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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 giv en? 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?

29. 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 eents 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 dividing 3 by 2-2, and the distance travelled in 11 hours will be 11 times-3-3-16.5 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 trayelling 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. 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 61.5, 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 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 denoted by four colons, thus, 4: 6: 12: 18, which is read, 4 is to 6 as 12 is to 18.

72

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 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 18 are also equal; and these, 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 7, 30, Ex. 6.) and if we multiply the terms of 18 by 4. the denominator of the first fraction, the product is also 7. 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 propor tional, the product of 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.

48

It

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. may stand, 4: 126: 18, 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 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 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 z, the proportion would stand

lem. lem. cts.
46

cts.

12:

x.

lem. cts. lem. cts. or 4 : 12 :: 6 : 3.

Now, since the product of the extremes is equal to that of the means, 4 times x equals 6 times 12, or, according to the second arrangement, 12 times 6. Hence, if 12 times 6, or 72, be divided by 4, the first extreme, the quotient/18, is evidently the other extreme, or the value of x.

196. 3. If 4 men can do a piece of work in 6 days, in how many days ean 8 men do it?

By analyzing the example, we find that 4 men 6 days 1 man 24 days, and 1 man 24 days=8 men 3 days. 8 then is the answer. Moreover it is obvious, that if 4 men ean do a piece of work in 6 days, twice the number or men will do it in half the time, or 3 days; and generally the greater the number of men, the less the time, and the reverse; and also, the the longer the time, the less the number of men, and the reverse. In the above example, the ratio of the men, 4 to 8-2, but the ratio of the times, 6 to 3. Now if we invert the first ratio, it becomes, 8 to 4; and we have two equal ratios, and consequently a proportion: i. e. 8:4:: 6 : 3, or 8: 6:4:3. By the question the proportion would stand, 8 : 6 : : 4:2; then 8x=4×6, and x=24—3, Ans. Where more requires less or less requires more, that is, when one of the ratios is inverted, as explained in this article, it is denominated inverse proportion; otherwise it is called direct proportion.

197. When three terms of a proportion are given, the operation by which the fourth is found, is called the Single Rule of Three. All questions, which can be solved by the single rule of three, must contain three given numbers, two of which are of the same kind, and the other of the kind of the required answer; and from an examination of the preceding analysis, it will be seen that the given number, which is of the same kind as the answer, may always be one of the means in the proportion; and, since the proportion is not altered by changing the places of the means, (195) it may always be regarded as the first mean, or the middle one of the three given terms. Now if the conditions of the question require the answer to be greater than the given number of the same kind, or first mean, the other mean must obviously be greater than the first extreme; but if the answer be required to be less, the second mean must be less than the first extreme. Hence we have the following general

RULE.

198. Write down the given number, which is of the same kind as the answer, or number sought, for the second term. Consider whether the answer ought to be greater, or less, than this number; and if greater, write the greater of the other two given numbers for the third term. and the less for the first term; but if less, write the least of the other two given numbers for the third term, and the greater for the first. Multiply the second and third terms together, and divide the product by the first, the quotient will be the answer.

NOTE. Before stating the question, the first and third terms must be reduced to the sae denomination, if they are not already so, and the middle term to the lowest denomination mentioned in it. The answer will be in the same denomination as the second term, and may be brought to a higher by reduction, if necessary.

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5. If $7.50 buy 15 bushels of corn, what will $12.50 buy? $cts. bu. $cts.

7.50 15: 12.50

15

6250

1250

bu.

7.50) 187.50 (25 Ans. This is the reverse of the prece ding example, and therefore proves

it.

6. If a family of 12 persons spend 5 bushels of wheat in 4 weeks, how much will last them a year, allowing 52 weeks to a year?

W. bu. W.

4: 5: 52 Ans. 65 bush. 7. If 9lb. of sugar 6s. what will 25lb. cost? Ans. 16s. 8d.

When there is a remainder after dividing the product of the second and third terms by the first, reduce it to the next lower denomination, and divide as before.

132oz. 66d.

24

396oz.

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Oz.

d. 132 66: 396

66

2376

2376

132) 26136 (198d.—

16s. 6d. Ans.

Here the several terms must be reduced to the lowest denominations mentioned, before stating the question.

9. If 8 acres produce 176 bushels of wheat, what will 34 acres produce?

Ans. 748 bushels.

10. A borrowed of B 250 dollars for 7 months; afterwards B borrowed of A 300 dollars; how long must he keep it to balance the former favor? Ans. 5mo. 25d.

11. A goldsmith sold a tankard weighing 39oz. 15pwt. for £10 12s. what was it per oz.? oz. pwt. £

39 15 10 12 : 1 Ans. 5s. 4d.

12. If the terest of $100 for 1 year be 6 dolls. what will be the interest of 336 dollars for the same time? $ $

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13. If 100 men can do a piece of work in 12 days, how many men can do the same in 3 days? Ans. 400 men.

14. If 100 dollars gain 6 dollars in one year, in what time will a sum of money double at that rate, simple interest? $ yr. $ 6:1:100 Ans. 163 yrs.

15. If $100 gain $6 in 12 months, in how many months will a sum of money double at that rate, simple interest? $ mo. $ 6:12

100 Ans. 200mo.

16. If $100 gain 6 dollars in 365 days, in how many days will a sum of money double at that rate, simple interest?

Ans. 6003 days.

17. A owes B £296 17s. but becoming a bankrupt, can pay

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366 365: 1 Ans. 23h. 56m. 4s. nearly.*

24. Bought 4 bales of cloth, each containing 6 pieces, and each piece containing 27 yds. at £16 4s. per piece; what is the value of the whole, and the price per yard?

Ans. £388 16s. and 128. per yard.

25. If a hogshead of rum cost $75.60, how much water must be added to it to reduce the

only 7s. 6d. on the pound; price to 1 dollar per gallon?

how much will B receive?

Ans. £111 6s. 4d. 2qrs. 18. If 1 dozen of eggs cost 10 cents, what will 250 eggs Cost? Ans. $2.187.

19. If a penny loaf weigh 9oz. when wheat is 6s. 3d. per bushel, what ought it to weigh when wheat is 88. 24d. per

bnshel?

Ans. 6oz. 13drs.

20. How many yards of flannel 5qrs. wide, will line 20 yards of cloth 3qrs. wide?

Ans. 12 yards.

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Ans. 12 gal.

26. If a board be 9 inches

wide, how much in length will make a square foot ?

Ans. 16in.

27. How many yards of paper 3 quarters of a yard wide, will paper a room that is 24 yards round, and 4 yds. high?

Ans. 128 yards.

28. If a man spend 75 cents per day, what does he spend per annum? Ans. $273.75.

29. A garrison of 500 men how many must depart that has provisions for 6 months; there may be provisions for those who remain, 8 months? Ans. 125.

*This is called a sideral day.

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