Algebra. 236. MISCELLANEOUS PROBLEMS. 1. In a school there are 896 pupils. There are three times as many boys as girls. How many girls? How many boys? 2. A man had 235 sheep. In the second flock there were 15 more sheep than in the first. In the third flock there were 20 fewer than in the first. How many sheep in each flock? NOTE.—Let x = the number in the first flock; then x + 15 = the number in the second, and x — 20, the number in the third. 3. A man owns three farms. In the second there are half as many acres as in the first. In the third there are twice as many acres as in the first. In all there are 560 acres. How many acres in each farm ? 4. In an apple and pear orchard containing 296 trees, there were 5 more than twice as many apple trees as pear trees. How many of each kind ? 5. To a number I add one half of itself and 15 and have 150. What is the number? 6. From three times a number I subtract s of the number and 5, and have 37 remaining. What is the number? NOTE. -Let x= the number; then 3 x +5) = 37. On 3 removing the parenthesis, what sign must be changed ? See page 67, II. 7. If to three times a number I add s of the number and 18, the sum will be 238. What is the number? 8. Two thirds of a number is equal to the number decreased by 56. What is the number? 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 2 x = 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 their 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 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 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 = 720°. But the angular space that measures 720o, 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. (a) The gain equals what part of the cost ? What %? (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” " 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? or 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. Explanation. Operation. $2457 .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. |