PRODUCING EQUATIONS OF THE FIRST DEGREE CONTAINING TWO UNKNOWN QUANTITIES. 116. Many of the problems given in Section XIII. contain two or more unknown quantities; but in every case these are so related to each other that, if one becomes known, the others become known also; and therefore the problems can be solved by the use of a single letter. But many problems, on account of the complicated conditions, cannot be performed by the use of a single letter. No problem can be solved unless the conditions given are sufficient to form as many independent equations as there are unknown quantities. 1. A grocer sold to one man 7 apples and 5 pears for 41 cents; to another at the same rate 11 apples and 3 pears for 45 cents. What was the price of each ? We multiply (2) by 5 and (1) by 3, and obtain (3) and (4); subtracting (4) from (3) we have (5), which reduced gives (6), or x= 3. Substituting this value of x in (1), we have (7), which reduced gives (8), or y = 4. 2. There is a fraction such that if 2 is added to the umerator the fraction will be equal to ; but if 3 is added to the denominator the fraction will be equal to . What is the fraction? Clearing (1) and (2) of fractions, we obtain (3) and (4); subtracting (4) from (3), we obtain (5), which reduced gives (6), or x = 7. Substituting this value of x in (4), we have (7), or y=18. 3. There are two numbers whose sum is 28, and one 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. 22. 7. What are the two numbers whose difference is to their sum as 12, and whose sum is to their product as 4:3? SOLUTION. Let x the greater and y = the less. Then x−y: +y=1:2 (1) x+y:xy= 4:3 (2) 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 7 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. Required 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? |