15. How many ells at Paris will make 24 yards at Boston, if 129 yards at Boston make 86 ells at Paris? 16. How many yards of cloth, 3 qrs. wide, are equal in measure to 30 yds., 5 qrs. wide? Ascertain the answer by inspection without any statement. 17. What is the width of a piece of cloth, 50 yards long, equal in measurement to 30 yards, 5 qrs. wide? 18. If 30 men can perform a piece of work in 11 days, how many men can accomplish a piece of work of the same kind, 4 times as large, and in part of the time? 1. In stating, multiply the time by 5 (Why ?), then ascertain the answer by inspection. 2. Ascertain the answer without statement. 19. A wall that is to be built to the height of 27 feet, was raised 9 feet by 12 men in 6 days. How many men must be employed to finish the wall in 4 days, at the same rate of working? How many feet in height remain to be done? In stating, divide the ratio of the height by 3 (why not by 9?), and ascertain the answer by inspection. 20. If 18 feet of a wall was built by 36 men in 4 days, how many days will 12 men take to finish it to the height of 27 feet? 21. A borrowed of his friend B $250 for 7 months, promising to do him the like kindness. Some time after, B had occasion for $300. How long may he keep it to receive full amends for the favor? Longer or shorter? How do you ascertain the answer to be & of 7 by inspection, without any statement? 22. If 6 pairs of gloves cost $7, what will be the cost of 50 pairs? What proportional part must be added to the perfect ratio? Get the answer without statement. Ans. $58. 23. If 6 men build a wall 20 feet long, 6 feet high, and 4 feet thick, in 16 days, in what time will 24 men build one 200 feet long, 8 feet high, and 6 feet thick? How often is the factor 2 repeated in the imperfect ratio? In writing the statement, then, preserve as many 2s in the consequent of the perfect ratio, endeavoring to dispense with as many of the other factors as may be. If properly stated, the answer will appear at a very slight inspection. If it does not, try to state in another form. 24. If 24 men can build a wall 200 feet long, 8 feet high, and 6 feet thick, in 80 days, how many men will be required to build one 20 feet long, 6 feet high, and 4 feet thick, in 16 days? 25. How many men may be hired for 16 days for $103.04, if the wages of 4 men for 3 days be $1104? State, and divide by 44.16. Why? 26. If the wages of 7 men for 16 days amount to $103.04, how many days will 4 men work for $11'04 at the same rate? State, and divide by 25'76. Why? 27. If 8 men mow 36 acres of grass in 9 days, by working 9 hours each day, how many men will be required to mow 48 acres in 12 days, by working 12 hours each day? If each part of the perfect ratio be written in its lowest terms, the answer can be found by inspection. 28. If 6 men can mow 48 acres in 12 days, working 12 hours a day, how many acres can 8 men mow in 9 days, working 9 hours a day? 29. If 10 cows eat 12 tons of hay in 9 weeks, how many cows will eat 56 tons in 21 weeks? State in lowest terms, and find the answer by inspection. 30. If 20 cows eat 56 tons of hay in 21 weeks, how long will 12 tons last 10 cows at the same rate? Write the perfect ratio in lowest terms, and increase it one half. Why? 31. If 4 cows eat 4 tons of hay in 8 weeks, how many cows will eat 93 tons in 4 weeks? In stating, multiply the first part of the perfect ratio by 5, the latter part by 3? Why? 32. If 16 cows eat 93 tons of hay in 4 weeks, how long will 4 tons last 4 cows? 33. If of a bushel of wheat cost $15, how much may be bought for $2? To what does the imperfect ratio relate? How much of wheat? 12.4 12.4 48 = 15 Ans. 1 of a bushel. Suggestive Questions.-Whence comes this 1 in the first form of the perfect ratio? By what number is the second multiplied? Whence comes the 4g in the third? The 1 of the answer? The first and second form of the perfect ratio are superfluous, except to a mere beginner. Solve the question three times; 1st, as above; 2d, by omitting the first form of the perfect ratio; 3d, omitting the first two forms; that is, performing them mentally. 34. If £g purchase 24 doz. steel pens, what will § of a penny purchase? Whence comes the 120 ? 5 720*8*288 •288 288. Ans. 13 pens. = 35. If 7 times of of an estate be worth $15,000, what is of of it worth? The perfect ratio is process. Try it. of by a mental Ans. $61213. 36. Bought of a yard of cloth for $11. What will a yard cost at the same rate? State the imperfect ratio in the form of an improper fraction, and find the answer $23 by inspection. 37. If of a yard of cloth cost $4, what will of a yard cost? Ans. $1 by inspection. 38. If yd. cost $3, what will 27 yds. cost? 39. If 27 yards of cloth cost $4015, what will yd. cost? 40. A merchant, owning of a vessel, sold of his share for $750. How much of the vessel did he own, and what was the value of the whole vessel at that rate? Value, $3000. 41. The purchaser mentioned in the last exercise sold of his share for $200. How much did he gain by the sale? Ans. $12.50. 42. A man bought of a barrel of flour, and sold of it to one of his neighbors at the same rate for $112. What would a barrel come to at that rate? 43. When flour is $6 a barrel, what is the value of 1⁄2 of of a barrel? 44. If of 3 of a yard of cloth cost $13, what will of yd. cost? 45. If of of a yard of cloth cost $2.40, what will of yd. cost? 46. If a train of cars move at the rate of 20 miles an hour, what portion of a mile do they travel in a second of time? 47. If a train of cars move uniformly of a mile per second, what is their rate per hour? 48. What would a pile of building stone cost, measuring 30 feet long, 26 feet broad, and 4 feet high, at $1.25 per perch of 16 feet long, 1 foot high, and 1 feet broad? Use no fractions in the perfect ratio. 49. A pile of building stone, measuring 30 feet long, 26 feet broad, and 4 feet high, was sold for $177; how much is that per perch of 16 feet long, 1 foot high, and 13 feet broad? 50. How many cords of wood are contained in a pile 200 feet long, 10 feet high, and 36 feet broad? The imperfect ratio here is 1. Ans. 562 cords. 51. A justice of the peace has an income of $1500 per annum, and the perquisites of his office average $7 per week. How much will he save per annum, if his expenses average $15 per week, counting 52 weeks to the year? Imperfect ratio, 8. Why? Solve this first by the aid of a statement; afterwards mentally, without the use of a slate or black-board. 52. The perquisites of the office of a justice of the peace are found to average $7 per week. His expenses average $15 per week, and yet he finds his savings at the end of each year of 52 weeks to amount to $1084. What must be the amount of his annual income from other sources ? 53. A contractor employed 400 workmen on a railroad for 4 weeks, paying them a ration of provisions every day, and 50 cents per working-day; that is, for 6 days in the week. At the end of the 4 weeks, he employed 200 additional hands at the same rate. The work was finished at the close of 12 weeks from its commencement. What was the whole expense, supposing the rations to cost 25 cents each? The rations and daily pay should be calculated separately. Why? Ans. $44,800. 54. Another railroad contractor employed 360 workmen for 12 weeks, and paid 85 cents per day, without rations; but, owing to rainy days and sickness, it was found that they only worked 18 days in four weeks on an average. How much had he to pay? Ans. $16,524. 55. A man left by will of his estate to his widow, of the remainder to a son, and the rest, amounting to $100, to his daughter. How much did he leave in all? This may be done mentally. Ans. $1600. 56. If of a pole stands in the mud, 3 feet in the water, and of its length above the water, what is the length of the pole? Ans. 8 feet. Interest, Discount, Commission, Insurance, and Percentage generally. DEFINITIONS. I. Interest is the sum of money given for the loan or use of another sum of money. II. Three elements enter into all calculations of interest, namely: the principal, or money lent; the interest paid for the use of the principal, sometimes called use; and the time for which the principal is lent. The III. The interest will evidently be proportional to the time; for, whatever interest is paid for the use of $100 for one year, twice that interest will be required for the use of the same sum for two years, and half that interest for half a year. interest is also proportional to the principal; since, if $6 be required for the use of $100 for any given time, $3 will evidently be required for the use of $50, and $12 for the use of $200 for the same time. IV. The whole sum paid back to the lender, which of course includes both principal and interest, is called the amount. V. In questions of interest, the Latin terms per cent. and per annum are very frequently used. The former means for a hundred; the latter, for a year. Thus, "at $6 per cent. per annum," means "at $6 for a hundred for a year." Sometimes one, at other times both these terms are omitted. But they must never be left out of the calculation. The interest paid per cent., or for a hundred, is frequently called the rate. VI. Interest is either simple or compound. Simple interest is that which is reckoned and allowed upon the principal only during the whole time of the loan; but compound interest is reckoned, not only on the principal, or sum lent, but also on the interest, if it remains unpaid after it becomes due. Thus, reckoning by simple interest, if $6 be the interest of any sum for one year, $12 will be the interest for two years; whereas by compound interest it will be $1236; for, in the former case, the same interest is charged in both years, whereas, in the latter, the interest is charged on $100 the first year, and on $106 (the amount of principal and unpaid interest) the second year. In order to discourage protracted settlements, the law does not allow compound interest on money lent; yet, in purchasing annuities, reversions, leases, &c., it is always allowed. VII. In calculating interest, a month is reckoned as 30 days, |