RULE. Use for the third term the number which is of the same kind as the answer required. Arrange the other couplets according to the relation of the third term to the answer sought. The product of the means divided by the product of the extremes will be the answer. Problems in compound proportion are readily solved by what is termed the cause and effect method. Example 1, stated by cause and effect, is as follows: 2. If 11 men build 45 rods of wall in 6 days of 10 hours each, how many men will be required to build 81 rods of wall in 12 days of 11 hours each? 3. Three workmen dig a ditch 20 rd. long and 3 ft. wide in 10 days. How long will it take 5 workmen to dig a ditch 45 rd. long and 4 ft. wide? 4. If 18 men can perform a piece of work in 12 days, how many men could perform another piece of work 4 times as great in of the time? 5. If the freight charges on 125 cattle, averaging 900 pounds, is $200 for 150 miles, what should be the charges on 275 cattle, averaging 1200 pounds, for 225 miles? 6. If a block of marble 7 ft. long, 3 ft. wide, and 2 ft. thick weighs 6930 lb., what will be the weight of a block of the same kind 10 ft. long, 4 ft. wide, and 3 ft. thick? 7. If the capacity of a bin 24 ft. long, 41⁄2 ft. wide, and 4 ft. deep is 405 bushels, what is the capacity of a bin 16 ft. long, 5 ft. wide, and 4 ft. deep? 8. If it costs $180 to build a wall 60 ft. long, 14 ft. high, and 1 ft. 6 in. thick, what will it cost to build a wall 200 ft. long, 18 ft. high, and 1 ft. 4 in. thick? STAND. AR.- - 21 9. If 15 men, working 12 hr. a day, can hoe 60 acres in 20 days, how long will it take 35 boys, working 10 hr. a day, to hoe 90 acres, the work of 5 men being equal to that of 7 boys? 10. If 16 men can excavate a cellar 40 ft. long, 36 ft. wide, and 8 ft. deep in 12 days of 8 hours each, in how many days of 10 hours each can 8 men excavate a cellar 30 ft. long, 27 ft. wide, and 6 ft. deep? 11. If 5 iron bars, 4 ft. long, 3 in. broad, and 2 in. thick, weigh 240 lb., what will be the weight of 20 bars, each 6 ft. long, 24 in. broad, and 14 in. thick? 12. If 9 bricklayers can lay a wall 80 ft. long, 20 ft. high, and 14 ft. thick, in 15 days of 9 hr. each, in how many days of 10 hr. each can 12 bricklayers lay a wall 100 ft. long, 25 ft. high, and 2 ft. thick? 13. If 240 men, in 11 days of 8 hours each, dig a ditch 350 ft. long, 11 ft. wide, and 23 ft. deep, in how many days of 9 hours each will 48 men dig a ditch 500 ft. long, 163 ft. wide, and 3 ft. deep? 14. If 54 men, in 28 days of 10 hours each, dig a trench 352 yards long, 2 yards broad, and 14 yards deep, how long a trench 2 yards broad, and 14 yards deep, will 112 men dig in 25 days of 84 hours each? 15. If a regiment of 1025 soldiers consumes 11,500 pounds of bread in 15 days, how many pounds will 3 regiments of the same size consume in 12 days? 16. If the water that fills a vat, which is 8 feet long, 4 feet wide, and 5 feet deep, weighs 10,000 pounds, what will be the weight of the water required to fill a vat, which is 10 feet long, 5 feet wide, and 6 feet deep? 17. If 5 horses eat as much as 6 cattle, and 8 horses and 12 cattle eat 12 tons of hay in 40 days, how much hay will be needed to keep 7 horses and 15 cattle 65 days? PARTITIVE PROPORTION. 438. The process by which a number is divided into parts, proportional to other given numbers, is called Partitive Proportion. EXPLANATION. WRITTEN EXERCISES. 439. 1. Divide 240 into parts proportional to 3, 4, and 5. Since the parts are proportional to 3, 4, and 5, out of every 12 (the sum of 3, 4, and 5), there is a 3, a 4, and a 5. Consequently one part will be of 240, or 60, another will be 240, or 80, and the other of 240, or 100. Therefore the parts are 60, 80, and 100. of 2. Divide $390 into parts proportional to,, and . EXPLANATION. - Since fractions have the ratios of their numerators when their denominators are the same, the fractions are changed to 12ths, and we have 2, 12, 12. 4 Therefore the problem may be expressed thus: Divide $390 into parts proportional to 6, 4, and 3. This is solved in the same way as example 1. 3. Divide 420 into three parts which shall be to one another as 2, 5, and 7. 4. Divide 750 into five parts which shall be to one another as 1, 2, 3, 4, and 5. 5. Divide 468 into three parts, such that they shall be proportional to, 1, and 1. 6. Divide $1596 into parts proportional to 3, 4, and . 7. A man bought three farms for $26,150, and the prices paid for them were in the proportion of the fractions ,, and . What did he pay for each farm? 8. A man bequeathed his property in such a way that his wife received $7 for every $5 received by each of his two sons and every $4 received by each of his three daughters. If his estate was worth $250,000, what was the sum bequeathed to each of the heirs? INVOLUTION. 440. 1. Of what number are 4 and 4 the factors? 5 and 5? 6 and 6? 3, 3, and 3? 4, 4, and 4? 5, 5, and 5? 2. What is the product of 6 used twice as a factor, or the second power of 6? 3. What is the second power of 7? Of 5? Of 9? Of 10? Of 12? 4. What is the third power of 2? Of 3? 5. What is the second power of ? Of ? 441. The product arising from using a number a certain number of times as a factor is a Power of the number. 442. The powers of a number are named from the number of times it is used as a factor. Thus, 4 is the second power of 2; 9 the second power of 3; 8 the third power of 2; 27 the third power of 3. The number itself is called its first power. 443. The number of times a number is used as a factor is indicated by a small figure, called an Exponent, written a little above and at the right of the number. Thus, 32 means the second power of 3; 34 the fourth power of 3, etc. Since the area of a square is the product of two equal factors, and the volume of a cube the product of three equal factors, the second power is called the square, and the third power the cube. 444. The process of finding the power of a number is called Involution. WRITTEN EXERCISES. 445. 1. Find the fourth power of 8. SOLUTION. -8 x 8 x 8 x8 = 4096, the fourth power of 8. 4. What is the square of 25? 32? 48? 66? 5. What is the cube of 22? 25? 54? 71? 6. What is the fourth power of 4? 6? 12? 19? 10. What is the cube of? ? ? ? Raise the following to the powers indicated: 446. 1. Find the square of 35 in terms of its tens and of tens and units, is equal to the tens2 + 2 times the tens the |