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4. Position.

288. Position is a rule by which the true answer to a certain class of questions is discovered by the use of false or supposed numbers.

289. Supposing A's age to be double that of B's, and B's age triple that of C's, and the sum of their ages to be 140 years; what is the age of each ?

Let us suppose C's age to be & years, then, by the question, B's age is 3 times 8-24 years, and A's 2 times 24-48, and their sum is (8+24748=) 80. Now, as the ratios are the same, both in the true and supposed ages, it is evident that the true sum of their ages will have the same ratio to the true age of each individual, that the sun of the supposed ages has to the supposed age of each individual, that is, 80: 8:: 140: 12, C's true age; or, 80: 24: 140:42, B's age, or 80: 48:: 140: 84, A's age. This operation is called Single Position, and may be expressed as follows:

290. When the result has the same ratio to the supposition that the given number has to the required one.

RULE. Suppose a number, and perform with it the operation described in the question. Then, by proportion, as the result of the operation is to the supposed number, so is the given result to the true number required.

2. What number is that, which, being increased by, and itself, will be 125.? Then 50: 24:: 125: 60 Ans. Sup. 24 Or by fractions. =12 Let 1 denote the

required

8

= 6

then

Result 50

number

4. A vessel has 3 cocks ; the first will fill it in 1 hour, the second in 2, the third in 3; in what time will they all fill it together?

Ans. hour.
5. A person, after spending
and of his money, had

1+1+1+1=125, $60 left; what had he at
or 13+++ first?
13=44, and 1—

125(60 Ans.

(See p. 104, Miscel.)

3. What number is that

whose 6th part exceeds its 8th

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Ans. $144.

6. What number is that, from which, if 5 be subtractof the remainder will

ed,

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II. When the ratio between the required and the supposed number differs from that of the given number to the required one.

291. RULE. Take any two numbers, and proceed with each according to the condition of the question, noting the

291.

POSITION.

125

errors. Multiply the first supposed number by the last error, and the last supposed number by the first error; and if the errors be alike (that is, both too great or both too small), divide the difference of the products by the difference of the errors; but if unlike, divide the sum of the products by the sum of the errors, and the quotient will be the answer.

NOTE. This rule is founded on the supposition that the first error is to the second, as the difference between the true and first supposed is to the difference between the true and second supposed number; when that is ne the case, the exact answer to the question cannot be found by this rule:

7. There is a fish, whose head is 10 inches long, his tail is as long as his head, and half the length of his body, and his body is as long as his head and tail both; what is the length of the fish?

Suppose the fish to be 40 inches long, then

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The above operation is called Double Position. The above question, and most others belonging to this rule, may be solved by fractions, thus:

The body of the whole length; the tail of +10= +10, and the head 10: then +1+10+10-the length; but +, and --4-10+10-20 in. and 20X480 in. Áns,

2. What number is that I double that of the second; but which being increased by its if it be put on the second, his 4, its and 5 more, will be value will be triple that of the Joubled? Ans. 20. first; what is the value of each 3. A gentleman has 2 hors-horse? ee, and a saddle worth $50; if the saddle be put on the Arst horse, his value will be

Ans. 1st horse, $30, 2d, $40 4. A and B lay out equal shares in trade: A gains $126,

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126

PERMUTATION OF QUANTITIES.

and B loses $87, then A's
money is double that of B;
what did each lay out?
Ans. $300.

5. A and B have both the same income; A saves one fifth of his yearly, but B, by spending $50 per annum more

292, 293, 294.

than A, at the end of 4 years
finds himself $100 in debt;
what is their income, and what
do they spend per annum?
Ans. $125 their inc. per ann.
A spends $100.
B spends $150 per ann.

Permutation of Quantities.

292. Permutation of Quantities is a rule, which enables us to determine how many different ways the order or position of any given number of things may be varied.

293. 1. How many changes may be made of the letters in the word and?

The letter a can alone have only one position, a, denoted by 1, a and n can have two positions, an and nu, denoted by 1x2=2. The three letters, a, n, and d, can, any two of them, leaving out the third, have two changes, IX2, consequently when the third is taken in, there will be 1×2×3=6 changes, which may be thus expressed: and, adn, nda, nad, dan and dna, and the same may be shown of any number of things. Hence,

294. To find the number of permutations that can be made of a given number of different things.

RULE.-Multiply all the terms of the natural series of numbers from 1 up to the given number, continually together, and the last product will be the answer required

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295-298

PERIODICAL DECIMALS.

Periodical Decimals.

127

295. The reduction of vulgar fractions to decimals (129) presents two cases, one in which the operation is terminated, as -0.375, and the other in which it does not terminate, as 0.272727, &c. In fractions of this last kind, whose decimal value cannot be exactly found, it will be observed that the same figures return periodically in the same order Hence they have been denominated periodical decimals.

296. Since in the reduction of a vulgar fraction to a decimal, there can be no remainder in the successive di visions, except in one of the series of the numbers, 1, 2, 3, &c. up to the divisor, when the number of divisions exceeds that of this series, some one of the former remainders must recur, and consequently the partial dividends must return in the same order. The fraction 0.333+. Here the same figure is repeated continually; it is therefore called a single repetend. When two or more figures are repeated, as 0.2727+ (295), or 324324, it is called a compound repetend. A single repetend is denoted by a dot over the repeating figure, as 0.3, and a compound repetend by a dot over the first and last of the repeating figures, as 0324324.

297. The fractions which have 1 for a numerator, and any number of 9's for the denominator, can have no significant figure in their periods except 1.

Thus

-0.1111+.0.01010+

0.001001001. This fact enables us easily to ascertain the vulgar fraction from which a periodical decimal is derived. As the 0.1111+ is the developement of, 0.22+=f, 0.3=f, &c.

Again, as 0.010101, or 0.01, is the developement of, 0.02, and so on, and in like manner of &c. Hence,

298. To reduce a periodical, or circulating decimal, to a vuilgar fraction.

RULE. Write down one period for a numerator, and as many nines for a denominator as the number of figures in a period of the decimal.

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1. What is an Arithmetical Progression? When is the series ascending When descending? What is meant by the extremes? The means? When the first and last terms are given, how do you find the common difference? How the number of teras? How the sum of the series?

2. What is a Geometrical Progression? What is an ascending series ? What a descending? What is the ratio? When the first term and the ratio are given, how do you find any other term? When the first and last term and the ratio

are given, how do you find the sum of the series?

3. What is annuity 1 When is in arrears? What does ap anuity at compound interest form? How do you find the amount of an aanmity at sompound interest 1

4. What is the common division of a foot? What are these called? What kind of series do these fractions form? What is the ratio? What is the rule for the multiplication of duodecimals? How are all denominations less than a foot to be regarded?

3. What is Position? What does it suppose when single? When double? What kind of questions may be solved by the former 7 by the latter?

6. What is meant by the permu tation of quantities? How do you find the number of permutations? Explain the reason.

7. What is meant by a periodical decimal? By a single repetend"! By a compound repetend? How is a repetend denoted? How is a poriodical decimal changed to equivalent vulgar fraction ?

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