3. There are two numbers whose sum is 28, and onc fourth of the first is 3 less than one fourth of the second. What are the numbers? Ans. 8 and 20. 4. The ages of two persons, A and B, are such that 5 years ago B's age was three times A's; but 15 years hence B's age will be double A's. What is the age of each? Ans. A's, 25; B's, 65. 5. There are two numbers such that one third of the first added to one eighth of the second gives 39; and four times the first minus five times the second is zero. What are the numbers? 6. Find a fraction such that if 6 is added to the numerator its value will be, but if 3 be added to the denominator its value will be ? Ans. 2 7. What are the two numbers whose difference is to their sum as 1:2, and whose sum is to their product as 4:3? SOLUTION. Let x the greater and y = the less. Then c−g: +y=1:2 (1) x y: 2x-2y=x+y (3) x+y:xy= 4:3 (2) 3x+3y=4xy (4) x=3y (5) 9y+3y=12y2 (6) x = 3 (7) 1=y (8) Having written (1) and (2) in accordance with the statement in the problem, we form from them (3) and (4) by Art. 106. Reducing (3), we obtain (5); substituting this value of x in (4), we have (6), which, though an equation of the second degree, can be at once reduced to an equation of the first degree by dividing each term by y; performing this division and reducing, we obtain (8) or y = = 1; substituting this value of y in (5) we obtain (7), or x = 3. 8. What are the two numbers whose difference is to their sum as 3: 20, and three times the greater minus twice the less is 35? 9. There is a number consisting of two figures, which is seven times the sum of its figures; and if 36 is subtracted from it, the order of the figures will be inverted. What is the number? Ans. 84. 10. There is a number consisting of two figures, the first of which is the greater; and if it is divided by the sum of its figures, the quotient is 6; and if the order of the figures is inverted, and the resulting number divided by the difference of its figures plus 4, the quotient will be 9. What is the number? Ans. 54. 11. As John and James were talking of their money, John said to James, "Give me 15 cents, and I shall have four times as much as you will have left." James said to John, "Give me 73 cents, and I shall have as much as you will have left." How many cents did each Ans. John, 45 cents; James, 30 cents. have? 12. The height of two trees is such that one third of the height of the shorter added to three times that of the taller is 360 feet; and if three times the height of the shorter is subtracted from four times that of the taller, and the remainder divided by 10, the quotient is 17. quired the height of each tree. Ans. 90 and 110 feet. 13. A farmer who had $41 in his purse gave to each man among his laborers $2.50, to each boy $1, and had $15 left. If he had given each man $4 and then each boy $3 as long as his money lasted, 3 boys would have received nothing. How many men and how many boys did he hire? 14. A man worked 10 days and his son 6, and they received $31; at another time he worked 9 days and his son 7, and they received $29.50. What were the wages of each? 15. A said to B, "Lend me one fourth of your money, and I can pay my debts." B replied, "Lend me $100 less than one half of yours, and I can pay mine." Now A owed $1200 and B $1900. each have in his possession ? How much money did Ans. A, $800; B, $1600. 16. If a is added to the difference of two quantities, the sum is b; and if the greater is divided by the less, the quotient will be c. What are the quantities? 17. A man owns two pieces of land. Three fourths of the area of the first piece minus two fifths of the area of the second is 12 acres; and five eighths of the area of the first is equal to four ninths of the area of the second. How many acres are there in each? Ans. 1st, 64 acres; 2d, 90 acres. 18. A and B begin business with different sums of money; A gains the first year $350, and B loses $500, and then A's stock is to B's as 9: 10. If A had lost $500 and B gained $350, A's stock would have been to B's as 1: 3. With what sum did each begin? Ans. A, $1450; B, $2500. 19. If a certain rectangular field were 4 feet longer and 6 feet broader, it would contain 168 square feet more; but if it were 6 feet longer and 4 feet broader, it would contain 160 square feet more. Required its length and breadth. 20. A market-man bought eggs, some at 3 for 7 cents and some at 2 for 5 cents, and paid for the whole $2.62; he afterward sold them at 36 cents a dozen, clearing $0.62. How many of each kind did he buy? 21. A and B can perform a piece of work together in 12 days. They work together 7 days, and then A finishes the work alone in 15 days. How long would it take each to do the work? Ans. A 36 and B 18 days. 22. "I was ten times as old as you 12 years ago," said a father to his son; "but 3 years hence I shall be only two and one half times as old as you." What was the age of each? 23. If 3 is added to the numerator of a certain fraction, its value will be ; and if 4 is subtracted from the denominator, its value will be . What is the fraction? 24. A farmer sold to one man 7 bushels of oats and 5 bushels of corn for $12.76, and to another, at the same rate, 5 bushels of oats and 7 bushels of corn for $13.40. What was the price of each? 25. Find two quantities such that one third of the first minus one half the second shall equal one sixth of a; and one fourth of the first plus one fifth of the second shall equal one half of a. Ans. 34 a 23 and 15 a 23 26. A person had a certain quantity of wine in two casks. In order to obtain an equal quantity in each, he poured from the first into the second as much as the second already contained; then he poured from the second into the first as much as the first then contained; and, lastly, he poured from the first into the second as much as the second still contained; and then he had 16 gallons in each cask. How many gallons did each originally contain? Ans. 1st, 22; 2d, 10 gallons. SECTION XV. EQUATIONS OF THE FIRST DEGREE CONTAINING MORE THAN TWO UNKNOWN QUANTITIES. 117. The methods of elimination given for solving equations containing two unknown quantities apply equally well to those containing more than two unknown quantities. Multiplying equation (1) by 2 gives equation (4), which we subtract from (2), and obtain (6); multiplying (1) by 3 gives (5), and subtracting (5) from (3) gives (7). We have now obtained two equations, (6) and (7), containing but two unknown quantities. Multiplying (6) by 5, we obtain (8), and subtracting (7) from (8), we obtain (9), which reduced gives z=1. Substituting this value of z in (6), and reducing, we obtain y = 3. Substituting these values of y and z in (1), and reducing, we obtain x = 2. |