It is obvious that the terms of this series converge very rapidly. If we use only one term of the series, we have for the ratio. If we use two terms, we find 33 19 for the ratio; and in this way, we may find the successive approximate ratios. If we endeavor to de-compound the fraction by this method, we shall find it equivalent to this of fractions: 1 + + 3 § 7 z 4 5 3 6 0 series 1 1 1.2.3.4.5.6 1 1 1 1 1.2.3.4.5.6.7 It will be observed that the factors constituting the denominators of these fractions are the successive digits, except that in the last term the digit 8 does not appear. Now, instead of this last term, we may write these two If we de-compound the fraction by this method, we shall find it equal to 4 6 0 8 0 1 1.2.3.4.5.6.7.8.9 18131 CHAPTER V. RULE OF THREE. 61. THE quotient arising from dividing one quantity by another of the same kind, is called a ratio. Thus, the ratio of 12 to 3, is 12-3-2-4. The ratio of 15 yards to 5 yards, is 3. So that the ratio of one number to another is nothing more than the value of a vulgar fraction, whose numerator is the first term, and denominator the last term. The ratio of 7 days to 5 days, is 7. 5 0 The ratio of 34 hours to 71 hours, is 31÷7} = 13. When there are four quantities, of which the ratio of the first to the second is the same as that of the third to the fourth, these four quantities are said to be in proportion. Thus, 4, 6, 8, and 12, are in proportion, since the ratio of 4 to 6, is the same as 8 to 12. That is, Hence, a proportion is nothing more than an equality of ratios. The usual method of denoting that four terms are in proportion, is by means of points, or dots. Thus, 4 6 8 12; where two points are placed between the first and second terms, and also between the third and fourth, and four points are placed between the second and third; which is read, 4 is to 6 as 8 is to 12. The first and fourth terms of a proportion are called The second and third terms are called the extremes. the means. The first and second constitute the first couplet. The two terms of a couplet must be of the same name, or kind; since two quantities of different kinds cannot have a ratio. There can be no ratio between yards and dollars; but the numbers which represent the number of yards and dollars may have a ratio. Since, in a proportion, the quotient of the first term, divided by the second, is equal to the quotient of the third, divided by the fourth, it follows that the product of the extremes is equal to the product of the means. Hence, if we divide the product of the means by the first term, we shall obtain the fourth term. This process of finding the fourth term by means of the other three terms, is called the Rule of Three. Suppose it is required to find what 156 yards of cloth will cost, if 22 yards cost $20.90. Had there been twice as many yards in 156 as in 22, they would obviously cost twice $20.90; were there three times as many yards, their cost would be three times $20.90, and so on for other ratios. Hence, the $20.90 must be repeated as many times as 22 yards is continued times in 156 yards; that is, 15 times. 2 2 So that, if 22 yards of cloth cost $20.90, 156 yards will cost 15 times $20·90=1 times $20′90—78 times 2 2 1 of $20·90=78 times $1.90=$14&20. Again, suppose 15 francs to equal $14:10, how many francs are there in $34 78 ? Since 15 francs is equal to $14:10, twice $14.10 would equal twice 15 francs; thrice $14:10 would equal thrice 15 francs, and so for other ratios. Therefore, 15 francs must be repeated as many times as $14:10 is contained times in $34.78; that is, 347833 times. Hence, the number of francs required is 7 of 15 francs-37 times 37 times of 15 francs-37 times 1 franc37 francs. 1 1 5 1 1 0 1 5 15 The above method of working questions of the Rule of Three is contained in the following simple RULE. Of the three terms which are given, one will always be of the same kind as the answer sought; this will be the third term. Then if, by the nature of the question, the answer is required to be greater than the third term, divide the greater of the two remaining terms by the less, for a ratio; but if the answer is required to be less than the third term, then divide the less of the two remaining terms by the greater, for a ratio. Having obtained the ratio, multiply the third term by it, and it will give the answer in the same denomination as the third term. NOTE. Before obtaining the ratio by means of the first two terms, we must reduce them to like denominations. EXAMPLES. 1. If in 7 weeks there are 49 days, how many days are there in 21 weeks? In this example, the answer is required to be in days; therefore, we must take 49 days for our third term. And, since in 21 weeks there must be more days than in 7 weeks, we get our ratio by dividing 21 by 7, which |