Geometry. REGULAR HEXAGON ? 1. Every regular hexagon may be divided into equal isosceles triangles. 2. The sum of the angles of one triangle is equal to right angles,* then the sum of the angles of 6 triangles equal to right angles. 3. But the sum of the central angles in Fig. 2 (a + b + c '+d+e+f) is equal to right angles; † then the sum of all the other angles of the six triangles is equal to 12 right angles less 4 right angles or 8 right angles = 720o. But the angular space that measures 720°, as shown in Fig. 2, is made up of 12 equal angles; so each one of the angles is one 12th of 720° or 60°. Two of these angles, as 1 and 2, make one of the angles of the hexagon; therefore each angle of the hexagon measures 2 times 60° or 120°. 4. Using the protractor, construct a regular hexagon, making each side 2 inches long. Observe that since all the angular space about a point is equal to 4 right angles, or 360°, and since the space around the central point of the hexagon is divided into 6 equal angles, each of these angles is an angle of (360°:-6), 60°. But each of the other angles of these triangles has been shown to be an angle of 60°; so each triangle is equiangular. Are the triangles equilateral ? * See page 59, 6 and 7. + See page 29, Art. 66. 239. MISCELLANEOUS REVIEW. 1. A man buys goods for $60 and sells them for $75. He gains dollars. (a) The gain equals what part of the cost ? What %? (b) The gain equals what part of the selling price ? What per cent ? (c) The cost equals what part of the selling price ? What per cent? 2. When the cost is 2 thirds of the selling price, what is the per cent of gain ? 3. When the selling price is 2 thirds of the cost, what is the per cent of loss ? 4. Bought for $200 and sold for $300. What was the per cent of gain ? 5. Bought for $300 and sold for $200. What was the per cent of loss? 6. A tax of 15 mills on a dollar was levied in a certain town, the assessed value of the taxable property being $475250. If 5% of the tax proves to be non-collectible and if the collector is allowed 2% of the amount collected, for his services, how much will be realized from the levy ? 7. Which is the greater discount, “20 and 10 and 5 off” or “35 off" ? 8. A sold goods for B on a commission of 15 %. His sales for a certain period amounted to $780. If the goods cost B exactly $600, was B’s net profit more or less than 10 %? 9. A offers rubber boots at “50 and 20 off"; B offers them at “20 and 50 off.” The quality and list price being the same, which offer shall I accept? (P. 347.) PART II. INTEREST. 240. Interest is compensation for the use of money. 241. The money for which interest is paid is called the principal. 242. The principal and interest together are called the amount. NOTE. — Interest is usually reckoned in per cent, the principal being the base; that is, the borrower pays for the use of money a sum equal to a certain per cent of the principal. “ The rate of interest" is the per cent per annum which the borrower agrees to pay. When a man loans money “ at 6%" he expects to receive back the principal, and a sum equal to 6% of the principal for every year the money is loaned and at that rate for fractions of years. EXAMPLE Explanation. The interest of any sum for 2 years at 6 % is 12 5.14 hundredths of the principal. One hundredth of $257 is $2.57, and 12 hundredths of $257 is 12 times $2.57 25.7 or $ 30.84. $30.84 1. Find the interest of $242 for 3 yr. at 7%. Interest. 243. To COMPUTE INTEREST FOR ANY NUMBER OF YEARS AND MONTHS. 1 year; NOTE. — The interest for 1 month is 1 twelfth as much as it is for for 2 months, 2 twelfths or 1 sixth, etc. EXAMPLE. Operation and Explanation No. 1. $38.940 8.1125 Explanation. 2 yr. 5 mo.= 2 years. The interest of any sum for 2 yr. 2 times .06 = .141 5 mo. is 141 hundredths of the prin cipal. $324.50 1 hundredth of $ 324.50 is $3.2450 .141 1 hundredth of $ 324.50 is $1.6225 1.6225 4 hundredths of $324.50 is 12.9800 12.9800 32.450 10 hundredths of $324.50 is 141 hundredths of $324.50 is $47.0525 32.450 $47.0525 PROBLEMS. 1. Find the interest of $325.40 for 1 yr. 6 mo. at 7%.* 2. Find the interest of $ 420.38 for 2 yr. 10 mo. at 6%. 3. Find the interest of $221.60 for 2 yr. 3 mo. at 6%. 4. Find the interest of $145.20 for 1 yr. 9 mo. at 5%. 5. Find the interest of $340.10 for 3 yr. 1 mo. at 4%. (a) Find the sum of the five results. * In an interest problem, if there is a fraction of a cent in the answer and this fraction is not greater than ž, it may be disregarded; if it is more than £, regard it as 1 cent. Interest. 244. To COMPUTE INTEREST FOR ANY NUMBER OF YEARS, MONTHS, AND DAYS. NOTE. - In computing interest, 30 days is usually regarded as 1 month, and 360 days as 1 year; so each day is 3d of a month or oto of a year. EXAMPLE. Operation and Explanation No. 1. $ 30.7440 $ 8.9670 $.04270 Interest of $ 256.20 for 13 da. at 6 % .5551 Interest of $ 256.20 for 2 yr. 7 mo. 13 da. at 6% = $ 40.2661 Operation No. 2. Explanation. 2yr. 7 mo.13 da.=23&yr. The interest of any sum for 2 yr. 2323 times .06 =.1548 7 mo. 13 da. at 6 % is .1548 of the prin$ 2'56.20 cipal. .1543 1 hundredth of $256.20 is $ 2.5620 1.8361 48 of 1 hundredth of $256.20 is $ 1.8361 5 hundredths of $256.20 is $ 12.81 12.8100 10 hundredths of $256.20 is $ 25.62 25.620 1548 hundredths of $256.20 is $ 40.2661 40.2661 60 PROBLEMS. 1. Find the interest of $350.40 for 2 yr. 5 mo. 7 da. at 6%. 2. Find the interest of $145.30 for 1 yr. 7 mo. 10 da. at 8%. 3. Find the interest of $174.20 for 2 yr. 3 mo. 15 da. at 7 %. 4. Find the interest of $253.40 for 3 yr. 6 mo. 20 da. at 5%. (a) Find the sum of the four results. (P. 348.) |