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lowing infinite continued fraction
2+&c. Some of the first approximative values of this fraction are # # # # ## ##, 38, 238, đc.
59. We will conclude this subject by pointing out some of the many remarkable properties which the approximative values of continued fractions possess. We will refer to the values just obtained, for the ratio of the side of a square to its diagonal.
1. These values are alternately too small and too large. Thus i, j, 14, and me, are too small, whilst i, , , and iff, are too large. .
II. Any of these values differ from the true value, by a quan. tity which is less than the reciprocal of the square of its denominator. Thus 14, which is the ratio much used by carpenters in cutting braces, differs from the true ratio by a quantity less than (14)=ado.
III. Any two conseculive terms of these approximative values, when reduced to a common denominator, will differ by a unit in their numerators. Thus , and 14, when reduced to a common denominator, become y, and .
IV. The numerator and denominator of all approximative values of continued fractions are prime to each other; that is, they have no common measure.
RULE OF THREE. 60. 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.
From which we learn that the ratio of one number to ano. ther, 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 1. The ratio of 37 hours to 74 hours, is 31:71=1%. 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, 4 =
Hence, a proportion is nothing more than an equality of ratios.
The usual method of denoting that four terms are in pro. portion 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 two terms of a couplet must be of the same name or kind; since two quantities of different kinds can not 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 the quotient of the first 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, which may be thus given:
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 it by the third term, and it will give the answer in the same denomination as was 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; there. fore 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 gives 2; this multipli. ed by the third term, 49 days, gives -**=147 days, for our answer.
2. If a person perform a journey in 20 days, by traveling 10 hours each day, how long would it take him to perform the same, if he travel 8 hours each day?
In this example our answer is required to be in days; there. fore we must take 20 days for our third term. And since it will evidently tåke more days when he travels 8 hours each day, than it did when he traveled 10 hours each day, we must divide 10 hours by 8 hours, for our ratio, which becomes 19; this multiplied by the third term, 20 days, gives which by canceling becomes 25 days, for the answer.
3. If i} of a pound of sugar cost 33 of a shilling, how much will as of a pound cost ? "
In this example, our third term is z of a shilling. And since
of a pound is less than it, we must obtain our ratio by divi. ding ts by t}, which gives 9 X 13; this multiplied by the third
term, of a shilling, will give 9 * 18% 28. To reduce this
23 11 x 26" with the least labor we must resort to the method of canceling. Thus canceling the 23, which occurs in both numerator and denominator, also 13 of the numerator against a part of the 26 of the denominator, our expression will by this means
9 become avo= of a shilling.
Note. This method of canceling should be used when the nature of the question will admit, since it will always simplify very much the operation.
4. If a tree 38 feet 9 inches in height give a shadow of 49 feet 2 inches, how high is that tree which at the same time casts a shadow 71 feet 7 inches ?
In this example our third term is the height of the first tree, which is 38 feet 9 inches=384 feet=115 feet : our ratio will be obtained by dividing 71 feet 7 inches=711, feet=302 feet, by 49 feet 2 inches=491 feet=295 feet; which becomes 3 x703; this multiplied by the third term 15 gives 859 X 6 X 155 cm
P. Canceling 6 of the numerator against a part 12 x 295 x 4 of the 12 of the denominator, also canceling 5, a factor of 155 of the numerator, against 5, a factor of 295 of the denomina
859 x 31 _
=26429=56147 feet, for the answer.
5. If 3 pounds of coffee cost 21 shillings, how much will 103 pounds cost ? ..
In this example 2}=1 shillings must be our third term; and since 101= pounds must cost more than 3}=1 pounds ; we must divide en by ; for the ratio ; making it 1 2, this
" 6 x multiplied by the third term, 1 shillings, will give
" 6 x7x3;
. 61. this becomes, after canceling, 4 67 shillings.
6. If 164 yards of calico cost 21} shillings, how much can be bought for 324 shillings?
Ans. 24,957 yards. 7. Sold a tankard for £5 6s. at the rate of 58. 4d. per ounce. What was its weight?
Ans. llb. 7oz. 17pwt. 12gr. 8. If 300 men consume 70 barrels of provisions in ten months, how much will 240 men consume in the same time?
Ans. 56 barrels. 9. Gave 72 dollars for 5 barrels of fish. How much will twenty barrels cost at the same rate ? Ans. 288 dollars.