184-187. PER CENT. 4. Loss and Gain. 69 184. If I buy a horse for $50, and sell it again for $56, what do I gain per cent.? Subtracting 50 dollars from 56 dollars, we find that 50 doilars gains & dollars, and dividing 6 dollars by 50 dollars, we find $0.12 to be the gain on $1, or 12 cents on 100 cents, or $12 on $100, or 12 per cent. Hence 185. To know what is gained or lost per cent. RULE. Find the gain or loss on the given quantity by subtraction. Divide this gain or loss by the price of the given. quantity, and the quotient will be the gain or loss per cent. QUESTIONS FOR PRACTICE. 2. If I buy cloth for $1.25 a yard, and sell it again for $1.30, what do I gain per ct.? 1.25).0500(0.04 per cent. 500 Ans.* 3. If I buy salt for 84 cents a bushel, and sell it for $1.12 a bushel, what do I gain per cent.? Ans. $0.33 per cent. 4. If I buy cloth for $1.25 a yard, and sell it for $1.37 a yard, what do I gain per cent.? Ans. $0.10 per cent. These answers properly express the number of cents, loss or gain, on the dollar. If the decimal point be taken away, they will express the number of dollars on the $100. 186. If I buy tea for 75 cents a pound, how must I sell it to gain 4 per cent. ? $0.75 at 4 per cent. is (.75X.04) $0.03, and .75+.03 $0.78, the selling price. The method in this case is precisely the same as that for interest for one year (160). If, instead of gaining, I wish to lose 4 per cent., the .03 must be subtracted from .75, leaving 72 for the selling price. Hence 187. To know how a commodity must be sold to gain or lose so much per cent. RULE.-Multiply the price it cost by the rate per cent., and the product added to, or subtracted from, this price, will be the gaining or losing price. QUESTIONS FOR PRACTICE. 2. If I buy cloth for $0.75, how must I sell it to gain 94 per cent.? Ans. $0.8211. 3. If I buy corn for $0.80 a bushel, how must I sell it in order to lose 15 per cent.? Ans. $0.68. 4. Bought 40 gals. of rum at 75 cents a gallon, of which 10 gallons leaked out: how must I sell the remainder, in order to gain 12 per cent. on the prime cost? Ans. $1.125 per gallon. . 5. Equation of Payments. 188. A owes B 5 dollars, due in 3 months, and 10 dollars due in 9 months, but wishes to pay the whole at once; in what time ought he to pay it? $5, due 3 months $1, due in 15 months, and $10, due in 9 months $1, due in 90 months; then (5+10=) $15, due $5 in 3 months, and 10 in 9 months $1 due in (15+90) 105 months. Hence, A might keep $1, 105 months, or $15, of 105 months, or 5-7 months. This method of considering the subject supposes that there is just as much gained by keeping a debt a certain time after it is due, as is lost by paying it an equal length of time before it is due. But this is not exactly true; for by keeping a debt unpaid after it is due, we gain the interest of it for that time; but by paying it before it is due, we lose only the discount, which has been shown to be somewhat less than the interest (181). The following rule, founded on the analysis of the first example, will, however, be sufficiently correct for practical purposes. 189. RULE.-Multiply each of the payments by the time in which it is due, and divide, the sum of the products by the sum of the payments; the quotient will be the equated time of payment. QUESTIONS FOR PRACTICE. NOTE. It will be observed that the result obtained by the second method differs very materially from the others. But that result is evidently erroneous and unjust; for the debtor being under no obligation to make payments before the time specified in the note, he might have let out these payments upon interest till that time, and then the amount of these taken from the amount of the principal, would leave the balance justly due, and which would be the same as that found by method III. Hence, in computing interest on notes, bonds, &c. the conditions of the contract should always be taken into consideration. The second method is applicable to notes which are payable on demand, especially after a demand of payment has been made, and also to other contracts after the specified time of pay ment is past. REVIEW. 1. What is meant by the term per cent.?-by per annum? 2. What is meant by Interest?by the principle ?-by the rate per cent. ?-by the amount? 3. Of how many kinds is Interest? 4. How is the rate per cent. expressed? What do decimals in the rate below hundreds express? Is rate established by law? What is it in New England? in New York? 5. What is Simple Interest? 6. How would you find the interest on any sum for one year? For more years than one? Repeat the rule for the first method. rate? How then would you cast the interest on a given sum for a given time at 12 per cent.? 9. What part of 12 per cent. is 6 per cent.? What then would be the monthly rate at 6 per cent.? 10. What is the second method of casting interest at 6 per cent.? What is done with the odd days, if any less than 6? Having found by this method the interest at 6 per cent., how may it be found for any other per cent.? What is the rule which is to be observed in all cases for pointing? (122) 11. The time, rate, and amount being given, how would you find the 7. How would you proceed, if the principal were in English Mo-principal? ney? 3. If interest be allowed at 12 per cent., what would be the monthly 12. The time, rate, and interest being given, how would you find the principal? 13. The principal, interest, and time being given, how would you find the rate? 14. The principal, rate, and interest being given, how would you find the time? NOTE. The pupils should be required to show the reason of these general rules, by the analysis of examples. 15. What is Commission? Insurance? Premium? A Policy? What sum should the policy always cover? 16. What is the rule for commission and insurance ? Does it differ from that for casting interest for one year? 17. Is there a uniform method of computing interest on notes and bonds? 18. What is the first method given? Is it correct? Why not? 19. What is the second method? What does this method suppose? Is it correct? Does it differ widely from the truth? Where is this method established? 20. What is the third method? Where is interest allowed upon these principles? What is the first rule founded upon it?-the second rule? 21. What is Compound Interest ? -the rule? 22. What is Discount? Does it differ from Interest? Which is most at the same rate per cent.? How would you find the present worth of a sum due some time hence ?-how the discount? 23. What is Loss and Gain? How would you proceed to find what is lost or gained per cent.? How would you find how a commodity must be sold to gain or lose so much per cent. ? SECTION VI. Proportion. ANALYSIS. 190. 1. If 4 lemons cost 12 cents, how many cents will 6 lemons cost? Dividing 12 cents, the price, by 4, the number of lemons, we find that 1. lemon cost 3 cents, (10, 134) and multiplying 3 cents by 6, the number of which we wish to find the price, we have 18 cents for the price of 6 lemons. (8, 136.) 2. If a person travel 3 miles in 2 hours, how far will he travel in 11 hours, going all the time at the same rate? The distance travelled in 1 hour, will be found by divid ing 3 by 2=3, and the distance travelled in 11 hours will be 11 times =42=16.5 miles, the answer. 191. All questions similar to the above may be solved in the same way; but without finding the price of a single lemon, or the time of trav elling 1 mile, it must be obvious that if the second quantity of lemons were double the first quantity, the price of the second quantity would also be double the price of the first, if triple, the price would be triple, if one half, the price would be one half, and, generally, the prices would have the same relation to each other that the quantities had. In like manner it must be evident, that the distances passed over by a uniform motion would have the same relation to one another, that the times have in which they are respectively passed over. 192-195. PROPORTION. 73 192. The relation of one quantity, or number, to another, is called the ratio (24). In the first example, the ratio of the quantities is as 4 to 6, or 1.5; and the ratio of the prices, as 12 to 18, or 12-1.5; and in the second, the ratio of the times is as 2 to 11, or -5.5, and the ratio of the distances, as 3 to 16.5, or 16-5-5.5. Thus we see that the ratio of one number to another is expressed by the quotient, which arises from the division of one by the other, and that, in the preceding examples, the ratio of 4 to 6 is just equal to the ratio of 12 to 18, and the ratio of 2 to 11 equal to the ratio of 3 to 16.5. The combination of two equal ratios, as of 4 to 6, and 12 to 18, is called a proportion, and is usually denoted by four colons, thus, 4:6:12: 18, which is read, 4 is to 6, as 12 is to 18. 193. The first term of a relation is called the antecedent, and the second, the consequent; and as in every proportion there are two relations, there are always two antecedents and two consequents. In the propor tion 46 12: 18, the antecedents are 4 and 12, and the consequents are 6 and 18. And since the ratio of 3 to 6 is equal to that of 12 to 18, (192) the two fractions and are also equal; and those, being reduced to a common denominator, their numerators must be equal. Now if we multiply the terms of by 12, the denominator of the other fraction, the product is, (30, Ex. 6.) and if we multiply the terms of 18 by 4, the denominator of the first fraction, the product is also . By examining the above operations, it will be seen that the first numerator, 72, is the product of the first consequent and the second antecedent, or the two middle or mean terms, and the second numerator, 72, is the product of the first antecedent and second consequent, or of the two extreme terms. Hence we discover that if four numbers are proportional, the product the first and fourth equals the product of the second and third, or, in other words, that the product of the means is equal to the product of the extremes. 194. In the proportion, 4:6:: 12: 18, the order of the terms may be altered without destroying the proportion, provided they be so placed, that the product of the means shall be equal to that of the extremes. It may stand, 4 12 6 13, or 18: 12: 6:4, or 18: 6:: 12:4, or 6:4:18:12, or 6: 18:4: 12, or 12: 4 :: 18: 6, or 12: 18 :: 4 : 6. By comparing the second arrangement with question first, it will be seen that the ratio of the first number of lemons to their price is the same as that of the second number to their price, and this must be obvious from what was said in article 191. 195. Since, in every proportion, the product of the means is equal to the product of the extremes, one of these products may be taken for the other. Now if we divide the product of the means by one of the means, the quotient is evidently the other mean, consequently if we divide the product of the extremes by one of the means, the quotient is the other mean. For the same reason, if we divide the product of the means by one extreme, the quotient is the other extreme. Hence if we have three terms of a proportion given, the other term may readily be found. Take the first example. We have shown, (192) that 4 lemons are to 6 lemons as 12 cents are to the cost of 6 lemons, or 18 cents, and also (194) that 4 lemons are to 12 cents as 6 lemons to their cost, or 18 cents. Now of the above proportion we have given by the question only three terms, and the fourth is required to be found. Denoting the unknown term by the letter x, the proportion would stand lem. lem. ets. cts. lem. or 4 : cts. lem. ets. |