Algebra, 237. MISCELLANEOUS PROBLEMS. 1. A is 50 years old. B is 20 years old. In how many years will A be only twice as old as B? NOTE.-Let x= =the number of years; then (20+ x) X 2=50+x. 2. Find four consecutive numbers whose sum is 150. NOTE.-Let x = the first; then x +1= the second; x + 2= the third, etc 3. Find three consecutive numbers whose sum is 87. 4. Two numbers have the same ratio as 2 and 3, and their sum is 360. What are the numbers ? NOTE.-Let 2x = the first, and 3 x = the second. 5. Two numbers have the same ratio as 3 and 4, and their sum is 168. What are the numbers ? 6. Two numbers have the same ratio as 2 and 5, and theii difference is 87. . What are the numbers ? 7. A has $350. B has $220. How many dollars must A give to B so that each may have the same sum? NOTE.—Let x = the number of dollars that must be given by A to B; then 220 + x= 350 X. 8. C has $560. D has $340. How many dollars must C give to D so that each may have the same sum ? 9. E has $630. F has $240. How many dollars must E give to F so that E will have exactly twice as many dol. lars as F? 10. The fourth and the fifth of a certain number are together equal to 279. What is the number? 11. The difference between 1 fourth and 1 fifth of a cer tain number is 28. What is the number? Geometry. 238. HOW MANY DEGREES IN EACH ANGLE OF A REGULAR 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 is 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 $475,250. If 5% of the tax proves to be non-collectable 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? 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 vear the money is loaned and at that rate for fractions of years. EXAMPLE. Find the interest of $257 for two years at 6 %. Explanation. Operation. $2 57 .12 5.14 25.7 The interest of any sum for 2 years at 6 % is 12 hundredths of the principal. One hundredth of $257 is $2.57, and 12 hundredths of $257 is 12 times $257 or $30.84. $30.84 1. Find the interest of $242 for 3 yr. at 7%. (a) Find the sum of the six results. Interest. 243. To COMPUTE INTEREST FOR ANY NUMBER OF YEARS AND MONTHS. NOTE.—The interest for 1 month is 1 twelfth as much as it is for 1 year; for 2 months, 2 twelfths or 1 sixth, etc. EXAMPLE. . Find the interest of $324.50 for 2 yr. 5 mo. at 6%. Operation and Explanation No. 1. $38.940 8.1125 = $47.0525 Operation No. 2. 2 yr. 5 mo. = 21 years. 2 times .06 = .144 $3'24.50 .144 1.6225 12.9800 32.450 Explanation. The interest of any sum for 2 yr. 5 mo. is 14} hundredths of the principal. 1 hundredth of $324.50 is $3.2450 į hundredth of $324.50 is $1.6225 4 hundredths of $324.50 is 12.9800 10 hundredths of $324.50 is 32.450 144 hundredths of $324.50 is $47.0525 $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 för 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 j, it may be disregarded; if it is more than , regard it as 1 cent. |