4x hence, the amount for which the eggs were sold. therefore Or, each sort. += 4; 3 5 15x + 10x 24x = 120. x = 120 the number of eggs of 3. A person possessed a capital of 30,000 dollars, for which he drew a certain interest per annum; but he owed the sum of 20,000 dollars, for which he paid a certain interest. The interest that he received exceeded that which he paid by 800 dollars. Another person possessed $35,000, for which he received interest at the second of the above rates; but he owed 24,000 dollars, for which he paid interest at the first of the above rates. The interest that he received exceeded that which he paid by 310 dollars. Required the two rates of interest. Let x and y denote the two rates of interest: that is, the interest of $100 for one year. To obtain the interest of $30,000 at the first rate, denoted by x, we form the proportion And for the interest $20,000, the rate being y, But from the enunciation, the difference between these two in terests is equal to 800 dollars. We have, then, for the first equation of the problem, 300x - 200y = 800. By expressing the second condition of the problem algebraically, we obtain the other equation, 350y - 240x 310. Both members of the first equation being divisible by 100, and those of the second by 10, we may put the following, in place of them: 3x 2y = 8, 35y - 24x = 31. To eliminate x, multiply the first equation by 8, and then add it to the second; there results 19y = 95, whence y = 5. Substituting for y its value in the first equation, this equation becomes Therefore, the first rate is 6 per cent., and the second 5. The second condition can be verified in the same manner. 4. There are three ingots formed by mixing together three metals in different proportions. One pound of the first contains 7 ounces of silver, 3 ounces of copper, and 6 ounces of pewter. One pound of the second contains 12 ounces of silver, 3 ounces of copper, and 1 ounce of pewter. One pound of the third contains 4 ounces of silver, 7 ounces of copper, and 5 ounces of pewter. • It is required to form from these three, 1 pound of a fourth ingot which shall contain 8 ounces of silver, 3 ounces of copper, and 44 ounces of pewter. Let x= the number of ounces taken from the first. Now, since 1 pound or 16 ounces of the first ingot contains 7 ounces of silver, one ounce will contain of 7 ounces: that 1 16 But since 1 pound of the new ingot is to contain 8 ounces of silver, we have As the co-efficients of y in these three equations, are the most simple, we will eliminate this unknown quantity first. Multiplying the second equation by 4 and subtracting the first, gives 5x + 24z = 112. Multiplying the third equation by 3 and subtracting the second, gives 15x + 8z = 144. Multiplying the last equation by 3 and subtracting the first, gives 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 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 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, 1 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 I 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? for the horse. for the harness. for the chaise. Ans. { £13 6s. 8d. £6 13s. 4d. £40 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 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. 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. |