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number, viz. a fraction of a yard; and inverting the first term the question will stand thus,

-

108

coft, The answer or Is. 114d.

Then multiplying the three uppermoft figures of the fractions together for a numerator, and the three undermoft figures for a denominator, the anfwer is 10 of a pound, which, reduced to its real value, is 1s. 11d. of a penny.

Here it must be obferved, that the firft and third fractions must be reduced to the fame denomination as in whole numbers, as feen in the foregoing example, where they are both fractions of a yard; and the fourth fraction is of the fame denomination with the second.

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Qu. 2. If of a gallon of brandy cost of a pound, what will 123 coft at that rate?—Anf. 2, or 71. os. Såd. §. Qu. 3. If of a bale of linen cost 14/. 145. what will 7 bales coft at that rate?-Anfewer 1227. 105.

Qu. 4. If 12 lb. of fugar coft 15s. 9d. what is the price of 481lb. -Anfwer 31. os. 91d. 1.

16

Rule of Three inverfe in Vulgar Fractions.

Rule. Prepare all the fractions as for the foregoing rule, then confider (as taught in the rule of three in whole numbers) whether the quefiion belongs to the inverfe or direct rule, and if it belong to the rule of three inverfe, the third fraction is to be inverted, by tranfpofing the numerator and denominator; and the work is wrought exactly in the fame manner as in the direct rule, by multiplying the three uppermoft terms of the fractions together for a numerator, and the undermoft terms of the three fractions for a denominator; and the fraction thus formed will be the aufwer.

Proof. As before in whole numbers.

Example 1. If A lent B4 of 10col. for 3 of a year, how much muft Blend A for of a year in return?

16

After

After difpofing of the fractions as before directed, I confider that of a year being a longer time than, it will not require fo much principal lent, therefore the greater of the firft and third numbers must be the divifor (as in whole numbers); the third fraction therefore must be inverted, and the queftion will stand thus:

If of a year.

of 1000!.

10 of a year? Anfier 3000, or 1581, equal to 15S. 14. 7d.

Questions both in this rule and the former are proved by back-ftating the queftion, as in whole numbers; thus the foregoing example may be proved by faying, if of cool. principal require of a year, what will of 10col. require? and the answer is 38, or of a year.

24.2. How much shalloon will it require at of a yard wide to line the garments made with 10 yards of cloth at 1 yard wide?-Auf. 53, or 49, equal to 24 yards.

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2. 3. If 12 men can mow 24 acres in 103 days, how : many days will 6 men require to do the fame?-Anfwer 21 days.

Qu. 4. If a board be of a foot in breadth, how many inches in length will make a square foot?—Anf. 16 inches. From what has been delivered in this fection concerning vulgar fractions, it is plain that every other rule in arith metic may be wrought by vulgar fractions as well as by whole numbers, as the operation in both cafes depends upon the fame principle; thus, in the rule of three direct in vulgar fractions, inverting the first fraction, and multiplying it by the fecond and third, is the fame as multiplying the second and third fractions together, and dividing by the first and in the inverse rule, inverting the third fraction and multiplying it by the first and fecond, is equal to dividing the product of the first and second fractions by the third, as the learner may prove at his leifure.

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SECT. XIV.

OF PRACTICE.

PRACTICE is the most expeditious rule in arithmetic, and is of general ufe among men of business, as it readily dif covers the value of any number of integers from having the value of one.

By this rule are anfwered all queftions in the rule of three direct that have an unit for their firft number.

Rule. Divide the given number of integers by one or more aliquot parts of a penny, fhilling, or pound, or any two or three of them; and the quotient will be the answer, and of the fame denomination of which the divifor is a part.

An aliquot part of a number is fuch a part, that being taken any number of times, will exactly measure that number without a remainder: thus 2 is an aliquot part of 6, for it is contained exactly 3 times in 6; and 5s. is an aliquot part of a pound, for it is contained exactly four times in a pound; but 5s. 2d. is not an aliquot part, for it is not exactly contained any number of times in a pound without leaving a remainder.

Before the learner can perform this rule, he must perfectly understand the following tables of aliquot parts, and retain. them in his memory.

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These tables are fo plain as to need no explanation; their ufe is to discover by what number to divide any given number of integers.

Cafe 1. When the price is less than a penny, divide the given number by the aliquot parts of a penny equal to the given price, and the quotient gives the answer in pence, which reduce into fhillings and pounds by divifion; except the given price be 3 farthings, then it is brought into fhillings, and anfwered at once by dividing by 16.

Example 1. What is the amount of 8047lb. of old iron, at a halfpenny per pound?

2)8047

12)4023 1

2,0) 33,5 3

16 15

Here I divide the given number S047 by 2, as 2 farthings is the half of a penny, and the quotient 4023 is the price of the iron in pence, and I remains, which is 1 halfpenny, for the remainder is always of the fame name with the divifor; I then reduce the pence into fhillings by dividing by 12, and the quotient is 335 fhillings, and 3 remains, which is pence; and then reducing the fhillings into pounds, the answer is 167. 155. 3 d.

Example 2. What is the value of 5763 yards of trimming, at 3 farthings per yard?

In this example I divide the given number by 16, as before directed, as 3 farthings is the fixteenth part of a fhilling, and the quotient is 360 fhillings, which reduced into pounds is 181. 03. 24d. for the 3 that remains in the first divifion is 3 fixteenths of a fhilling, or 3 times

3 farthings, equal to 24d.

2,0) 16)5763 (36,0

48

18

96

96

Qu. 3. What comes 445, at §d. ?——Anf. 9s. 3‡d...
24. 4. What is the value of 3370, at Id. ?—Ans. 71. 52.

Cafe

• Cafe 2. When the price is an aliquot part of a fhilling, divide the given number by fuch aliquot part, and the quotient is the anfwer in fhillings, which must be reduced into pounds!

Example 5. What is the value of 879lb. of cheese, at 48. per lb. ?

Here the given number of pounds is divided by 3, as 4d, is of a shilling, and it quotes 293 fhillings, which are brought into pounds; and the Anfwer is 14. 13

3)879 2,0)29,3 fhillings

14

13

Qa. 6. What is the value of 297lb. of tallow at 3d. per Ib. ?-Here the given number, must be divided by 4, as 3d. is of a frilling, and the Anfwer is 3l. 145. 3d.

Example 7. What is the value of 3cwt. of fugar, at 6d.

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Example 8. What is the value of 2178lb. of alum, at 144

per lb. Anf. 137. 125. 3d.

8)2178

2,0) 27,2 2

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Example 9. What is the value of 4861b. at 2d. per lb.

Anf. 41. 15.

6)486 2,0) 8,1

4 I

Cafe 3. When the price of the integer is pence and farthings, and not an aliquot part of a fhilling, find what aliquot put of a fhilling is the nearest to the given price, and lefs than it, and divide the given number by that aliquot part; and for the remainder of the price confider what part it is of

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