a+b'a-b

- b
(3x— бу

2x+y
+3=
2

5
x-2y_2 , y
8

==+
4

2 3
(7+8x3x-6y 9-4x
10 2.-8

5
6y+9 3y+4 3y + 5x

=31+
4

2 4-6

Q

x=12 v=6.

=4

Ex. 10.

Ans.

x=9. ?y=7.

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x=3.

Ans. y=4.

256.

x=13. Ans. {y=4

2=7%.

y + 2=5

y
%+ -4

5x—6y+4z=15 Ex. 12. 7x+4y-3z=19

2.0+ y +62=46) x=21-4y

2x Ex. 13.

2=9

3
y=64–772

+ Ex. 14. 3x - 5y+z=75

9x-112+10=0

3x—54+4z=5 Ex. 15. 7x+2y-32=2

4x+3y- z=7

2—2y+3z=6 Ex. 16. 2x+3y-4z=20

x=5. Ans. y=-5.

2=5.

x=1. Ans. y=2.

2=3.

X=8.

Ans. y=4..

2=2.

Ans.

x=4. y=9. 2=16. U=25.

=

=12 Ans.

3x-2y +52=26
7x-3y=1

42-7y=1 Ex. 17.

112-7u=1 192-3u=1

Ex. 18.

3u-2y=2
2x+3y=39
5x - 72511
4y + 3x=41
2x-3y+22=13
4y+22=14
4u-2x=30
5y+3u=32

y=5..
2=7.
u=4.
x=3.
y=1.
2=5. .
u=9.

Ex. 19.

Ans.

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PROBLEMS INVOLVING EQUATIONS OF THE FIRST DEGREE WITH

SEVERAL UNKNOWN QUANTITIES. Prob. 1. Two sums of money, which were put out at interest, the one at 5 per cent., the other at 41 per cent., yielded in one year $284.40 interest. If the former sum had been put out at 44 per cent., and the latter at 5 per cent., they would have yielded $4.50 less interest. What were the two sums of money?

Ans. One was $3420, the other $2520. · Prob. 2. There is a number consisting of two digits; the number is equal to three times the sum of its digits, and if it be multiplied by three, the result will be equal to the square of the sum of its digits. Find the number.

Ans. 27. Prob. 3. A merchant sold two bales of goods for the sum of $9875, the first at a loss of 84 per cent., the second at a loss of 117 per cent. If he had sold the first at a loss of 117 per cent., and the second at a loss of 84 per cent., he would have received the sum of $992. How much did each bale cost?

Ans. The first $455, the second $645. Prob. 4. Two messengers, A and B, from two towns distant 574 miles from each other, set out to meet each other. If A starts 5$ hours earlier than B, they will meet in 63 hours after B starts; but if B starts 54 hours earlier than A, they will meet in 55 hours after A starts. How many miles does each travel in an hour?

Ans. A 3 miles, and B 34 miles. Prob. 5. A jeweler has two masses of gold of different degrees of fineness. If he mixes 10 ounces of the one with 5 ounces of the other, he obtains gold 11 carats fine; but if he

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