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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 not the case, the exact answer to the question cannot be found by this rulei
7. There is a fish, whose head is 10 inches long, his tail is as long as his hcad, 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
X body d=. ... 20 =
30 tails of 1+10=20 of 1 +10=25 60 5 head 105 .. 10
10=10 10 40
The above operation is called Double Position. The above question, and most others belonging to this rule, may be solved by fractions, ihus :
The body=} of the whole length; the tail=} of 3+10=} +10, and the head 10 : then 3+1+10+10=the length; but +1=\, and --=1=10+10=20 in. and 20X4=80 in. Ans.
2. What number is that | double that of the second ; but which being increased by its if it be put on the second, his s, its 4 and 5 more, will be value will be triple that of the doubled ?
Ans. 20. first; what is the value of each 3. A gentleman has 2 hors- 1 horse? es, and a saddle worth $50; Ans. Ist horse, $30, 22, $40 if the saddle be put on the 4. A and B lay out equal first horse, his value will be shares in trade: A gains $126,
and B loses $87, then A's | than A, at the end of 4 years money is double that of B; finds himself $100 in debt; what did each lay out? what is their income, and what
Ans. $300. do they spend per annum? 5. A and B have both the Ans. $125 their inc. per ann. Eame income; A saves 'one A spends $100 fifth of his yearly, but B, by spending $50 per annum more
B spends $150 per ann.
Detinutation of Ottantities.
£92. Permutation of Quantities is a rule, which enables us to determine how inany different ways the order or position of any given number of things may be varied.
293. 1. Ilow 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 nå, denoted by 1X2=2. The three letters, a, n, and d, can, any two of them, leaving out the third, have two changes, 1x, consequently when the third is taken in, there will be 1x2x3=6 changes, which may be thus expressed: and, adn, ndo, nud, 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.
Rulr.-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
2. Ilow many days can 7 5. How many changes may persons be placed in a differ- | be rung on 12 bells, and how ent position at dinner? 5040. I long would they be in ringing,
3. How many changes may supposing 10 changes to be be rung on 6 bells?
rung in one minute, and the Ans. 720.
year to consist of 365 days, 5 4. Ilow many changes can hours and 49 minutes ? be made in the position of the Ans. 479001600 changes, 8 notes of music?
and 91 years, 26d. 22h. Alm. Ans. 40320. time.
Periodical Decimals. 295. The reduction of vulgar fractions to decimals (129) presents two cases, one in which the operation is terminated, as g=0.375, and the other in which it does not terminate, as i=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 }=0.1111t: o's=0.01010+. t=0.001001001. This fact enables us easily to ascertain the vulgar fraction from which a periodical decimal is derived. As the 0.111lt is the developement of }, 0.22+=5, 0.3=1, &c.
Again, as 0.010101, or 0.01, is the developement of gly, 0.02=5, and so on, and in like manner of yfy, &c. Hence,
298. To reduce a periodical, or circulating decimal, to a vulgar 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.
1. What is the vulgar frac 5. Reduce 0.769230 to the tion of 0.is ?
form of a vulgar fraction.
6. What vulgar fraction is
equal to 0.138 ?
9x13+8=125=numerator. 3. Reduce 0.83 to the form 900=denominator. of a vulgar fraction.
7. What vulgar fraction is Here 0.8 is 8 tenths, and 3
equal to 0.53 ? is 3 9ths=} of 1 10th, or 1 8. What is the least vulgar 30th; then +36=3+34 fraction equal to 0.5025 ? 4. Reduce 275463 to the
9. What finite number is form of a vulgar fraction.
equal to 31.62 ? Ans. 312.6.
1. What is an Arithmetical Pro 4. What is the coinmoy division gression ? When is the series as of a foot ? What are these called? cending? When descending ? What What kind of series do these frac. is meant by the extremes? The tions forin? What is the ratio ? means ? When the first and last | What is the rule for the multiplicaterms are given, how do you find tion of duodecimals? How are all the common dillerence? How the denominations less than a foot to be number of terras ?
How the sum regarded ? of the series ?
3. What is Position ? What does 2. What is a Geometrical Pro it
suppose when single ? When gression? What is an ascending double? What kind of questions series ? What descending ? may be solved by the fornier ? by What is the ratio ? When the tirst the latter ? term and the ratio are given, how
6. What is meant by the permudo you find any other term? Whentation of quantities? How do you the first and last term and the ratio find the number of pcrmutations ? are given, how do you tind the sum Explain the reason. of the series?
7. What is meant by a periodical 3. What is annuity ? When is decimal ? By a single repetend ? it in arrears ? What does an annu By a compound repetend ? How is ity at compound interest form ? a repetend denoted? How is a poHow do you find the amount ribdical decimal changed lo an annuity at compound interest? equivalent vulgar fraction ?
Exchange of Currencies.
299. In £13, how many dollars, cents and mills ?
Now, as the pound has different values in different places, the amount in Federal Money will vary according to those values. In England, $1=4s. 6d.=4.5s.==L0.225, and there £13=13:-0.225=$57.777. In Canada, $1=53.=* s =£0.25, and there £13=13:-0.25=$52. In New England, $1=6s.=£=£0.30, and there, £13=13-:-0.3=$43.333. In New York, $1=8s.=*=£0.4, and there, £13–13:0.4= 32.50. In Pennsylvania, $1=7s. 6d.—7.53.=0/75=£0.375, and there, £13=13--0.375=$34.666. And in Georgia, $1=
4.6+ 43.8d.=4.6+s=£20.0=£0.2333+, and there, £13=13:0.2333 =$55.722.
300. In £16 7s. 8d. 2qr., how many dollars, cents and mills ?
Before dividing the pounds, as above, 7s. 8d. 2gr., must be reduced to a decimal of a pound, and annexed to £16. This may be done by Art. 143, or by inspection, thus, shillings being 20ths of a pound, every 23. will be 1 tenth of a pound: therefore write half the even number of shillings for the tenths= £0.3. One shilling being 1 20th=£0.05; hence, for the odd shilling we write £0.05. Farthings are 96Cths of a pound, and if 960ths be increased by their 24th part, they are 1000tis. Hence 8d. 2qr.(334qr.+1)==0.035; and 16-10.3+0.05+0.033 =£16.385, which, divided as in the preceding example, give for English currency, $72.822, Can. $65.54, N. Y. $40.902, &c. Hence,
301. To change pounds, shillings, pence and farthings to Federal Money, and the reverse.
RULE.—Rcduce the shillings, &c. to the decimal of a pound; then, if it is English currency, divide by 0.225; if Canada, by