5. The sum of two numbers is 53. Four times the first is 20 more than twice the second. Find the numbers.

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

third takes balance, $24.

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

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 13% acres more than the third.

10. A merchant has sold of a piece of cloth, and has remaining 16 yards more than of the piece. How many 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?

MULTIPLICATION OF ALGEBRAIC QUANTITIES.

1225. Multiply x + 3 by x +4.

The product is equal to x times (x+3) + 4 times (x+3).

x(x+3)= x2+3x

4(x+3)= 4x+12

(x+3)(x+4)= x2+7x+12 Ans.

NOTE.2 is read x square. The 2 is called an exponent.

9. (x-5)(x+5) 12. (x-6) (2x+1) 15. (2x+3)(2x-3)

The product is equal to x(x-5)-4(x-5); that is, that 4(x-5) is to be subtracted from x(x-5).

x2

x(x-5)=x2-5 x; 4(x-5)=4x-20.

Placing the subtrahend under the minuend, and changing the signs of the former (Art. 1209), we have

1230. (x-7)(x-9)=?

=

Using either as a multiplier, place one under the other. Commencing with x, say x×x = -9xx-9x. Taking - 7 as a multiplier, say xX(-7)=-7x, (-9)x(-7)=63. Combining, we get the product.

1231. Note that the multiplication of a + (positive) quantity by a (positive) quantity gives a +(positive) product; that (+) × (−) or (−) × (+) gives a (negative) product; and that (-) × (-) gives a + (positive) product. This is usually stated as follows:

1232. Like signs produce +, and unlike signs produce

1233. Give results:

16. (x −7) (x −7) 20.(x+7)( x −6) 24. (2x+7)(3x+3) 17. (x-5)(x-9) 21. (x-4)(x-7) 25. (2x-3)(3 x − 2) 18. (x+5)(x+5) 22. (2x-4)(3x-6) 26. (2 x − 3)(2x+3) 19. (x −3)(x+8) 23. (2x+6)(3 x −7) 27.(2x+9)(4 x − 6)

- 384

x2 = 64

X= +8.

1235. Since (- 8) × (− 8) = 64, the square root of 64 may be either + 8 or - 8. It is written 8, and is read "positive or negative 8." (It is sometimes less correctly called plus or minus 8.)

1. Find the dimensions of a field, the length of which is wice its breadth, its area being 1,800 square rods.

2. The surface of the six equal faces of a cube contains 96 square inches. Find the length of one edge.

3. One number is fourth-fifths of another, and their product is 80. What are the numbers?

4. One-third of a number multiplied by two-fifths of the same number gives a product of 270. Find the number.

5. Thirty per cent of a number multiplied by forty per cent of the same number gives a product of 300. What is the number?

6. Thirty per cent of forty per cent of a number is 300. What is the number?

7. The base of a right-angled triangle is as long as the perpendicular, and the area of the triangle is 96 square rods. Find the length of the base. What is the length of the hypotenuse?

3 x 4

8. The base of a right-angled triangle measures x yd., the perpendicular measures yd. What is the length of the hypotenuse? If the hypotenuse measures 15 yd., find the length of the base.

9. The base of a right-angled triangle measures x ft., the hypotenuse measures (x+9) ft., the perpendicular measures 15 ft. What is the length of the base?

10. The difference between the squares of two consecutive numbers is 49. What are the numbers?

AFFECTED QUADRATICS.

1238. Preliminary Exercises.

of the first

(x+1)(x+1)= x2+2x+1

The square of the sum of two quantities is equal to the square twice the product of the first and the second + the square of the second.

The square of the difference of two quantities is equal to the twice the product of the first and the second

square of the first

the square of the second.