CHAPTER XIII THE EQUATION 230. The Equation Method. There is a better method for solving many of the problems of arithmetic than any that have thus far been given. It is known as the equation method. 231. Equation. An indicated equality between two expressions is called an equation. For example, 2 + x = 5 is an equation. 232. Unknown Quantity. In the equation 2 + x = 5, x is called the unknown quantity. We may indicate an unknown quantity by x, or by some initial, as d for dollars, or n for number. We may also use x%, or any other convenient symbol. 233. Indicated Multiplication. We indicate the multiplication of 5 x x by writing simply 5 x. 234. Solving an Equation. To find the value of the unknown quantity in an equation is to solve the equation. 235. How to Solve. In the equation 2 + x = 5, we may subtract 2 from these two equal quantities, and we have = 5 — 2, or x = 3. To solve an equation, do to one side what is necessary to leave a by itself, and then perform the same operation on the other side. Thus, if 3 x = 15, divide both sides by 3. Then 5. If { x = 7, multiply both sides by 3. Then = 21. 236. Applications of the Equation. The following are illustrations of the special application of the equation to percentage: 1. 25 is what per cent of 225 ? Let x% the required rate per cent. Then 2% X 225 = 25. 25 Dividing by 225, X% 225 = }, or 113%. 2. $8.40 is 12% of what number? Let X = the required number. Then 0.12 X = $8.40. Dividing by 0.12, $8.40 : 0.12 = $70. = EXERCISE 135 Find what per cent the first number is of the second : 1. 25, 750. 6. 11, 71. 2. 37, 148. 7. $1.80, $3. 3. 5 ft., 37 ft. 8. 7 ft., 7 yd. 4. 528 ft., 2 mi. 9. $4.90, $29.40. 5. $8.60, $77.40. 10. 73 ft., 194 yd. 2 ft. 11. A man's income is $1650 a year, and he spends $693. What per cent of his income does he save ? 12. In a certain village 576 out of the 1200 pupils in school are boys. What per cent are boys ? girls ? 13. The purity of gold is measured in carats, or 24ths, 18 carats meaning pure gold. What is the per cent of pure gold in an 18-carat ring ? 14. What is the per cent of pure gold in a watch case that is 16 carats fine ? in a chain that is 12 carats fine ? in a ring that is 14 carats fine ? X = 15. A horse and carriage together cost $375, and the carriage cost twice what the horse did. What did the horse cost? We may explain the work as follows: the number of dollars the horse cost. Then 2 x = the number of dollars the carriage cost, and X + 2 x = the number of dollars both cost, which is 375. Therefore 3 x = 375, because x (that is, 1x) and 2 x are 3x. Therefore x = 125, by dividing both sides by 3. Therefore the horse cost $125. The actual work of this problem is as follows: 3x = 375. : 125. X = 16. After increasing 6%, a certain sum amounts to $318. What is the sum ? We may explain the work as follows: x = the number of dollars in the sum. Then 0.06 x = the number of dollars increase.. Adding, 1.06x = the number of dollars after the increase. Therefore 1.06 x = 318. Therefore - 318 : 1.06 = 300. Therefore the sum is $300, as is easily proved. The actual work of this problem is as follows: 1.06 x = 318. = 300. = 17. A man sold a carriage for $120, which was 20% less than it cost. How much did it cost? We may explain the work as follows: X = the number of dollars of cost. Then 0.20 x = the number of dollars lost, and X – 0.20 x the number of dollars received. Therefore 0.80 x = 120, because 13 -- 0.20 x = 0.80 x. Therefore - 120 ; 0.80 = 150. Therefore the cost was $150, as is easily proved. The actual work of this problem is as follows: 0.80 x = 120. 18. Of what length is 112 ft. 6 in. exactly 45% ? 19. After gaining 12%, a certain sum amounts to $392. What is the sum ? 20. If to { of a certain number 730 is added, the result is 855. What is the number? 21. A boy lost i of his marbles and had 18 left. How many marbles had he at first? 22. A man sold of his cattle and had 68 left. How many cattle had he at first ? 23. A certain sum was increased 50% and then amounted to $505.50. What was the sum ? 24. A man sold a house for $2250, which was 20% less than it cost. How much did it cost? 25. A man saves $675.20 a year, which is 32% of his income. How much is his income? 26. A school has 20% of its pupils in the sixth grade, which numbers 29. How many are there in the school ? 27. If a school is in session 50% of the days of a certain year, and is in session 183 days, is that a leap year or not? 28. With $2250 a man bought 20 acres of land from Mr. A, who thereby gained 25% on what it cost him. How much did Mr. A pay per acre for the land ? 29. A man sold a farm of 100 acres at the rate of $56.25 an acre. He lost 10% on the cost of the farm. How much did the farm cost him ? 30. A man buys some lumber and sells it for $60, thereby gaining 20% on the cost. The $60 is what per cent of the cost ? What is the cost ? 31. A certain school has 119 boys, which is 85% of the number of girls. How many girls are there in the school ? What is the total number of pupils ? 32. A man sells 41 sheep, which is 331% of all he owned. He received $5.50 a head. At this rate, what is the value of the sheep remaining in his flock ? 33. A man's salary has been increased 22% this year. It is now $1525. What was it last year ? 1.22 = what number? 34. If I have added 60% to the size of my farm, and now have 240 acres, did I have before the increase ? 35. If you weigh 78.1 lb., and have increased 42% in weight in the last 5 years, how much did you weigh five years ago ? 36. If our regular army has been increased 21%, and now numbers 73,810, how many did it number before the increase ? 37. If a library has increased in size 163% in the past five years, and now has 8323 volumes, how many volumes had it before the increase ? 38. A dealer bought two horses at the same price. He sold one, at a profit of 20%, for $102. The other he sold at a loss of 10%. How much did he receive for the latter ? 39. A certain village, after gaining 10% on its population in 1910, had 10% less population than another village of 4180 inhabitants. What was the population of the first village in 1910 ? 40. A furniture dealer bought 5 sets of furniture at $65 a set, 3 sets at $90, and 2 sets at $40. The $40 sets he sold at a profit of 30%, and the $65 sets at a profit of 20%, but on the $90 sets he lost 10%. What per cent did he gain on the lot? |