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 7 × 8 expressed by 16 12 x 5 and 4 x 3 will express the quan In like manner, tity of silver contained in 5 ounces of the second ingot, and 3 ounces of the third. Now, we have therefore, a 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 sum 33 ? difference is 7, and 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 gallons 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 were 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; for each? £13 6s. 8d. Ans. £ 6 13s. 4d. what did he give for the horse. for the harness. 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 the income of each? 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 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? 17. To divide the number 90 into four such Ans. £4 16s. parts, that if the by 2, the third first be increased by 2, the second diminished 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? X Ans. 1 h. 5 min. 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 be in drinking it alone? 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 143 days, B 1723, 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 time d; a third, the work e in a time f. Required the time it would take the three laborers, working together, to perform the work g. 22. 1f 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 lbs. 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 possessed 10,000 dollars more than the first, and putting out his capital 1 per cent. more advantageously, had an income greater by 800 dollars. A third, possessed 15,000 dollars more than the first, and putting out his capital 2 per cent. more advantageously, had an income greater by 1500 dollars. Required the capitals, and the three rates of 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? 27. Find what each of three persons, A, B, C, is worth, knowing, 1st, that what A is worth added to 7 times what B and C are worth, is equal to p; 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. If we denote by x what A, B, and C, are worth, we introduce into the calculus an auxiliary unknown quantity, and resolve the question in a very simple manner. The term calculus, in its general sense, denotes any operation performed on algebraic quantities. 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 F is worth p times that of B. This problem may be resolved by means of a single equation, involving but one unknown quantity. Explanation of Negative Results. 104. The algebraic signs are an abbreviated language. They indicate certain operations which are to be performed on the quantities before which they are placed. The operation indicated by a particular sign, must 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 means always 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 the sign is -, we understand that it is to be subtracted. For example, if we have 4, it indicates that this 4 is to be subtracted from some other number, or that it is the result of a subtraction but partially made. If it were required to subtract 20 from 16, the subtraction could not be made by the rules of arithmetic, since 20 is greater than 16. By observing that 2016+ 4, we may express the subtraction thus, 16 2016 16 4=- 4. We thus make the subtraction of 20 from 16 as far as it is possible, and obtain a remainder 4 with a minus sign, which indicates that 4 is still to be treated as a subtractive quantity. To show the necessity of giving to this remainder its proper sign, let us suppose that 10 is to be added to the difference of 16-20; or what is the same thing, that 20 is to be subtracted from 26. The numbers would then be written and had the sign not been preserved in the first subtraction, the second result would have been + 14 instead of +6. 105. If the sum of the negative quantities in the first member of the equation, exceeds the sum of the positive quantities, the second member of the equation will be negative, and the verification of the equation will show it to be so. For example, if α b = c, and we make a 15 and b = 18, c will be = 3. Now, the essential sign of c is different from its algebraic sign in the equation. This arises from the circumstance, that the equation expresses generally, the difference between a and b, without indicating which of them is the greater. When, therefore, we attribute particular values to a and b, the sign of c, as well as its value, becomes known. We will illustrate these remarks by a few examples. |