Cases in Equations of the First Degree. which reduces itself to B = 0, an obvious absurdity, unless B is zero. The sign∞ is, therefore, rather to be regarded as the expression of the peculiar species of absurdity which arises from diminishing the denominator of a fraction till it becomes zero. 125. Corollary. When we have A=0, and B = 0, which is equal to any quantity whatever, and is called an indeterminate expression. The given equation is, indeed, in this case an equation which is satisfied by any value whatever of x, and is called an identical equation. Equations of the First Degree with one unknown quantity. 7. Two capitalists calculate their fortunes, and it appears that one is twice as rich as the other, and that together they possess $38 700. What is the capital of each? Ans. The one has $ 12 900, the other $25 800. 8. To find two such numbers, that the one may be m times as great as the other, and that their sum = a. 9. The sum of $1200 is to be divided between two persons, A and B, so that A's share is to B's as 2 to 7. How much does each receive? Ans. A $266, B $933. Equations of the First Degree with one unknown quantity. 10. To divide a number a into two such parts, that the first part is to the second as m to n. 4th and 5th 11. How much money have I, when the parts of it amount together to $2,25? Ans. $5. 12. Find a number such, that when it is divided successively by m and by n, the sum of the quotients = a. 13. Divide the number 46 into two parts, so that when the one is divided by 7, and the other by 3, the sum of the quotients 10. Ans. 28 and 18. = 14. All my journeyings taken together, says a traveller, amount to 3040 miles; of which I have travelled 34 times as much by water as on horseback, and 24 times as much on foot as by water. How many miles did he travel in each of these three ways ? Ans. 240 miles on horseback, 840 miles by water, and 1960 miles on foot. 15. Divide the number a into three such parts, that the second may be m times, and the third n times as great as the first. a Ans. 1+m+n' 1+m+n' 1 + m + n2 16. A bankrupt leaves $21 000 to be divided among four creditors A, B, C, D, in proportion to their claims. Now A's claim is to B's as 2:3; B's claim : C's = 4:5; and C's claim: D's 6:7. How much does each creditor receive? Ans. A $3200, B 8 4800, C $ 6000, D $7000. Equations of the First Degree with one unknown quantity. 17. Divide the number a into three such parts, that the 1st shall be to the 2d as m to n; and the 2d part: the 3d=p: q. Ans. mpa nqa пра mp+np+ng' mp+np+ng' mp+np+ng 18. There are two numbers whose sum is 96, and difference 16; what are they? Ans. 56 and 40. 19. A father gives to his five sons $1000, which they are to divide according to their ages, so that each elder son shall receive $20 more than his next younger brother. What is the share of the youngest? Ans. 160. 20. One has six sons, each whereof is 4 years older than his next younger brother; and the eldest is three times as old as the youngest. What is the age of the eldest? Ans. 30 years. 21. There is a certain fish whose head is 9 inches; the tail is as long as the head and half the back; and the back is as long as both the head and the tail together. What is the length of the fish? Ans. 72 inches. 22. Five gamesters have lost jointly $40§; B's loss amounts to dollar more than triple A's; C's loss is $2 less than twice B's; D lost dollar less than A and B together; and E twice as much as B less dollar. How much did each of them lose? Ans. A $2, B $ 61, C $ 11, D $ 81, E $ 123. 23. A mason, 12 journeymen, and 4 assistants, receive together $72 wages for a certain time. The mason receives $1 daily, each journeyman dollar, and each assistant dollar. How many days must they have worked for this money Ans. 9 days. ? Equations of the First Degree with one unknown quantity. 24. Find a number such that if you multiply it by 5, subtract 24 from the product, divide the remainder by 6, and add 13 to the quotient, you will obtain this number. Ans. 54. 25. A courier left this place n days ago, and makes a miles daily. He is pursued by another making 6 miles daily. In how many days, will the second overtake the 26. A courier started from a certain place 12 days ago, and is pursued by another, whose speed is to that of the first as 8:3. In how many days will the second overtake the first? Ans. 7 days. 27. A courier started from this place n days ago, and is pursued by another whose speed is to that of the first as p is to q. In how many days will the second overtake the first? Ans. ng 28. Two bodies move in opposite directions; one moves c feet in a second, the other C feet. The two places, from which they start at the same time, are distant a feet from one another. When will they meet? 29. Two bodies move in the same direction from two places at a distance of a feet apart; the one at the rate of c feet in a second, the other pursuing it at the rate of C feet in a second When will they meet? 30. At 12 o'clock, both hands of a clock are together. When and how often will these hands be together in the next 12 hours? |
