306. Laying off a Rectangle.

1. When the two sides of a rectangle are in the ratio of 3 units to 4 units, the diagonal is one fourth more than the longer side, thus: If the sides of a rectangle are 18 ft. and 24 ft., the diagonal is 24 ft. plus 6 ft., or 30 ft. Prove that this is correct and that it holds with various rectangles when the ratio of the sides is as 3 to 4.

2. A farmer asked two schoolboys to lay off a rectangle 16 ft. by 24 ft. to mark the foundation of a carriage house. The boys used two

h

i e

15ft.

20ft.

25ft.

24ft.

S

16ft.

d pieces of cord and a measure. They tied the cords to two stakes so that they crossed at c, a corner of the rectangle, and extended one cord in the direction of cd and the other in the direction of ce. To make the

angle at e a right angle and thus to get the positions of the two sides of the rectangle, they measured off from c . 15 ft. on one cord and 20 ft. on the other cord. With these as the legs of a right triangle, they adjusted the position of the cords so as to make the distance hl 25 ft. Having determined the direction of the sides cs and ce, they measured 24 ft. on cs and 16 ft. on ce and marked the corners s and i. From s they extended a line in the direction so, and from i a line in the direction io. They measured 16 ft. on the line so and 20 ft. on the line io. They marked the point o where the two measured lines met. They tested their work, finding that the diagonals co and si were equal.

3. Using cords and a measure, lay out on the school grounds rectangles 12 ft. by 16 ft.; 20 ft. by 30 ft. 4. Lay off a baseball "diamond" whose side is 60 ft.

SIMILAR SURFACES AND SOLIDS

307. 1. Draw squares whose sides are 1 inch; 2 inches; 3 inches. Find their areas. The areas of the three squares are to each other as 12, 22, and 32.

2. Express the ratio of the areas of a 4-inch

a 6-inch square.

square and

3. Draw circles whose diameters are 4 inches; 6 inches. As the area of a circle is πr2, the ratio of the areas of these two circles is as 22 is to 32. Explain.

4. Is the ratio of the squares of the diameters of two circles the same as the ratio of the squares of their radii?

5. Is the ratio of the squares of the diagonals of two squares the same as the ratio of the squares of their sides?

6. Figures that are of exactly the same shape are called similar figures. Draw two similar figures. Are similar figures necessarily the same in size?

The areas of similar plane figures are proportional to the squares of their corresponding lines.

7. Find the volume of a 2-inch cube; of a 3-inch cube. Their volumes are in the ratio of 23 to 33.

The volumes of similar solids are proportional to the cubes of their corresponding lines.

8. Compare the volumes of two spheres, one 5 inches in diameter and the other 10 inches in diameter.

As the diameter of the larger sphere is twice the diameter of the smaller, the volume of the larger is 28 times the volume of the smaller. Explain.

9. Compare the weights of two solid iron spheres of the same density, if one is 2 inches in diameter and the other is 4 inches in diameter.

MISCELLANEOUS EXERCISES

308. 1. A merchant gained $28 by selling some goods at a profit of 20%. Find the cost of the goods.

2. An agent who canvassed for a book received 40% of the amount of the sales for selling and delivering the books. Find the amount of his commission for selling and delivering 60 copies at $1.25 per copy.

3. If the amount received for goods sold averages 20% more than the cost of the goods, find the net profit for a month when the sales amounted to $24,000 and the expenses for the month amounted to $3000.

4. 20% of the pupils enrolled in a certain school were absent one stormy day. Twenty-four pupils were present. Find the number enrolled.

5. A commission merchant sold $6000 worth of produce at a commission of 11%. Find the amount of his commission.

6. If a collector received 20% commission for collecting a bill of $17.75, what was the amount of his commission? What per cent of the amount of the bill did the creditor receive?

7. A man paid $42 taxes when the tax rate was 2%. What was the assessed valuation of his property?

8. If a person pays $50 tax on property when the rate is 11%, what is the assessed valuation of his property?

9. A miller bought a ton of wheat through a broker, who charged a commission of 2%. What was the amount paid for the wheat if the cost of the wheat and the brokerage amounted to $25.50?

The cost of the wheat plus 2% of the cost of the wheat, or 102% of the cost of the wheat, was $25.50. Prove your answer.

10. A fruit grower shipped 25 boxes of apples to a commission merchant, who sold them at 85 per box, charging 4% commission. He was directed to invest the deducting a commission of 2% for making the purchase. Find the amount expended for the groceries.

proceeds in groceries, after

The net proceeds of the sale of the apples was $20.40, which was 102% of the amount invested in groceries. Prove your answer.

11. After selling 25% of his interest in a flour mill, a man considered his remaining interest worth $9000. At this rate, what was the value of his interest before making the sale?

12. A merchant pays $240 a month rent, in advance, for his store. Money is worth 6%. This is equivalent to what sum paid at the end of each month? in the middle of each month?

13. When money is worth 6%, what sum should be deducted from a debt of $240 paid 90 da. before it is due?

14. A man bought a lot for $1200 and sold it for $1400. He bought it back for $1500 and resold it for $1600. How much did he make on the lot?

15. A man bought four 50-ft. lots and divided the land into 40-ft. lots, which he sold at the same price per lot as he had paid. Find his gain per cent.

16. At 22 a square foot, find the cost per front foot of paving a street 60 ft. wide. Find the cost per front foot to a property owner who pays for half the width of the street. Find the cost to the property owner of paving the street in front of a 45-ft. lot.

17. If 27 tons of coal cost $243, how many tons can be bought for $189?

PART VI

APPENDIX

CORPORATIONS, STOCKS, AND BONDS

309. 1. Corporations. A large business enterprise frequently requires more capital than one person may care to invest in it. Provision is made in the laws of the various states whereby a number of persons may organize a company, called a corporation, to engage in business as one body. Sometimes all the necessary capital is subscribed by the persons who organize the corporation, but often the organizers of a company subscribe only a part of the capital.

The laws regulating the incorporation of companies differ considerably in the several states. Frequently a corporation intending to transact business in one state will incorporate in another state, because of certain advantages to be derived thereby.

2. Railway companies, mining companies, express companies, oil companies, and banking institutions are among the largest business corporations.

310. Shares of Stock. 1. Each corporation is capitalized for a special amount, as $25,000, $50,000, $1,000,000, etc. The capital is divided into shares, usually of $100 each or of $1 each. Thus, a corporation that is capitalized for $100,000 may issue 1000 shares of the face value of $100 each, or 100,000 shares of the face value of $1 each, etc. These shares are bought by persons who invest in the enterprise. Each person who owns one or more shares of stock is called a stockholder. The several stockholders constitute the corporation.