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Hence, by successive substitutions, we find,

351 1 1
96522x3'2x3x5 2x3x5x11

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Here we observe that the successive terms are alternately plus and minus. We also see that the successive factors of the different denominators may be found by continually dividing the denominator, 965, by the numerator, 351, and the successive remainders.

To make this more clear, we will give the above method of division at full length.

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As a second example, de-compound the fraction , which is nearly the ratio of the English foot to the French metre.

Operation.
1200)3937(3

3600
337)3937(11

3707
230)3937(17

3910
27)3937(145
3915
22)3937(178
3916
21)3937(187
3927
10)3937(393
3930

7, &c. Hence, the value of art!

1 1 1 -3 3.11 3.11.17 3.11.17.145 3.11.17.145.178

1200 3937

3.11.17.145.178.187+3.11.17.145.178.187.393 - &c

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 is for the ratio ; and in this way, we may find the successive approximate ratios.

If we endeavor to de-compound the fraction 2013 by this method, we shall find it equivalent to this series of fractions :

1 1 1 1 1 1
1.27 1.2.3-1.2.3.4+ 1.2.3.4.

5 1.2.3.4.5.6
1

1
1.2.3.4.5.6.7 1.2.3.4.5.6.7.9°

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 terms

1.2.3.4.5.6.7.871.2.3.4.5.6.7.8.9% which are obviously equivalent in value.

This change being made, we have 28673_1 1 1 1 1 1 45360-1-1.27 1.2.3 1.2.3.4 1.2.3.4.5 1.2.3.4.5.6 1

1
+1.2.3.4.5.6.7-1.2.3.4.5.6.7.87 1.2.3.4.5.6.7.8.9

If we de-compound the fraction 18 131 by this method, we shall find it equal to 1 1 1 1

1

1 . . 2 2.4-2.4.

6 2.4.6.8 2.4.6.8.10 2.4.6.8.10.12

10

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= =4.
The ratio of 15 yards to 5 yards, is 1=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 3.
The ratio of 3; hours to 7 hours, is 3; -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, Ariz.

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 extremes. The second and third terms are called the means.

The first and second constitute the first couplet.
The third and fourth constitute the second 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, 10 times.

So that, if 22 yards of cloth cost $20.90, 156 yards will cost of times $20.90=1 times $20.90=78 times ii of $20.90=78 times $1.90=$148:20.

Again, suppose 15 francs to equal $14:10, how many francs are there in $34:78 ?

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