cent.? der per gallon to gain upon the whole cost at the rate of 10 per Ans. $1'265 per gallon. 25. A merchant bought 10 tons of iron for $950; the freight and duties came to $145, and his own charges to $25; how must he sell it per lb. to gain 20 per cent. by it? Ans. 6 cents per lb. EQUATION OF PAYMENTS. T 92. Equation of payments is the method of finding the mean time for the payment of several debts, due at different times. 1. In how many months will $1 gain as much as 5 dollars will gain in 6 months? 2. In how many months will $1 gain as much as $40 will gain in 15 months? Ans. 600. 3. In how many months will the use of $5 be worth as much as the use of $1 for 40 months? 4. Borrowed of a friend $1 for 20 months; afterwards lent my friend $4; how long ought he to keep it to become indemnified for the use of the $1 ? 5. I have three notes against a man; one of $12, due in 3 months; one of $9, due in 5 months; and the other of $6, due in 10 months; the man wishes to pay the whole at once; in what time ought he to pay it? $12 for 3 months is the same as $1 for 36 months, and He might, therefore, have $1 141 months, and he may keep 27 dollars part as long; that is, 141 5 months, 6+ days, Answer. Hence, To find the mean time for several payments,-RULE :Multiply each sum by its time of payment, and divide the sum of the products by the sum of the payments, and the quotient will be the answer. Note. This rule is founded on the supposition, that what is gained by keeping a debt a certain time after it is due, is the same as what is lost by paying it an equal time before it is due; but, in the first case, the gain is evidently equal to the interest on the debt for the given time, while, in the second case, the loss is only equal to the discount of the debt for that time, which is always less than the interest; therefore, the rule is not exactly true. The error, however, is so trifling, in most questions that occur in business, as scarce to merit notice. 6. A merchant has owing him $300, to be paid as follows: $50 in 2 months, $100 in 5 months, and the rest in 8 months; and it is agreed to make one payment of the whole: in what time ought that payment to be? Ans. 6 months 7. A owes B $136, to be paid in 10 months; $96, to be paid in 7 months; and $260, to be paid in 4 months: what is the equated time for the payment of the whole? Ans. 6 months, 7 days +. 8. A owes B $600, of which $200 is to be paid at the present time, 200 in 4 months, and 200 in 8 months; what is the equated time for the payment of the whole? Ans. 4 months. 9. A owes B $300, to be paid as follows: in 3 months, months, and the rest in 6 months: what is the equated time?" in 4 Ans. 4 months. RATIO; OR THE RELATION OF NUMBERS. ¶ 93. 1. What part of 1 gallon is 3 quarts? 1 gallon is 4 quarts, and 3 quarts is of 4 quarts. Ans. of a gallon. 2. What part of 3 quarts is 1 gallon? 1 gallon, being 4 quarts, is of 3 quarts; that is, 4 quarts is 1 time 3 quarts and of ano ther time. 3. What part of 5 bushels is 12 bushels? Ans. 13. Finding what part one number is of another is the same as finding what is called the ratio, or relation of one number to another; thus, the question, What part of 5 bushels is 12 bushels? is the same as, What is the ratio of 5 bushels to 12 bushels? The Answer is 12 22. 1 Ratio, therefore, may be defined, the number of times one number is contained in another; or, the nuinber of times one quantity is conlained in another quantity of the same kind. 4. What part of 8 yards is 13 yards? or, What is the ratio of 8 yards to 13 yards? 13 yards is of 8 yards, expressing the division fractionally. If now we perform the division, we have for the ratio 13; that is, 13 yards is I time 8 yards, and § of another time. We have seen, (T 15, sign,) that division may be expressed fractionally. So also the ratio of one number to another, or the part one number is of another, may be expressed fractionally; to do which, make the number which is called the part, whether it be the larger or the smaller number, the numerator of a fraction, under which write the other number for a denominator. When the question is, What is the ratio, &c. ? the number last named is the part; consequently it must be made the numerator of the fraction, and the number first named the denominator. 5. What part of 12 dollars is 11 dollars? or, 11 dollars is what part of 12 dollars? 11 is the number which expresses the part. To put this question in the other form, viz. What is the ratio, &c. ? let that number which expresses the part be the number last named; thus, What is the ratio of 12 dollars to 11 dollars? Ans. 6. What part of 1 £. is 2s. 6d. ? or, What is the ratio of 1 £. to 2 s. 6 d.? 1 £. 240 pence, and 2 s. 6 d. 30 pence; hence, 30= the Answer. }, is 7. What part of 13 s. 6 d. is 1 £. 10 s.? or, What is the ratio of 13 s. 6 d. to 1 £. 10 s.? Ans. 20 20. 9 of 7 of 90 to 15? of 615 to 1107 ? Ans. to the last, §. PROPORTION; OR THE RULE OF THREA. T94. 1. If a piece of cloth, 4 yards long, cost 12 dollars, what will be the cost of a piece of the same cloth 7 yards long? Had this piece contained twice the number of yards of the first piece, it is evident the price would have been twice as much; had it contained 3 times the number of yards, the price would have been 3 times as much; or had it contained only half the number of yards, the price would have been only half as much; that is, the cost of 7 yards will be such part of 12 dollars as 7 yards is part of 4 yards. 7 yards is of 4 yards; consequently, the price of 7 yards must be of the price of 4 yards, or of 12 dollars. of 12 dollars, that is, 12 x 21 dollars, Answer. 84. 2. If a horse travel 30 miles in 6 hours, how many miles will he travel in 11 hours, at that rate? 11 hours is 11 of 6 hours, that is, 11 hours is 1 time 6 hours, and of another time; consequently, he will travel, in 11 hours, 1 time 30 miles, and of another time, that is, the ratio between the distances will be equal to the ratio between the times. 11 of 30 miles, that is, 30 x 11380=55 miles. If, then, no error has been committed, 55 miles must be 11 of 30 miles. This is actually the case; for. Ans, 55 miles. Quantities which have the same ratio between them are said to be proportional. Thus, these four quantities, hours. hours. miles. miles. 6, 11, 30, 55, written in this order, being such, that the second contains the first as many times as the fourth contains the third, that is, the ratio between the third and fourth being equal to the ratio between the first and second, form what is called a proportion. It follows, therefore, that proportion is a combination of two equal ratios. Ratio exists between two numbers; but proportion requires at least three To denote that there is a proportion between the numbers 6, 11, 30, and 55, they are written thus: 6: 11: 30: 55 which is read, 6 is to 11 as 30 is to 55; that is, 6 is the same part of 11 that 30 is of 55; or, 6 is contained in 11 as many times as 30 is contained in 55: or, lastly, the ratio or relation of 11 to 6 is the same as that of 55 to 30. 95. The first term of a ratio, or relation, is called the antecedent, and the second the consequent. In a proportion there are two antecedents, and two consequents, viz. the antecedent of the first ratio, and that of the second; the consequent of the first ratio, and that of the second. In the proportion 6 11 :: 30: 55, the antecedents are 6, 30; the consequents, 11, 55. The consequent, as we have already seen, is taken for the numerator, and the antecedent for the denominator of the fraction, which expresses the ratio or relation. Thus, the first ratio is, the second 1; and that these two ratios are equal, we know, because the fractions are equal. The two fractions and being equal, it follows that, by reducing them to a common denominator, the numerator of the one will become equal to the numerator of the other, and, consequently, that 11 multiplied by 30 will give the same product as 55 multiplied by 6. This is actually the case; for 11 x 30 330, and 55 X 6 = 330. Hence it follows,-If four numbers be in proportion, the product of the first and last, or of the two extremes, is equal to the product of the second and third, or of the two means. Hence it will be easy, having three terms in a proportion given, to find the fourth. Take the last example. Knowing that the distances travelled are in proportion to the times or hours occupied in travelling, we write the proportion thus: hours. hours. miles. miles. 6 : 11: 30 : Now, since the product of the extremes is equal to the product of the means, we multiply together the two means, 11 and 30, which makes 330, and, dividing this product by the known extreme, 6, we obtain for the result 55, that is, 55 miles, which is the other extreme, or term, sought. 3. At $54 for 9 barrels of flour, how many barrels may be purchased for $186 ? In this question, the unknown quantity is the number of barrels bought for $186, which ought to contain the 9 barrels as many times as $186 contains $54; we thus get the following proportion : Any three terms of a proportion being given, the operation by which we find the fourth is called the Rule of Three. A just solution of the question will sometimes require, that the order of the terms of a proportion be changed. This may be done, provided the terms be so placed, that the product of the extremes shall be equal to that of the means. 4. If 3 men perform a certain piece of work in 10 days, how long will it take 6 men to do the same? The number of days in which 6 men will do the work being the term sought, the known term of the same kind, viz. 10 days, is made the third term. The two remaining terms are 3 men and 6 men, the ratio of which is . But the more men there are employed in the work, the less time will be required to do it; consequently, the days will be less in proportion as the number of men is greater. There is still a proportion in this case, but the order of the terms is inverted; for the number of men in the second set, being two times that in the first, will require only one half the time. The first number of days, therefore, ought to contain the second as many times as the second number of men contains the first. This order of the terms being the reverse of that assigned to them in announcing the question, we say, that the number of men is in the inverse ratio of the number of days. With a view, therefore, to the just solution of the question, we reverse the order of the two first terms, (in doing which we invert the ratio,) and, instead of writing the proportion, 3 men : 6 men, (,) that is, we write it, 6 men : 3 men, (,) men. men. days. days. 6 : 3 :: 10 : .. Note. We invert the ratio when we reverse the order of the terms in the proportion, because then the antecedent takes the place of the consequent, and the consequent that of the antecedent; consequently, the terms of the fraction which express the ratio are inverted; hence the ratio is inverted. Thus, the ratio expressed by 62, being inverted, is = Having stated the proportion as above, we divide the product of the means, (10 X 330,) by the known extreme, 6, which gives 5, that is, 5 days, for the other extreme, or term sought. Ans. 5 days. From the examples and illustrations now given we deduce the following general RULE. Of the three given mumbers, make that the third term which is of the same kind with the answer sought. Then consider, from the nature of the question, whether the answer will be greater or less than this term. If the answer is to be greater, place the greater of the two remaining numbers for the second term, and the less number for the first term; but if it is to be less, place the less of the two remaining numbers for the second term, and the greater for the first; and, in either case, multiply the second and third terms together, and divide the product by the first for the answer, which will always be of the same denomination as the third term. Note 1. If the first and second terms contain different denominations, they must both be reduced to the same denomination; and if the third term be a compound number, it either must be reduced to integers of the lowest denomination, or the low denominations must be educed to a fraction of the highest denomination contained in it. Note 2. The same rule is applicable, whether the given quantities be integral, fractional, or decimal. EXAMPLES FOR PRACTICE. 5. If 6 horses consume 21 bushels of oats in 3 weeks, how many bushels will serve 20 horses the same time? Ans. 70 bushels. • The rule of three has sometimes been divided into direct and inverse, a distinction which is totally useless. It may not however be amiss to explain, in this place, in what this dis tinction consists. The Rule of Three Direct is when more requires more, or less requires less, as in this example:-If 3 men dig a trench 48 feet long in a certain time, how many feet will 12 men dig in the same time? Here it is obvious, that the more men there are employed, the more work will be done; and therefore, in this instance, more requires more. Again-If 6 men dig 49 feet in a given time, how much will 3 men dig in the same time? Here less requires less, for the less men there are employed, the less work will be done. The Rule of Three Inverse is when more requires less, or less requires more, as in this example: If 6 men dig a certain quantity of trench in 14 hours, how many hours will it require 12 men to dig the same quantity? Here more requires less; that is, 12 men being more than 6, will require less time. Again:-if 6 men perform a piece of work in 7 days, how long will 3 men be in performing the same work? Here less requires more; for the number of men, being less, will require more time. |