184-187. PER CENT. 69 Ans.* 4. Loss and Gain. 184. If I buy a horse for $50, and sell it again for $56, what do I gain per centa? Subtracting. 50 dollars from. 56 dollars, we find that 50 doilars gains 6 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 centa QUESTIONS FOR PRACTICE. 2. If I buy cloth for $1.25 5. If I buy cloth at $1.02 a. a yard, and sell it again for yard, and sell it at $0.903. $1.30, what do I gain per ct. ? | what do I lose per cent. ? 1.25).0500(0.04 per cent. Ans. $0.1111 500 6. If corn be bought for 3. If I buy salt for 84 cents $0.75, and sold for $0.80, a. a bushel, and sell it for $1.12 bushel, what is gained per ct. a bushel, what do I gain per Ans. $0.06% cent.? Ans. $0.33 per cent. 4. If I buy cloth for $1.25 a * These answers properly express yard, and sell it for $1.37} a the number of cents, loss or gain, on the dollar If the decimal point be yard, what do I gain per cent. ? taken away, they will express the Ans. $0.10 per cent. number of dollars on the $100. 186. If I buy tea for 75 cents a pound, how must I sell it to gain 4 $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 musi be subtracted from .75, leaving 12 for the selling price. Hence 187. To know how a commodity must be sold to gain or lose 90 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. per cent. ? per. cent.? QUESTIONS FOR PRACTICE. 2. If I buy cloth for $0.75, 4. Bought 40 gals. of rum how must I sell it to gain 9. at 75 cents a gallon, of which Ans. $0.821. 10 gallons leaked out: how 3. If I buy corn for $0.80 a must I sell the remainder, in bushel, how must I sell it in order to gain 12, per cent. on order to lose 15 per cent. ? the prime cost ? Ans. $0.68. Ans. $1.125 per gallon • pay it? 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 $5, due in 3 months=$1, due in 15 months, and $10, due in 9 months= $1, due in 90 months; then (5+105) $15, due $5 in 3 months, and 10 in nionths=$1 due in (15+10=) 105 months. Hence, A mighi keep $1, 105 months, or $15, T's of 105 onths, or 1837 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. 2. A owes B $380, to be 4. Bowes C $190, to be paid $100 in 6 months, $120 paid as follows, viz. $50 in 6 in 7 months, and $160'in 10 months, $60 in 7 months, and months; what is the equated $80 in 10 months ; what is the time for the payment of the equated time to pay the whole? debt? Ans. 8 months. Ans. 8 mos. 3. A owes B $750, to be 5. Cowes D a certain sum paid as foNows, viz. $500 in 2 of money, which is to be paid anonths, $150 in 3 months, and 1 in 2 months, in 4 months, $100 in 4 months; what is and the remainder in 10 mos.; the equated time to pay the what is the equated time to whole ? pay the whole ? Ans. 2438=285 mo. Ans, 4 mos. MISCELLANEOUS. 1. What is the interest of interest of $125 for 2 years, $223.14, for 5 years, at 6 per at 6 per cent. ? cent. ? Ans. $66.942. 4. What is the amount of $760.50, for 4 years, at 4 per 2. What is the amount of 125 cents, for 500 years, at 6 cent., compound interest? 5. What is the amount of Ans. $3,87). $666 for 2 years, at 9 per centos 3. What is the compound compound interest? per cent. 6. What is the present worth 9. Supposing a note for 317 of 426 dollars, payable in 4 dollars and 19 cts. to be dated years and 12 da. at 5 per cent.? | July 12, 1822, payable Sept. Ans. $354.409. 18, 1826, upon which were the 7. What is the present worth following endorsements, viz. of 960 dollars, payable as fol- Oct. 17, 1822, $61.10 lows, viz. } in 3 months, $ in March 20, 1823, 73.61 6 months, and the rest in 9 Jan. 1, 1825, 84. months, discount to be made what was due when the time at 6 per cent ? Ans. $936.70. of payment arrived ? 8. A buys a quantity of rice for $179.56; for what must he By meth. 1. [178) $139.655 meth. II. sell it to gain 11 per cent. ? $144.363 Aus meth. Ill. $ 139.653 Ans. $199.311. Note.--It will be observed that the result obtained by the second meihod differs very materially from the others. But that result is evidently erroneous and unjust; for the debtor being under no obligation to mako payments before the time specified in the note, he might have let out ihese payınents 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 Ill. 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 meat is past. REVIEW. 1. What is meant by the term rate ? How then would you cast per cent. ?-by per onnum? the interest on a given sum for a 2. What is meant hy. Interest ? -given time at 12 per cent. ? by the principle by the rate per 9. What part of 12 per sent. is 6 cent. ?-by the amount ? per cent.? What then would be the 3. Of how many kinds is Interest ? monthly rate at 6 per cent. ? 4. How is the rate per cent. ex- 10. What is the second method pressed? What do decimals in the of casting interest at 6 per cent. ? rate below hundreds express ?. Is What is done with the odd days, if rate established by law? What is it any less than 6? Having found' by in New England ? in New York ? this methiod the interest at 6 per 5. What is Simple Interest ? cent., how may it be found for any 6. How would you find the inte- other per cent. ? What is the ruke rest ou any sum for one year ? For which'is to be observed in all cases more years than one ? Repeat the for pointing ? (22) rule for the first niethod. II. Thu time, rate, and amount 7. How would you proceed, it being given, how would you find the the priucipal were in English Mo- priucipal ? Dey 12." The time, rate, and interest 8. If interest be allowed at 12 per being given, how would you find the vent., what would be the monthly principal ? 13. The principal, interest, and What does this method suppose ? time being given, how would you Is it correct? Does it differ widely find the rate ? from the truth? Where is this me14. The principal, rate, and inte, thod established ? rest being given, how would you find 20. What is the third method ? the time? Where is interest allowed upon NOTE.-The pupils should be re- these principles ? What is the first quired to show the reason of these rule founded upon it ?—the second general rules, by the analysis of rule ? examples. 21. What is Compound Interest ? 15. What is Commission ? Insu- -the rule ? rance ? Premium? A Policy? What 22. What is Discount? Does it suin should the policy always cover ? differ from Interest? Which is most 16. What is the rule for commis- at the same rate per cent. ? How sion and insurance ? Does it differ would you find the present worth of from that for casting interest for one a sum due some time hence ?-how the discount? 17. Is there a uniform method of 23. What is Loss and Gain ? computing interest on notes and How would you proceed to find bonds ? what is lost or gained per cent. ? 18. What is the first method giv- How would you find how a comen? Is it correct? Why not? modity must be sold to gain or lose 19. What is the second method ? so much per cent. ? year ? SECTION VI: Proportion. 1 ANALYSIS. 190. 1. If 4 lemons cost 12 cents, how many cents will 6 lemons cost? Dividing 12 cents, the price, hy 4, the number of lemons, we find that ! lemon cost 3 cents, (10, 131) 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 dividing 3 by 25, and the distance travelled in 11 hours will be 11 times=*=*=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 trave 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 ihe 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—196. 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, o f=1.5; and the ratio of the prices, as 12 to 18, or 1=1,5; and in the second, the ratio of the times is as 2 to 11, or y=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 ralios, 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 4:6 :: 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 of and the are also equal; and thuse, being reduced to a common denominator, their numerators must be equal. Now if wo multiply the terms of by 12, the denominator of the other fraction, the product is 73, (30, Ex. 6.) and if we multiply the terms of 1 by 4, the denominator of the first fraction, the product is also me. By examining the ab ve 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 ve altered without destroying the proportion, provided they be so placod, that the product of the means shall be equal to that of the extremes. Ii 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 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. l'or the same reason, if we divide the product of the means by one extreme, the quotient is the ot'ler 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 io 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. lem. 6 :: 12 : 12 :: 6 cts. ct or 4 |