25 (9)

140

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

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

There are many useful Questions may be answered 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. 6 d. 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 ch

Anfw. 781. 1 s. 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 Ell Flemish. qrs 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 1. Sterl. for the Value delivered at Amfterdam in Flemish Money at 1 1. 13 s. 6 d. for 1 Pound Sterl. How much Flemish Money was delivered at Amfterdam?

Firft, 11. 13 s. 6 d. 402 d. the Value of one Pound Sterl at Amfterdam.

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

CHAP.

CHAP IV.

of Uulgar Fractions.

Sect. 1. Of otation.

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

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 less than the Denominator.

As {& is one Fourth Part is one Third Part

is one Half.

And { is two Thirds, &c.

is

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 Denominator.

As for or &c.

A Compound Fraction is a Part of a Part, confifting of feveral 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 firft divided into any Number of equal Parts, and each of those Parts are

fubdivided

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

As for inftance, fuppofe a Pound Sterling (or 20 s.) be the Unit or Whole; then is 8 s. the of it, and 6 s. the of thofe two Fifths, and 2 s. is the of thofe three Fourths; viz. 2s. } of of of one Pound Sterling.

=

All Compound Fractions are reduced into fingle ones, Thus,

RULE.

6

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 . For 1×3×26 the Numerator, and 3 x 4 x 560 the Denominator, but or o of a 1. Sterl. is 2 s. As above.

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

IN 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× 2 4

As in thefe,

= Or 3x2 6

2x5 ΙΟ

Or

&c.

2×3 6 3×3 9 That is, and t. Or and. Or and Value, in respect to the Whole or Unit.

3x5 15
are of the fame

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 Denominator, and ftill retain the fame Value.

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

According to the foregoing Propofition, if

ed with 7, it will become , viz.

2 x 7

=

14

3 × 7 21

be equally multipli

equally multiplied with 3, it will become, viz.

H

Again, if be

3×3 9

=

7×3 21 And

And by this means I have obtained two new Fractions, 1 and 2, that are of one Denomination, and of the fame Value with the two first propofed, vizit and 2=}.

And from hence 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.

7.

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

A new Denominator will be thus found.

3

5

And the new Numerators will

15

4

4

95 57

64

18

5

90

4

420

140. 168 315⚫ 360 Hence 420 is the common Denominator; and 140. 168.315. 360, are the new Numerators, which being placed Fraction-wife are 149.168.1.128 the New Fractions required.

That is,

Sect. 3.

To bring mixed Jumbers into Fractions, and the contrary.

MIX'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 2. For 9 × 5.
And, +2 the improper Fraction required.

Again, 13 will become . For 13 x 15.
And 22h. And fo 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 +29, as before is evident:

Then

Then it follows that if 49 be divided by 5, the Quotient will give 9. And if 206 be divided by 15, it will give 131, &c. confequently it follows, that

If the Numerator of any improper Fraction be divided by it's Denominator, the Quotient will difcover the true Value of that. Fraction.

EXAMPLES.

=

5. And 44. And 26. Or 123, &c. When whole Numbers are to be expreffed Fraction-wife, it is but giving them an Unit for a Denominator. Thus 45 is 2 92, and 25 is 22, &c.

Sect. 4.

To Abbreviate or Reduce Fractions into their Lowest or Leaf Denomination.

THIS is done, not out of any neceffity, but for the more convenient managing of fuch Fractions as are either propofed in large terms; or fwell into fuch, either by Addition or otherwife: befides it is most like an Artist to exprefs or fet down all Fractions in the lowest Terms poffible; and to perform that, it will be neceffary to confider thefe following Propofitions.

Numbers are either Prime or Composed.

1. A Prime Number is that which can only be measured by an Unit. Euclid 7. Defin. 11.

That is, 3, 5, 7, 11, 13, 17, &c. are faid to be Prime Numbers, because it is not, poffible to divide them into equal Parts by any other Number but Unity or 1.

2. Numbers Prime the one to the other, are fuch as only an Unit doth measure, being their common Measure. Euclid 7. Defin. 12.

For instance, 7 and 13 are Prime Numbers to each other, because they cannot be divided by any Number but an Unit. And 9 and 14 are alfo Prime Numbers to each other, for altho' 3 will meafure, or divide 9 without leaving a Remainder; yet 3 will not meafure 14 without leaving a Remainder; Again, altho' 2 will measure 14 without any Remainder, yet 2 will not measure 9 without leaving a Remainder, &c.

3. A compofed Number is that which fome certain Number measureth. Euclid 7. Defin. 13.

For instance, 15 is a compofed Number of 3 and 5, for 5 x 3 =15, confequently 3 or 5 will juftly measure 15. Allo 20