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them to the decimal of a pound by inspection, (See Case III. Reduc'tion of Decimals,) and write the decimal at the right hand of the pounds; multiply the sum by 4, and the product will be dollars and decimal parts.
Examples. 1. In €6 4s 6d 2qrs how ma 2. In £125 9s 7d 3qrs how ny dollars ? 6.227
Ans. $501.92% 4
many dollars ?
Rule. Divide the given sum by 4, and the quotient will be pounds and the decimal of a pound. The value of the decimal must be found by inspection. (See Case V. Reduction of Decimals.)
Examples. 1. Reduce $24.908 to pounds, 2. In 8501.928 how many shillings, pence and farthings. pounds, shillings, pence and far 4)24.908
Ans. £125 9s. 74d. 6.227 Ans. 26 4s. 6 d.
Additional Rules in Exchange. VII. To change N. E. currency to N. Y, currency; add one third. VIII.
N. Y. to N. E. currency; subtract one fourth. IX.
N. E. to Penn. currency; add one fourth. 3
Pen. to N. E. currency; subtract one fifth, XI.
N. Y. to Penn. currency; subtract one sixteenth. XII.
Penn. to N. Y. currency; add one fifteenth. XIII.
N. E. to Canada currency; subtract one sixth. XIV.
Canada to N. E. currency; add one fifth.
Miscellaneous Examples. 1. In 21 is 104d N. E. currency, how many dollars ? i
Ans. 83.646.. 2. In £1 1s 104d N. Y. currency, how many dollars ?
Ans. 82.735. 3. In £i 1s 10dPenn. currency, how many dollars ?
Ans. $2.916.. 4. In £i is 1034 Canadá currency, how many dollars ?
Ans. 84.376. 5. In $255.406 how many pounds, shillings, pence and farthings i
276 12s 5d N. E. cur.
£102 3s 3d N. Y. cur.
£63 175 04d Canada cur. 6. Change £240 15s N. E. currency to the several other curren cies.
€321 Os Od N. Y. cur. Ans.
300 18s 9d. Penn. cur. 200' 12s 6d Canada curs. 8802.50 Federal Money.
1. How do you reduce N. E. and N. rency the same as Pennsylvania! Y. currencies to Federal Mon. 7. How do you change Federal Mo
ney to Pennsylvania currency? 2. In what other States is the cur 8. How is Canada currency reduced
rency the same as in New-Eng. to Federal Money?
9. How is Federal Money reduced 3. In what the same as New-York? to Canada
currency? 4. How do you change Federal Mo. 10. How is N. E. currency changed
ney to N. E. and N. Y. curren to N. Y currency ? -N. Y. to cies?
N. E.?-N: E. to Penn.?-Penn. 5. How do you change Pennsylvania to N. E.?-N. Y. to Penn. ? currency to Federal Money?
Penn. to N. Y.?-N. E. to Cape 6. In what other States is the cur ada Capada to N. E.?
The SINGLE RULE OF Three is known by having three numbers given to find a fourth, which shall bear the same proportion to the second that the third has to the first.
It is sometimes called Simple Proportion, and, on account of its importance, the Golden Rule.
Rule.* 1. Write down that number which is of the same kind with the answer, or number sought, for the second term.
2. Consider whether the answer ought to be greater or less than this number, and if greater, place the greater of the other two given numbers for the third term, and the less for the first term ; but if less, write the less of the other two given numbers for the third term, and the greater for the first.
3. Multiply the second and third terms together, and divide the product by the first, the quotient will be the answer.
* Proportion is of two kinds; one arises from considering the differences of numbers, and is called Arithmetical Proportion ; the other from considering their quotients, and is called Geometrical Proportion. The latter is that with which we are at present concerned. Four numbers are said to be in geometrical proportion when the first has the same proportion to the second which the third has to the fourth ; that is, when the quotient of the second, divided by the first, is the same as the quotient of the fourth, divided by the third, and the
Thus 2:4::6: 12 are in geometrical proportion, because 2 is to 4, as 6 to 12, that is, 4 divided by 2 gives the same quotient as 12 divided by 6 : viz. 2; and if 2 be divided by 4, and 6 by 12, the quotient is in each case .5. la the same way it will appear that the fourth term has the same proportion to the second as the third has to the first, and the reverse. It also appears, that the product of the Grst and fourth terms, or extremes, (2x12=24,) is equal to the product of the second and third, or means (4>x6=24.) This holds true in all cases where the numbers are proportional, and upon this fact, all the operations in the rule of three are founded.
In order to compare numbers together, it is necessary to consider them abstractly, or as applied to things of the same kind. We cannot compare 2 men and 4 days, but we may compare 2 and 4, or 2 men and 4 men, or 2 days and 4 days. In the Rule of Three, we have three terms given to find a fourth, which shall have the same relation to one of the given terms which exists between the other two. Two of the given terms will therefore apply to things of the same kind, so as to be compared ; "and the other known term and the unknown term will also apply to similar objects, so that a like comparison may be instituted between them. EXAMPLE.- If 2lb. of sugar cost 14cts. what will 12lb. cost? Here 2 and 12 apply to pounds of sugar, they may therefore be compared ; and
N. B. Before stating the question, the first and third terms must be reduced to the same denomination, if they are not already so, and the middle term to the lowest denomination mentioned in it. . The answer will be in the same denomination as the second term, and may be brought to a higher by reduction if necessary.
Examples. 1. If 15 bushels of corn cost $7.50, what will 25 bushels cost? bu. $cts. bu. 15:7.50 : : 25
Here it will be seen at once that the 25
answer is to be in money, and therefore,
$7.50 must be the second term. It is also 3750
evident the 25 bushels will cost more than 1500
15, and therefore that the answer will be $cts.
more than $7.50, and consequently, 25 must 15)187.50(12.50 Ans. be the third term, and 15 the first.
the required number must be cents in order to compare with 14. Now it is evident that the cost of 12lb, will be as many times 14cts. as 12 is times 2, and therefore, the number expressing the value of 12lb. will bear the same proportion to 14 that 12 does to 2; and 2, 14 and 12 will be the three first terms of a geometrical proportion; that is, 14 and 12 will be the two means, and 2 the first extreme. Now since the product of the two means is equal to the product of the extremes, it is plain that if the product of the means be divided by one extreme, the quotient will be the other extreme ; thus 1412=168, product of means, and 168:2=84, the other extreme, which is precisely the rule. If from the nature of the question, the answer is required to be greater than the given number of the same kind, that number must evidently be multiplied by the greater of the other two given numbers, and the product divided by the less, and the reverse when the answer is required to be less. Hence the direction for stating is obviousa
Besides the method given above for performing the operation in the Rule of Three, there are the four following.
1. Divide the second term by the first, multiply the quotient by the third, and the product will be the answer.
2. Divide the third term by the first, multiply the quotient by the second, and the product will be the answer.
3. Divide the first term by the second, divide the third term by the quotient, and the last quotient will be the answer.
4. Divide the first term by the third, divide the second by the quotient, and the last quotient will be the answer.
The Single Rule of Three is usually divided into Direct and Inverse, both of which are included in the general rule given above.
But for the sake of such as may wish to acquaint themselves with proportion, considered under these two divisions, they are here subjoined.
1. Single Rule of Three Direct teaches by having three numbers given to find a fourth, which shall have the same proportion to the third as the second has to the first.
Role.-1. State the question by making that number which asks the ques
2. If 87.50 buy 15 bushels of 4. If 8lb. 4oz. of tobacco cost corn, what will $12.50 buy? 58. 6d. what will 241b. 12oz. cost? $cts. bu. $ cts.
lb.oz. s. d. . Ib.oz. Here the sey. 77.50 : 15 :: 12.50
This is the 84 5 6 24 12 eral terms must 15 reverse of the 16
16 be reduced to first example,
the lowest deno. 6250 and therefore 132oz. 66d. 156 minations men. 1250
24 tioned, before bu.
stating the ques: 7.50 187.50(25 Ans.
d. 3. If a family of 12 persons 132 : 66:: 396 spend 5 bushels of wheat in 4
66 weeks, how much will last them
2376 a year, allowing 52 weeks to a
2376 year? w. bu. w.
d. 4:5 :: 52 Ans. 65 bush.
132)26136(198=16s. 6d. Ans.
tion, the third term, that which is of the same kind, the first term, and that which is of the same kind as the answer, the second term. 2. Multiply the second and third terms together, and divide the product by the first, the quotient will be the answer. EXAMPLE.--If 81b. of sugar cost $1.00, what will 40lb, of sugar cost? 8:1.00 ::40 40
Here 40 asks the question, it is therefore the third
teim ; 8 being of the same kind, is the first; and 1.00 8 ( 40.00 being of the same kind as the answer, is the second.
$5 00 Ans. II. The Single Rule of Three Inverse teaches by having three numbers given to find a fourth, which shall bear the same proportion to the second that the first has to the third.
RULE.-State the question as in the rule of three direct. Multiply the first and second terms together, and divide the product by the third, the quotient will be the answer.
EXAMPLE.--How many yards of sarcenet 3qs. wide, will line 9 yards of cloth 8qrs. wide ?
8:9::3 Here 3 asks the question, 8 is of the same kind, and 9 the 8
same as the answer sought. Therefore the product of 8 and
9 divided by 3, is the answer. 3) 72
24 Ans. Having stated the question, to know whether it belongs to inverse or direct proportion.-1. If the third term be greater than the frst, and the fourth term is required to be greater than the second, or is the third term be less than the
first, and the fourth term is required to be less than the second, the question * belongs to the rule of three direct. 2. If the third term be greater than the first,
and the fourth term is required to be less than the second, or if the third term be less than the first, and the fourth term is required to be greater than the seconding the question belongs to the role of three inverse. .