# Loss and Gain. 184. If I buy a horse for $50, and sell it again for $56, what do I gain per cent ? Subtracting 50 dollars froin 58 dollars, we find that 50 dollars 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 cent. 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 $0.90; $1.30, what do I gain per cent? | what do I lose per cent? 1.25 .0500 ( 0.04 per cent. Ans. $0.1111. 500 Ans.* 6. If corn be bought for 3. If I buy salt for 84 cents i $0. $0.75, and sold for $0.80 a a bushel, and sell it for $1.12 bushel, what is gained per cent? a bushel, what do I gain per Ans. $0.063. cent? Ans. $0.33} per cent. | * These answers properly ex 4. If I buy cloth $1.25 a press the number of cents, loss or yard, and sell it for $1.37], a ! % "I gain, on the dollar. If the decinjal point be taken away, they will exyard, what do I gain per cent! | press the number of dollars on the Ans. $0.10 per cent. 1 $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 (.75*.043) $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, 1. 4. Bought 40 gals. of rum how must I sell it to gain 93 at 75 cts. a gallon, of which 10 per cent ? Ans. $0.8214. gallons leaked out, how must I 3. If I buy corn for $0.80 a sell the remainder in order to bushel, how must I sell it in gain 12, per cent on the prime order to lose 15 per cent ? cost? Ans. $1.125 per gal. Ans. $0.68. 188, 189. EQUATION OF PAYMENTS. 125 5. Equation of Payments. 188. A owes B 5 dollars, due in 3 inonths, 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=81, due in 15 months, and $10, due in 9 months =$1, due in 90 months; then (57-10=) $15, due $5 in 3 months, and 10 in 9 monihs=$1 due in (15-+90=) 105 months. Hence A might keep 1, 105 months, or $15, 1 of 105 mo. or 49,5–7 mo. 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 exacily true; for hy 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 paid | 4. B owes C $190, to be $100 in 6 months, $120 in 7 paid as follows, viz. $50 in 6 months, and $160 in 10 months, months, $60 in 7 months, and what is the equated time for $80 in 10 months; what is the the payment of the debt ? equated time to pay the wbole? Ans. 8 months. I Ans. 8 months. 3. A owes B $750, to be 5. Cowes D a certain sum paid as follows, viz. $500 in 2 of money, which is to be paid months, $150 in 3 months, and in 2 months, in 4 months, $100 in 4 months; what is and the remainder in 10 mo. the equated time to pay the what is the equated time to whole ? pay the whole ? Aps, 4 mo. Ans. 24%8=2,15 mo. | MISCELLANEOUS. 1. What is the interest of | terest of $125 for 2 years, at $223.14 for 5 years, at 6 per 6 per cent ? Ans. $66.942. 4. What is the amount of 2. What is the amount of | $760.50 for 4 years, at 4 per 12, cents, for 500 years, at 6 | cent, compound interest ? | - 5. What is the amount of per cent? ANS. $3.0/2: $666 for 2 years,at 9 per cent. · 3. What is the compound in. compound interest ? 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. $354.489. 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. l of payment arrived ? 8. A buys a quantity of rice By meth. 1.(178) $139.655) for $179.56; for what must be meth. II. $144 363Ans. sell it to gain 11 per cent ? I meth. III. $139.653) Ans. $199.311. NOTE.-It will be observed that the result obtained by the second me. thod differs very materially from the others. But that result is evidently erroneous and unjust; for the debtur, being under no obligation to make payments before the time specified in the note, he might have let out ihese 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 ou demand, especially after a demand of payment has been made, and also to other contracts after the specified time of payment is past. REVIEW 1. What is ineant by the term, I ly rate? How then would you cast per cent ?- by per annum ? the interest on a given sum for a 2. What is meant by Interest ? -- given tinje at 12 per cent ? by the principal ?-by the rate per 9. What part of 12 per cent 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 rale per cent ex 10. What is the second method pressenl? What do decimals in the uf casting interest at 6 per cent ? rate below hundreds express? Is i What is done with the odd days, if rate established by law? What is any, less than 6? Having found by it in New-England ? in New-York? | this method 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 rule rest on any sum for one year? For which is to be observed in all cases more years than one? Repeat the for pointing ?(122) sule for the first method. ii. The time, rate, and amount 7. How would you proceed, if being given, how would find ide the principal were in English Mo- principal ? 12. The time, rate, and interest 8. If interest be allowed at 12 ) being given, how would you find per cent, what would be the month. I the principal ? ney? 190, 191. * PROPORTION. 127 13. The principal, interest, and | What does this method suppose ? time being givea, 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 20. What is the third method ? find the time? | Where is interest allowed upon Note.-- The pupils shnuld be these principles ? What is the first required 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- 1 --the rule ? rance ? Premium? A Policy? What 22. What is Discount? Does it sum 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 a sum due some time hence?--how one year? 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 give | How would you find how a comen? Is it correct? Why not? | modily must be sold to gain or lose 19. What is the second method ? | so much per cent ? SECTION VI. ANALYSIS 190. 1. If 4 lemons cost !2 cents, how many eents will 6 lemons cost? Dividing 12 cents, the price, ty 4, the number of lemons we find that I 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 li hours, going all the time at the same rate ? The distance travelled in 1 hour, will be found by dividing 3 by 2=3, 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 lemoas were double the first quantity, the price of the second quantity would also be double the price of the first, if iriple, 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 hy a uniform motion, would have the same relation to one another, that the times have in which they are respectively passed over. 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 6=1.5, and the ratio of the prices, as 12 to 18, or 18=1.5; and in the second, the ratio of the times is as 2 to 11, or 41=5.5, and the ratio of the distances, as 3 to 16.5, or 16.3=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 denoteil 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 proportion 4:6:: 12 : 18, the antecedents are 4 and 12, and the conseguents are 6 and 18. And since the ratio of 3 to 6 is equal to that of 12 to 18, (192) the two fractions 6 and 18 are also equal; and these, being reduced to a common denominator, their numerators must be equal. Now if we multiply the terms of 6 by 12, the denominator of the other fraction, the product is 72, 730, Ex. 6.) and if we multiply the terms of 18 by 4, the denominator of the first fraction, the product is also z. By examining the above operations, it will be seen that the first numerator, 72, is the product of the first consequent and the second aniecedent, 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 of the first and fourth equals the product of the second and third, or in other words, thai 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 placer, that the product of the means shall be equal to that of the extremes. It may stand, 4:12 :: 6:18, or 18: 12:: 6:4, or 18: 6 :: 12 : 4, or 6:4 :: 18 : 12, or 6.: 13 :: 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 nuniber 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 means, 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 ex. treme, 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 e, the proportion would stand lem. lem. cts. cts. lem. cts. Jem. cts. 4. :: 6 :: 1% : *. or 4 : 12 ;;6 ;. *.. |