Let x represent the number of gallons; then 12 8 3 +25 represents the number of gallons of wine; and -5 represents the number of gallons of cider; and the Equation will be 14. After a person had expended $10 more than of his money, he had $15 more than of it remaining. What sum had he at first? Then, by the conditions of the problem, we shall have · 15. Divide the number 91 into two such parts that if the greater be divided by their difference, the quotient may be 7. Let x represent the greater part; then 91-x represents the less; and x-(91-x), or 2x-91, represents their difference; This Equation will give x=49 the greater part; hence 91-49-42, is the less part. 16. A and B had equal sums of money; the first paid away $25, and the second $60, when it appeared that A had twice as much left as B. What sum had each? Let x represent the sum each had; then x-25 represents the sum A had after spending $25; and X- -60 represents the sum B had after spending $60. Then, by the conditions of the problem, we shall have x-25-2(x-60); from which we shall find x=$95. 17. After paying away of my money, and then of what was left, I had $180. What sum had I at first? 18. A line 37 feet in length is to be divided into 3 parts, so that the first may be 3 feet less than the second, and the second 5 more than the third. What are the parts? Let x represent the length of the third part; then x+5 represents the length of the 2d part; and The Equation will then be x+(x+5)+(x+2)=37; from which a will be found = 10 feet, the third part; hence 10+5=15 feet, the second part; and 10+2=12 feet, the first part. 19. A can perform a piece of work in 12 days, and B can perform the same in 15 days. In what time could both together do the work? Let x represent the number of days. Then, since A could do of the work, and B of it in 1 day, This Equation will give x=63 days. 20. If A can mow a certain field in 6 days, B in 8 days, and C in 5 days, in what time could the three together do it? Let x represent the number of days. Then, since A could mow of the field, B of it, and Cof 1 day, 6 represents the part A could mow in x days. 810 8100 8150 represents the part B could mow in x days; and represents the part C could mow in x days. We shall then have the Equation 21. Out of a cask of wine which had leaked away a third part, 20 gallons were afterwards drawn, and the cask was then found to be but half full. How much did it hold? 22. It is required to divide $300 between A, B, and C, so that A may have twice as much as B, and C as much as the other two together. Let x represent B's share; then 2x represents A's share; and 3x represents C's share. We shall then have the Equation x+2x+3x=300; which will give x=$50, B's share; hence 2×50-$100 is A's share; and 3 x 50-$150 is C's share. 23. A gentleman spends of his yearly income in board and lodging, of the remainder in clothes, and then has $20 left. What is the amount of his income? from which x will be found equal to $180. 24. A person, at the time he was married, was three times as old as his wife, but 15 years afterwards, he was only twice as old. What were their ages on their wedding day? Let x represent the wife's age; then 3x represents the husband's age; +15 represents the wife's age 15 years afterwards; and 3x+15 represents the husband's age 15 years afterwards. Then, by the conditions of the problem, we shall have 3x+15=2(x+15); from which x will be found = 15 years, the wife's age; hence 3 × 15=45 years, was the husband's age. 25. Two persons, A and B, lay out equal sums of money in trade; the first gains $126, and the second loses $87, and A's money is now double of B's. What did each lay out? Let a represent the sum each laid out; then x+126 represents the sum A had after gaining $126; and 26. A courier who travels 60 miles a day had been despatched 5 days, when a second is sent to overtake him, who goes 75 miles a day. In what time will he overtake him? Let x represent the number of days; then x+5 represents the number of days the first traveled ; 60(x+5) represents the number of miles the first traveled; and 75x represents the number of miles the second traveled. Since both traveled the same number of miles, we shall have 60(x+5)=75x; from which we shall find x=2 20 days. 27. An island is 60 miles in circumference, and two persons, A and B, start together to travel the same way around it; A goes 15 miles a day, and B 20. In what time would the two come together again? Let x represent the number of days; then 15x represents the distance A traveled; and 20x represents the distance B traveled. Since the distance B traveled exceeds the distance A traveled, by the circumference of the island, the Equation will be 20.r-15x=60; from which x will be found equal to 12 days. 28. A man and his wife usually drank out a cask of beer in 12 days, but when the man was from home it lasted the woman 30 days. How many days would the man alone be in drinking it? |