2. A horse and carriage are together worth $450; but the horse is worth twice as much as the carriage; what is each worth? Ans. Carriage, $150; horse, $300.

All problems should be verified to see if the answers obtained fulfil the given conditions. In each of the preceding problems there are two conditions, or statements. For example, in Prob. 2 it is stated (1st) that the horse and carriage are together worth $450, and (2d) that the horse is worth twice as much as the carriage; both these statements are fulfilled by the numbers 150 and 300.

3. The sum of two numbers is 72, and the greater is seven times the less; what are the numbers?

4. A drover being asked how many sheep he had, said that if he had ten times as many more, he should have 440; how many had he?

5. A father and son have property of the value of $8015, and the father's share is four times the son's; what is the share of each?

Ans. Father's, $6412; son's, $1603.

6. A farmer has a horse, a cow, and a sheep; the horse is worth twice as much as the cow, and the cow twice as much as the sheep, and all together are worth $490; how much is each worth?

7. A man has three horses which are together worth $540, and their values are as the numbers 1, 2, and 3; what are the respective values?

Let x, 2x, and 3x represent the respective values.

Ans. $90, $180, and $270.

8. A man has three pastures, containing 360 sheep, and the numbers in each are as the numbers 1, 3, and 5; how many are there in each?

9. Divide 63 into three parts, in the proportion of 2, 3, and 4.

Let 2x, 3x, and 4x represent the parts.

10. A man sold an equal number of oxen, cows, and sheep for $1500; for an ox he received twice as much as for a cow, and for a cow eight times as much as for a sheep, and for each sheep $6; how many of each did he sell, and what did he receive for all the oxen?

Ans. 10 of each, and for the oxen, $960.

11. Three orchards bore 872 bushels of apples; the first bore three times as many as the second, and the third bore as many as the other two; how many bushels did each bear?

12. A boy spent $4 in oranges, pears, and apples; he bought twice as many pears and five times as many apples as oranges; he paid 4 cents for each pear, 3 for each orange, and 1 for each apple; how many of each did he buy, and how much did he spend for oranges? how much for pears, and how much for apples?

Ans. {25 oranges, 50 pears, and 125 apples.

Spent for oranges, $0.75; pears, $2; apples, $1.25.

13. A farmer hired a man and two boys to do a piece of work; to the man he paid $12, to one boy $6, and to the other $4 per week; they all worked the same time, and received $264; how many work?

weeks did they Ans. 12 weeks.

14. Three men, A, B, and C, agreed to build a piece of wall for $99; A could build 7 rods, and B 6, while X C could build 5; how much should each receive?

15. Four boys, A, B, C, and D, in counting their money, found they had together $1.98, and that B had twice as much as A, C as much as A and B, and D as much as B and C; how much had each?

Ans. A 18 cents, B 36, C 54, and D 90.

16. It is required to divide a quantity, represented by a, into two parts, one of which is double the other.

17. If in the preceding example a = 24, what are the required parts?

Ans.

α 3

24
3

2 a

48

3

= = 16.
3

18. It is required to divide c into three parts so that the first shall be one half of the second and one fifth of the third.

Ans. 20, and

с

8

5 c

8

19. Divide n into three parts, so that the first part shall be one third the second and one seventh of the third.

20. A is one half as old as B, and B is one third as old as C, and the sum of their ages is p; what is the age of each? 2 Ans. A's, B's, and C's T

17. THE last letters of the alphabet, x, y, z, &c., are used in algebraic processes to represent unknown quantities, and the first letters, a, b, c, &c., are often used to represent known quantities.

NUMERICAL QUANTITIES are those expressed by figures, as 4, 6, 9.

LITERAL QUANTITIES are those expressed by letters, as a, x, y.

MIXED QUANTITIES are those expressed by both figures and letters, as 3a, 4x.

18. The sign plus, +, is called the positive or affirmative sign, and the quantity before which it stands a positive or affirmative quantity. If no sign stands before a quantity, is always understood.

19. The sign minus,, is called the negative sign, and the quantity before which it stands, a negative quantity.

20. Sometimes both and are prefixed to a quantity, and the sign and quantity are both said to be ambiguous; thus, 8 ± 3 = 11 or 5, and ab=a+b, or a b, according to circumstances.

21. The words plus and minus, positive and negative, and the signs and, have a merely relative signification; thus, the navigator and the surveyor always represent their northward and eastward progress by the sign +, and their southward and westward progress by the sign —, though, in the nature of things, there is nothing to prevent representing northings and eastings by and southings and westings by +. So if a man's prop

erty is considered positive, his gains should also be considered positive, while his debts and his losses should be considered negative; thus, suppose that I have a farm worth $5000 and other property worth $3000 and that I owe $1000, then the net value of my estate is $5000 + $3000 $1000 $7000. Again, suppose my farm is worth $5000 and my other property $3000, while I owe $12000, then my net estate is worth $5000 + $3000 $12000 = $4000, i. e. I am worth $4000, or, in other words, I owe $4000 more than I can pay. From this last illustration we see that the sign - may be placed before a quantity standing alone, and it then merely signifies that the quantity is negative, without determining what it is to be subtracted from.

-7y,

22. The TERMS of an algebraic expression are the quantities which are separated from each other by the signs or —; thus, in the equation 4 a b = 3x + c the first member consists of the two terms 4 a and and the second of the three terms 3x, c, and

7y.

b,

23. A COEFFICIENT is a number or letter prefixed to a quantity to show how many times that quantity is to be taken; thus, in the expression 4x, which equals xx +x+x, the 4 is the coefficient of x; so in 3 ab, which equals ab+ab + ab, 3 is the coefficient of ab ; in 4 ab, 4a may be considered the coefficient of b, or 4b the coefficient of a, or a the coefficient of 4b.

Coefficients may be numerical or literal or mixed; thus, in 4 ab, 4 is the numerical coefficient of ab, a is the literal coefficient of 4b, 4 a is the mixed coefficient of b.

If no numerical coefficient is expressed, a unit is understood; thus, x is the same as 1x, bc as 1bc.

24. An INDEX or EXPONENT is a number or letter placed after and a little above a quantity to show how many times that quantity is to be taken as a factor; thus, in the ex