RULE OF THREE INVERSE. Q. What is the Rule of Three Inverse? A. The Rule of Three Inverse teaches, by having three numbers given to find a fourth, which shall have the same proportion to the second, as the first has to the third. EXPLANATIONS. The Rule of Three Inverse, like the Rule of Three Direct, is merely an application of Multiplication and Division. The only difference, or distinction between Rule of Three Direct and Inverse, is, that in direct proportion more requires more, or less requires less in every question belonging to that rule; but, in inverse proportion, more requires less, and less requires more. More requiring less, is when the third term is greater than the first, and requires the fourth term, or answer, to be less than the second. Less requiring more, is when the third term is less than the first, and requires the fourth term, or answer, to be greater than the second. The greatest difficulty, and, in fact, the only difficulty, which you will have to encounter, will be to distinguish inverse from direct proportion; but if you will pay particular attention to the rule for stating the questions, which is the same in both, you will, with the greatest ease, be able to decide to which it belongs, after having considered whether less requires more, or more requires less. RULE. Q. How do you state and work the questions in Rule of Three Inverse? A. The questions must be stated, and the terms reduced, if of different denominations, the same as in Rule of Three Direct. You must then multiply the first and second terms together, and divide the product by the third, and the quotient will be the answer, in a denomination of the same name to which the second term was reduced.. EXAMPLES. 1. If 6 men can build a house in 96 days, in how many days can 72 men build it? Ans. 8da. EXPLANATIONS. men. da. men. 6: 96: 72 .6 72)576 (8da.A. 576 In this example, 72 men, which moves the question, is the third term, 6 men, the same kind, is the first, and 96 days, the second, the same as the answer. Here, more requires less, and it is, therefore, very evident, from the conditions of the question, that this sum belongs to the Rule of Three Inverse; for, it is perfectly plain, that the more men there are employed, the less time it will require to build the house. In this example, also, the third term is larger than the first, and requires the fourth term, or answer, to be less than the second; and it is very evident, that the fourth term, or answer, should be only one twelfth part as large as the second term, because it must require only one twelfth the number of days for 72 men to do a piece of work, that it would require 6 men to do the same piece of work. It will also appear, that the fourth term, or answer, 8 days, bears the same proportion to the second term, 96 days, that the first term, 6 men, bears to the third term, 72. You must, therefore, multiply the first and second terms together, and divide by the third, and the quotient will be the answer. 2. If $100, in 12mo. bring $6 interest, what sum, or principal, will bring the same in 8mo.? Ans. $150. 3. If 30bu. of grain, at 50c. a bushel, will pay a debt, how many bushels, at 75c. a bushel, will pay the debt? Ans. 20bu. 4. If 8 men can mow a piece of meadow in 24 days, how many men can do it in 4 days? Ans. 48 men. 5. If I lend my friend $100 for 180 days, how long ought he to lend me $450 to return the kindness? Ans. 40da. 6. If a traveller performs a journey in 10 days, when the day is 12 hours long, in how many days would he perform the same journey, when the day is but hours long? Ans. 15da. 7. How much land, at $2,50c. an acre, should be given in exchange for 360 acres, at $3,75c. an acre? Ans. 540a. 8. A garrison of 1200 men has provisions for 9 months, at the rate of 14oz. per man a day, how long will the provisions last, at the same allowance, if the garrison be re-enforced by 400 men? Ans. 63mo. 9. The imperial canal, which intersects China from north to south, employed 30,000 men 43 years in its construction, how many men must have been employed to have completed it in 1 year? Ans. 1,290,000 men. 10. If, when wheat is 83c. a bushel, the cent loaf weighs 9oz., how many ounces should it weigh when wheat is $1,24,5m. a bushel? Ans. 6oz. 11. How many yards of paper, 3gr. wide, will paper a room that is 24yd. round, and 4yd. high? Ans. 128yd. 12. If, for $6, a merchant has 4cwt. of goods carried 160m., how far can he have 20cwt., or 1 tun, carried for the same money? Ans. 32m. 13. How many yards of baize, 3qr. wide, will line a cloak, which has in it 12yd. of camlet, a yard wide? Ans. 8yd. 14. If a board be 9in. in width, how much in length will make a square foot? Ans. 16in. 15. How many yards of sarcenet, that is 3gr. wide, will line 18yd. of cloth, 2yd. wide? Ans. 48yd. 16. A cistern has a pipe which will empty it in 15 hours, how many pipes of the same capacity will empty it in 45min. Î Ans. 20 pipes. DISCOUNT. Q. What is Discount? A. Discount is an allowance made for the payment of any sum of money before it becomes due, or upon advancing ready money on notes, bills, obligations, &c., which are payable at some future day or period. EXPLANATIONS. As was stated on page 168, Discount is only an application of the Rule of Three Direct, and has this name given to it merely as applicable to its application and use in this par ticular transaction of business. RULE. Q. How do you state and work the terms to find the discount of any given sum? A. As the amount of $100, or £100, at the given rate and time, is to the interest of $100, or £100, at the same rate and time, so is the given sum to the discount. To find the present worth of a given sum, you must substract the discount from the given sum, and the remainder will be the present worth, or such a sum as if put to interest would, in the given time, and at the given rate per cent., amount to the sum or debt then due. Or, to find the present worth of any given sum, you may state it thus: as the amount of $100, or £100, is to $100, or £100, so is the given sum or debt to the present worth. EXAMPLES. 1. What is the discount of $100, due 1 year hence, at 7 per cent. a year? Ans. $6,54,2m. EXPLANATIONS. In this example, you multiply the given sum, $100, by 7, the rate per cent., and divide the product by $107, the amount of $100 and the rate per cent., $7, added together, and the quotient is the answer, or discount of $100 for 1 year at 7 per cent. 107 $ 7 100 $c.m. 107) 700 (6,54,2 discount. 642 580 535 450 428 220 214 2. What is the present worth of $100, due 1 year hence, at 7 per cent.? Ans. $93,45,7m. EXPLANATIONS. In this example, as before, multiply the second and third terms together, and divide the product by the first term, and the quotient will be the present worth of $100, due a year hence, at 7 per cent. You will at once perceive, that $100, the second term, is the present worth of $107, due a year hence; for $100 put to interest at 7 per cent., in one year, amounts to $107. It is also perfectly evident, that the fourth term, or 750 749 101 answer, bears the same proportion to $100, the third term, that $100, the second term, bears to $107, the first term. By the preceding examples and EXPLANATIONS, you will readily see, that, in discount, money is supposed not to bear interest until it is due; and it is perfectly reasonable and just, that a discount should be made for the payment of money before it is due; for the debtor, by retaining the money until it is due, may put it to interest for the time; but, should he pay it before it becomes due, he will give that advantage and benefit to another. Many persons have very incorrectly supposed, that the interest on any given sum, for a given rate and time, was the discount, and that this interest, taken from the given sum, or principal, would give the present worth. This supposition is, however, very erroneous; for, if that were true, the discount on $100 would be $7, and the present worth of $100, due, or payable, one year hence, at 7 per cent., would be $93; but $93. put to interest for one year, will not amount |