Therefore, in order to form a pound of the fourth ingot, we must take 8 ounces of the first, 5 ounces of the second, and 3 of the third. VERIFICATION. If there be 7 ounces of silver in 16 ounces of the first ingot, in 8 ounces of it there should be a number of ounces of silver expressed by will express the quantity of silver contained in 5 ounces of the second ingot, and 3 ounces of the third. Therefore a pound of the fourth ingot contains 8 ounces of silver, as required by the enunciation. The same conditions may be verified with respect to the copper and the pewter. 3. A's age is double B's, and B's is triple C's, and the sum of all their ages is 140. What is the age of each? Ans. A's, 84; B's, 42; and C's, 14. 4. A person bought a chaise, horse, and harness for £60. The horse came to twice the price of the harness; and the chaise, to twice the cost of the horse and harness. What did he give for each? Ans. £13 6s. 8d. for the horse; £6 13s. 4d. for the harness; £40 for the chaise. 5. Divide the number 36 into three such parts that of the first, of the second, and of the third, may be all equal to each other. Ans. 8, 12, and 16. 6. If A and B together can do a piece of work in 8 days, A and C together in 9 days, and B and C in 10 days, how many days would it take each to perform the same work alone? Ans. A, 14 days; B, 17; C, 23. 7. Three persons, A, B, and C, begin to play together, having among them all $600. At the end of the first game A has won one half of B's money, which, added to his own, makes double the amount B had at first. In the second game A loses and B wins just as much as C had at the beginning, when A leaves off with exactly what he had at first. How much had each at the beginning? Ans. A, $300; B, $200; C, $100. 8. Three persons, A, B, and C, together possess $3640. If B gives A $400 of his money, then A will have $320 more than B; but if B takes $140 of C's money, then B and C will have equal sums. How much has each? Ans. A, $800; B, $1280; C, $1560. 9. Three persons have a bill to pay, which neither alone is able to discharge. A says to B, "Give me the fourth of your money, and then I can pay the bill." B says to C, Give me the eighth of yours, and I can pay it." But C says to A, "You must give me the half of yours before I can pay it, as I have but $8." What was the amount of their bill, and how much money had A and B? Ans. Amount of the bill, $13; A had $10, and B $12. 10. A person possessed a certain capital, which he placed out at a certain interest. Another person, who possessed $10,000 more than the first, and who put out his capital 1 per cent more advantageously, had an annual income greater by $800. A third person, who possessed $5000 more than the first, putting out his capital 2 per cent more advantageously, had an annual income greater by $1500. Required the capitals of the three persons, and the rates of interest. Ans. Capitals, $30,000, $40,000, $45,000; rates of interest, 4%, 5%, 6%. 11. A widow receives an estate of $15,000 from her deceased husband, with directions to divide it among two sons and three daughters so that each son may receive twice as much as each daughter, she herself to receive $1000 more than all the children together. What was her share, and what the share of each child? Ans. The widow's share, $8000; each son's, $2000; each daughter's, $1000. 12. A certain sum of money is to be divided between three persons, A, B, and C. A is to receive $3000 less than half of it; B, $1000 less than one third part; and C, $800 more than the fourth part of the whole. What is the sum to be divided, and what does each receive? Ans. Sum, $38,400; A receives $16,200; B, $11,800; C, $10,400. 13. A person has three horses, and a saddle which is worth $220. If the saddle be put on the back of the first horse, it will make his value equal to that of the second and third; if it be put on the back of the second, it will make his value double that of the first and third; if it be put on the back of the third, it will make his value triple that of the first and second. What is the value of each horse? Ans. 1st, $20; 2d, $100; 3d, $140. 14. The crew of a ship consisted of her complement of sailors, and a number of soldiers. There were 22 sailors to every three guns, and 10 over; also the whole number of hands was five times the number of soldiers and guns together. But after an engagement, in which the slain were one fourth of the survivors, there wanted 5 men to make 13 men to every two guns. Required the number of guns, soldiers, and sailors. Ans. 90 guns, 55 soldiers, and 670 sailors. 15. Three persons have $96, which they wish to divide. equally between them. In order to do this, A, who has the most, gives to B and C as much as they have already; then B divides with A and C in the same manner, that is, by giving to each as much as he had after A had divided with them; C then makes a division with A and B; when it is found that they all have equal sums. How much had each at first? Ans. 1st, $52; 2d, $28; 3d, $16. 16. Divide the number a into three such parts that the first shall be to the second as m to n, and the second to the third as p Ans. x = anp 2= ang to q. y 17. Three masons, A, B, and C, are to build a wall. A and B together can do it in 12 days; B and C, in 20 days; and A and C, in 15 days. In what time can each do it alone, and in what time can they all do it if they work together? Ans. A, in 20 days; B, in 30; and C, in 60; all, in 10. INEQUALITIES. 117a. An inequality is an algebraic expression of two unequal quantities, connected by the sign of inequality. Thus, a>b is an inequality, showing that a is greater than b. Of two negative quantities, that one is the greater algebraically which has the fewer units. The part on the left of the sign is called the first member, and the part on the right the second member, of the inequality. Two inequalities subsist in the same sense when the greater quantity is in the first member of both or in the second member of both; they subsist in a contrary sense when the greater quantity is in the first member of one and in the second member of the other. Thus, the inequalities 35 30 and 18> 10 subsist in the same sense, and the inequalities 15> 13 and 12 <14 subsist in a contrary sense. The following principles enable us to transform inequalities: (1) If we add the same quantity to, or subtract the same quantity from, both members of an inequality, the resulting inequality will subsist in the same sense. Thus, if we add 5 to, and subtract 5 from, both members of the inequality we have 4>2, This principle enables us to transpose a term from one member of an inequality to the other by simply changing its sign. Thus, from the inequality 3x-b>2x+a, we find, by transposition, x> a+b. (2) If two members of an inequality be multiplied or divided by a positive quantity, the resulting inequality will subsist in the same sense. Thus, if we multiply or divide both members of the inequality 12 > 8 by +4, we have 48 32 and 3>2. |