Multiplying the second equation by 4, and subtracting the first Multiplying this last equation by 3, and subtracting the preced. ing one from the product, we obtain 40x=320, whence x=8. Substitute this value for x in the equation 15x+8z=144; it be comes 120+8z=144, whence z=3. Lastly, the two values x=8, z=3, being substituted in the equation 6x+y+5z=68, give 48+y+15=68, whence y=5. 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 ex In like manner 12x5 and 4x3 will express the quantity of silver contained in 5 ounces of the second ingot, and 3 ounces of the third. Now, we have 7×8 12×5 4×3 128 pound of the fourth ingot contains 8 ounces of silver, as required by the enunciation. The same conditions may be verified relative to the copper and pewter. 5. What two numbers are those, whose difference is 7, and sum 33? Ans. 13 and 20. 6. To divide the number 75 into two such parts, that three times the greater may exceed seven times the less by 15. Ans. 54 and 21. 7. In a mixture of wine and cider, of the whole plus 25 gal. lons was wine, and part minus 5 gallons was cider; how many gallons were there of each? Ans. 85 of wine, and 35 of cider. 8. A bill of £120 was paid in guineas and moidores, and the number of pieces of both sorts that were used was just 100; if the guinea be estimated at 21s. and the moidore at 27s. how many were there of each? Ans. 50 of each. 9. Two travellers set out at the same time from London and York, whose distance apart is 150 miles; one of them goes 8 miles a day, and the other 7; in what time will they meet? Ans. In 10 days. 10. At a certain election, 375 persons voted for two candidates, and the candidate chosen had a majority of 91; how many voted for each? Ans. 233 for one, and 142 for the other. 11. A's age is double of B's, and B's is triple of 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. 12. 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 price of the horse and harness; what did he give for each? X 13. Two persons, A and B, have both the same income: A saves } of his yearly, but B, by spending £50 per annum more than A, at the end of 4 years finds himself £100 in debt; what is their income? Ans. £125. 14. A person has two horses, and a saddle worth £50; now if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the first; what is the value of each horse? Ans. One £30, and the other £40. 15. To divide the number 36 into three such parts that of the first, of the second, and 4 of the third, may be all equal to each other. Ans. 8, 12, and 16. 16. A footman agreed to serve his master for £8 a year and a livery, but was turned away at the end of 7 months, and received only £2. 13s. 4d. and his livery; what was its value? Ans. £4. 16s. 17. To divide the number 90 into four such parts, that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, difference, product, and quotient so obtained, will be all equal to each other. Ans. The parts are 18, 22, 10, and 40. 18. The hour and minute hands of a clock are exactly together at 12 o'clock; when are they next together? Ans. 1h.5min. 19. 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 ? Ans. 20 days. 20. If A and B together can perform 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 person to perform the same work alone? Ans. A 140 days, B 174, and C 23. 21. A laborer can do a certain work expressed by a, in a time expressed by b; a second laborer, the work c in a timed; a third, the work e, in a time f. It is required to find the time it would take the three laborers, working together, to perform the work g. a=27; b=4 | c=35; d=6|e=40; f=12 | g=191; x will be found equal to 12. 22. If 32 pounds of sea water contain 1 pound of salt, how much fresh water must be added to these 32 pounds, in order that the quantity of salt contained in 32 pounds of the new mixture shall be reduced to 2 ounces, or of a pound? Ans. 224 lb. 23. A number is expressed by three figures; the sum of these figures is 11; the figure in the place of units is double that in the place of hundreds; and when 297 is added to this number, the sum obtained is expressed by the figures of this number reversed. What is the number? Ans. 326 24. A person who possessed 100,000 dollars, placed the greater part of it out at 5 per cent. interest, and the other part at 4 per cent. The interest which he received for the whole amounted to 4640 dollars. Required, the two parts. Ans. 64,000 and 36,000. 25. A person possessed a certain capital, which he placed out at a certain interest. Another person who possessed 10,000 dollars more than the first, and who put out his capital 1 per cent. more advantageously than he did, had an income greater by 800 dollars. A third person who possessed 15,000 dollars more than the first, and who put out his capital 2 per cent. more advantageously than he did, had an income greater by 1500 dollars. Required, the capitals of the three persons, and the three rates of interest. Sums at interest, Rates of interest, $30,000, $40,000, $45,000. 4 5 6 per cent. 26. A banker has two kinds of money; it takes a pieces of the first to make a crown, and b of the second to make the same sum. Some one offers him a crown for c pieces. How many of each kind must the banker give him? Ans. 1st kind, a(c-b) b(a-c) a-b 27. Find what each of three persons A, B, C, is worth knowing; 1st, that what A is worth added to I times what B and Care worth is equal top; 2d, that what B is worth added to m times what A and C are worth is equal to q; 3d, that what C is worth added to n times what A and B are worth is equal to r. This question can be resolved in a very simple manner, by intro ducing an auxiliary unknown quantity into the calculus. This unknown quantity is equal to what A, B and C are worth. 28. Find the values of the estates of six persons, A, B, C, D, E, F, from the following conditions: 1st. The sum of the estates of A and B is equal to a; that of C and D is equal to b; and that of E and F is equal to c. 2d. The estate of A is worth m times that of C; the estate of D is worth n times that of E, and the estate of Fis worth p times that of B. This problem may be resolved by means of a single equation, involving but one unknown quantity. Theory of Negative Quantities. Explanation of the terms, Nothing and Infinity. 104. The algebraic signs are an abbreviated language. They point out in the shortest and clearest manner the operations to be performed on the quantities with which they are connected. Having once fixed the particular operation indicated by a parti. cular sign, it is obvious that that operation should always be performed on every quantity before which the sign is placed. Indeed, the principles of algebra are all established upon the supposition, that each particular sign which is employed always means the same thing; and that whatever it requires is strictly performed. Thus, if the sign of a quantity is +, we understand that the quantity is to be added; if it is, we understand that it is to be subtracted. For example, if we have -4, we understand that this 4 is to be subtracted from some other number, or that it is the result of a subtraction in which the number to be subtracted was the greatest. If it were required to subtract 20 from 16, the subtraction could not be made by the rules of arithmetic, since 16 does not contain |