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?

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 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 travelling 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 =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 165-5.5. 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, 46:12:18, 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 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 12 by 4, the denominator of the first fraction, the product is also 18. 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 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.

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, 4 126: 13, or 18: 12: 6:4, or 18 6: 12:4, or 64 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, the proportion would stand

lem. lem. cts. cts.
4 : 6 : : 12

:

x.

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

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 can 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 can do a piece of work in 6 days, twice the number of 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 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:x; then 8x= 4X6, and 24-3. Ans. Where more requires less or less requires that is, when one of the ratios is inverted, as explained in this article, it is denominated inverse proportion; otherwise it is called direct propor

more,

tion.

1. Single Rule of Three.

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 kird 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 same 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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