EXAMPLE.

Let it be required to find how many Shillings and Pounds are contained in 85680 Pence.

The Pence in Is. are 12) 85680 (7140 s.85680 d.

Again, the Shillings in 1 l, are 20) 7140 (3571. the Answer required.

Another Example in Coin.

How many Pence, Shillings, and Pounds, are contained in 264859 Farthings.

12) 20)

4) 264859 (66214d. (5517s. (275%.

Remains (3) 9. Note, the Remainder is always of the fame

Denomination with the Dividend.

The last Quotient 2751. together with the feveral Remainders give the Answer required.

Viz. 2751. 175. 10d. 39. = 264859 Farthings.

Example in Troy Weight.

Suppose it were required to find how many Pwts. Ozs. and lbs, are contained in 171333 Grains.

20)

12)

24) 171333 gr. (7138 pw. (356 (29 lb.

Anfw. 29lb. 8 oz. 18 pwt. 21 grs. This and the last Example are the Reverse or Proof of thofe in Pages 43, 45.

&c.

1. In 42905 Ounces, Averdupois Weight; How many Pounds,

Thus

(9)

(21) Anfw. 23 C. 3 qrs. 21 lb. 9oz.

2. In 15966720 Inches; How many English Miles, &c. Anfw. 252 Miles, &c. as occafion may require.

There are many useful Queftions may be anfwered by the help of Reduction only: As the changing one fort of Coin for another; and comparing one fort of Measure with another, &c.

For Inftance: Suppofe one had 347 Rixdollars, at 4 s. 6d. per Dollar; and defired to know how many Pounds Sterling they

make.

347

54 the Pence in one Dollar, viz. 4 s. 6 d. 54 d.

(6) d.

(1) s.

Anfw. 781. 1s. 6 d. Sterl. are 347 Rixdollars.

Queft. 2. In 645 Flemish Ells; How many Ells English? Note, 3 Quarters of a Yard English make one Ell Flemish, and 14, or 5 Quarters of a Yard, is an English Ell.

Therefore, 645

3 the grs. of a Yard in 1 Ell Flemish. grs. in 1 Ell=5) 1935 (387 English Ells for the Answer.

Queft. 3. Suppofe a Bill of Exchange were accepted at London, for the Payment of 400 l. Sterl. for the Value delivered at Amfterdam in Flemish Money at 11. 135. 6d. for I Pound Sterl. How much Flemish Money was delivered at Amfterdam?

Firft, 11. 135. 6d. 402 d. the Value of one Pound Sterl. Amfterdam.

Then, 402d. x 400 = 160800d. — 6701. Flemish, and so much was delivered at Amfterdam.

CHAP.

CHAP. IV.

Of Mulgar Fractions.

Sect. 1. Of Notation.

A Fraction, or Broken Number, is that which represents a Part or Parts of any thing proposed, (vide Page 3.) and is expreffed by two Numbers placed one above the other with a Line drawn betwixt them:

Thus, {3 Numerator,

4 Denominator.

The Denominator, or Number placed underneath the Line, denotes how many equal Parts the thing is fuppofed to be divided into (being only the Divifor in Divifion). And the Numerator, or Number placed above the Line, fhews how many of those Parts are contained in the Fraction (it being the Remainder after Divifion). (See Page 29.) And thefe admit of three Diftinctions:

A proper, pure, or Simple Fraction, is that which is less than an Unit. That is, it reprefents the immediate Part or Parts of any thing less than the whole, and therefore it's Numerator is always lefs than the Denominator.

As

is one Fourth Part.
is one Third Part.

And {is two Thirds, &c.
one Half.

An Improper Fraction is that which is greater than an Unit. That is, it reprefents fome Number of Parts greater than the whole thing; and it's Numerator is always greater than the De

nominator.

Asoror &c.

A Compound Fraction is a Part of a Part, confifting of several Numerators and Denominators connected together with the Word [of]..

As of of, &c. and are thus read, The one Third of the three Fourths of the two Fifths of an Unit.

That is, when an Unit (or whole thing) is first divided into any Number of equal Parts, and each of thofe Parts are

fubdivided

fubdivided into other Parts, and fo on: Then those laft Parts are called Compound Fractions, or Fractions of Fractions.

43

As for instance, fuppofe a Pound Sterling (or 20 s.) be the Unit or Whole; then is 8s. the of it, and 6 s. the of those two Fifths, and 2 s. is the of those three Fourths; viz. 2s. = = +of of of one Pound Sterling. All Compound Fractions are reduced into fingle ones, Thus,

RULE.

Multiply all the Numerators into one another for a Numerator, and all the Denominators into one another for the Denominator. Thus the of of will become. Or.

43

For 1 x 3 x 2-6 the Numerator, and 3 x 4 x 5 = 60 the Deno. minator, but or of a 1. Sterl. is 2s. As above.

6 66

Sect. 2. To Alter or Change different Fractions into one Denomination retaining the fame Value.

IN

N order to gain a clear Understanding of this Section, it will be convenient to premife this Propofition, viz. If a Number multiplying two Numbers produce other Numbers, the Numbers produced of them fhall be in the fame Proportion that the Numbers multiplied are, 17 Euclid 7.

That is to fay, If both the Numerator and Denominator of any Fraction be equally multiplied into any Number, their Products will retain the fame Value with that Fraction.

2 x 5-10, &c.
Or
3×5 15

2× 2 4. 2×3 6 As in thefe, Or 3×2 6 That is, and. Or and. Or and are of the fame Value, in refpect to the Whole or Unit.

3×3 9
43

From hence it will be eafy to conceive, how two or more Fractions that are of different Denominations, may be altered or changed into others that fhall have one common Denominater, and fill retain the fame Value.

Example. Let it be required to change and into two other Fractions that shall have one common Denominator, and yet retain the fame Value.

According to the foregoing Propofition, if be equally multipli

ed with 7, it will become,
14 viz.

2X7

14. Again, if be

3×7 21

equally multiplied with 3, it will become, viz.

H

And by this means I have obtained two new Fractions,

that are of one Denomination, and of the fame Value with the two firft proposed, viz. 4;= 3 and 2 = 4.

And from hen. e doth arife the general Rule for bringing all Fractions into one Denomination.

RULE.

Multiply all the Denominators into each other for a new (and common) Denominator. And each Numerator into all the Denominators but it's own, for new Numerators.

Example. Let the propofed Fractions be,,, and .
Then, by the Rule,

420

140. 168. 315.

Hence 420 is the common Denominator; and 140. 168.315. 360, are the new Numerators, which being placed Fraction-wife . the New Fractions required.

are 45

168 315 ༡༨༡ 420 420 420

That is,

168

31s3 3. and

360 6

420

3

420 5 420, 4

420 7

7

7

360

Sect. 3. To bring mixed Numbers into Fractions, and

the contrary.

M IX'D Numbers are brought into improper Fractions by the following Rule.

RULE.

Multiply the Integers, or whole Numbers, with the Denominator of the given Fraction, and to their Product add the Numerator, the Sum will be the Numerator of the Fraction required.

Example. 9 by the Rule will become . For 9 x 5 = 45. And, the improper Fraction required. Again, 13 will become 206. And+. And to for any other as occafion requires. To find the true Value of any improper Fraction given, is only the Converfe of this Rule. For if 99, as before is evident:

Then