11. The sum of two numbers is 19. The sum of the second number and ten times the first, minus the sum of the first and ten times the second, equals 45. What are the numbers?

12. Reduce to an equivalent fraction, the sum of whose numerator and denominator shall be 126.

13. What fraction equivalent to has 147 for the difference between its numerator and denominator?

14. 10 pounds of coffee at 30 per pound are mixed with x pounds of coffee at 25¢ per pound. What is x equal to, when the mixture is worth 26 per pound?

25x + (10 × 30) = 26 (10 + x).

15. A grocer mixes green tea costing 60 per pound with black tea costing 40 per pound. He uses 100 pounds in all, and the mixed tea costs him 48¢ per pound. How many pounds of each does he use?

Let x = number of pounds of black tea; y = number of green. Then x + y = number of pounds of mixed tea.

an equation containing only two unknown quantities.

Compare the two equations (d) and (e), which contain the same

Substituting values of y and z in (a), we have

2. Find the values of the unknown quantities in the following equations:

x-3y+2x= 3, (a)

2x+y+32=22, (b)

5x+2y+7z=51. (c)

Multiply (a) by 2, and subtract from (b). Multiply (a) by 5, and subtract from (c). This gives two equations, each of which contains two unknown quantities.

Compare these two resulting equations, and eliminate y.

Eliminate z by comparing (a) and (b), multiplying the former by 5.

Compare (a) and (c), multiplying the former by 2.

556. Written Problems.

1. A man placed of his capital at 5% and the other third at 6%. At the end of a year, capital and interest amounted to $31,600. What was his capital?

2. A has 18 chestnuts more than B. If each finds 4 more, A will have four times as many as B. How many chestnuts has each?

3. Two mechanics earn together $8 per day. One works 23 days and the other 17 days, for which they receive together $166. What does each earn per day?

4. The sum of the first and the second of three numbers is 55, of the first and the third 62, of the second and the third 83. What are the numbers?

SUGGESTION. - Add together the three equations.

5. The sum of two numbers is

is 20 more than twice the second.

sons.

53.

Four times the first Find the numbers.

6. A certain sum of money is divided among four perThe first takes of it, the second takes of the remainder, the third takes of what then remains, the fourth receives the balance, $24. What is the share of each of the other three?

7. A merchant sold a lot of goods for $510, thereby losing of their cost. What did the goods cost?

3

20

8. A man collected a bill for a physician and deducted of the amount for his services. If he gave the physician $147, what was the amount collected?

9. Divide 130 acres of land among three persons, giving the first 27 acres more than the second, and the second 133 acres more than the third.

of a piece of cloth, and has of the piece. How many

10. A merchant has sold remaining 16 yards more than yards did the piece contain originally?

11. A servant is engaged for a year for $280 and a suit of clothes; he leaves at the end of six months, and receives $130 and the suit. What is the value of the clothes?

Yearly wages = 280+x. Wages for six months

EXPONENTS.

557. (1) 22 means x times x, or xx.

(2) y3 means y times y times y, or yyy.
(3) a means aaaa.

In (1) 2 is an exponent.

In (2) 3 is an exponent.

In (3) 4 is an exponent.

Notice that the exponent is written above the quantity to which it belongs.

On which side of the quantity is it written?

How many times is the quantity used as a factor if the exponent is 2? If the exponent is 3? If the exponent is 5?

Tell two facts concerning the location of the exponent. Tell one fact concerning the meaning of the exponent. What is an exponent?

NOTE.—When no exponent is written, 1 is understood.