6. There are 106lb. of filver, the property of 3 men; of which A receives 171b. 10oz. 19pwts. 19grs. of what remains, B fhares 1oz. 7grs. fo often as C fhares 13pwts. What are the fhares of B and C ?

Answer, B's hare 53lb. 8oz. 5pwts. 5grs. C's fhare 34lb. 4oz. 15pwts.

§ 2. Fractions.

WHEN the thing or things fignified by figures are whole ones, then the fig. ures which fignify them are called Integers or whole numbers. But when only fome parts of a thing are fignified by figures, as two thirds of any thing, five fixths, feven tenths, &c. then the figures which fignify these parts of a thing being the expreffion of fome quantity lefs than one, are called FRAC

TIONS.

FRACTIONS are of two kinds, Vulgar and Decimal; they are diftinguished by the manner of representing them; they also differ in their modes of op. eration.

VULGAR FRACTIONS.

To understand Vulgar Fractions, the learner muft fuppofe an integer (or the number 1) divided into a number of equal parts; then any number of thefe parts being taken, would make a fraction, which would be represented by two numbers placed one directly over the other, with a fhort line between them, thus two thirds, three fifths, Seven eights, &c.

EACH of thefe figures have a different name and a different fignification. The figure below the line is called the Denominator and fhews into how many parts an integer, or one individual of any thing is divided-the figure above the line is called the numerator and fhews how many of thofe parts are fignified by the fraction.

FOR illuftration, fuppofe a filver plate to be divided into nine equal parts. Now one or more of thefe parts make a fraction, which will be represented by the figure 9 for a denominator placed underneath a fhort line fhewing the plate to be divided into nine equal parts; and fuppofing two of thofe parts to be taken for the fraction, then the figure 2 must be placed directly above the 9 and over the line (3) for a Numerator, fhewing that two of those parts are fignified by the fraction, or two ninths of the plate. Now let 5 parts of this plate, which is divided into 9 parts, be given to John, his fraction would be five ninths; let 3 other parts be given to Harry, his fraction would be 3 three ninths; there would then be one part of the plate remaining still (5 and 3 are 8) and this fraction would be expreffed thus one ninth.

In this way all vulgar fractions are written; the Denominator, or number below the line fhewing into how many parts any thing is divided, and the numerator, or number above the line, fhewing how many of those parts are taken, or fignified by the fraction.

To afcertain whether the Learner underftands what has now been taught him of fractions, let us again fuppofe a dollar to be cut into 13 equal parts ;— let 2 of thofe parts be given to A; 4 to B; and 7 to C.

A's fractionREQUIRED of the Learner that he should write B's fraction

C's fraction▬▬▬▬。

Ir is from Divifion only that fractions arife in Arithmetical operations; the remainder after divifion is a portion of the Dividend undivided; and is always the Numerator to a fraction of which the Divifor is the DenominaThe Quotient is fo many integers.

tor.

THE Arithmetic of Vulgar Fractions is tedious and even intricate to begin. ners. Befides, they are not of neceffary use. We shall not, therefore, enter into any further confideration of them here. This difficulty arifes chiefly from the va riety of denominators; for when numbers are divided into different kinds, ot

C Millions.

parts they cannot be cafily compared. This confideration gave rife to the invention of

DECIMAL FRACTIONS.

Decimal fractions are alfo expreffions of parts of an integer; or, are in value fomething icfs than one of any thing, whatever it may be, which is fignified by them.

In decimals, an integer, or the number one, as 1 foot, 1 dollar, 1 year, &c. is conceived to be divided into ten equal parts, (in vulgar fractions an integer may be divided into any number of parts) and each of these parts is fubdivided into ten leffer parts, and so on. In this way, the denominator to a decimal fraction in all cafes, will be either 10, 100, 1000, or unity (1) with a number of cyphers annexed; and this number of cyphers will always be equal to the number of places in the numerator. 685 Thus, are Decimal Fractions, of which the cyphers in the denominator of each are equal to the number of places in its own numerator.

6

100 T000

"As the denominator of a decimal fraction is always 10, 100, 1000, &c. "the denominators need not be expreffed; for the numerator only may be "made to exprefs the true value; for this purpose it is only required to "write the numerator with a point, (,) before it, called a feparatrix, at the "left hand to diftinguish it from a whole number; thus, is written,6; 66 21,27 685 ,685, &c.

100

;

1005

When integers and decimals are expreffed together in the fame fum, that fum is called a mixed number: Thus, 25,63 is a mixed number; 25, or all the figures to the left hand of the feparatrix being integers, and ,63 or all thẹ figures to the right hand of the fame point being decimals.

The firft figure on the right hand of the decimal point fignifies tenth parts, the next hundredth parts, the next thousandth parts, and fo on.

,7 fignifies feven tenth parts.

,07-feven hundredth parts.

,27

,357

5,7

-two tenth parts and feven hundredth parts; or twenty-feven hundredths.

-three tenth parts, five hundredth parts, and feven thoufandth parts; or, 327 thousandths.

-five and feven tenth parts.

5,007-five and feven thoufandths.

The value of each figure from unity, and the decrease of decimals toward the right hand, may be feen in the following

9 8 7 6 5 4 3 2

1 2, 3 4

7 8 9

Cyphers placed to the right hand of decimals do not alter their value; placed at the left hand, they diminifh their value in a tenfold proportion.

Hundredth parts

Thoufandth parts
X Thousandth parts
C Thousandth parts

Tenth parts

· Millionth parts ∞ X Millionth parts

C Millionth parts

ADDITION OF DECIMALS.

RULE.

"1. PLACE the numbers whether mixed or pure decimals, under each "other, according to the value of their places.”

"2. Find their fum as in whole numbers, and point off fo many places for “decimals as are equal to the greatest number of decimal places in any of the "given numbers."

EXAMPLES.

1. What is the amount of 73,612 guineas, 436 guineas, 3,27 guineas, ,8632 of a guinea, and 100,19 guineas when added together?

NOTE. When the numerator has not fo many places as the denominator has cyphers, prefix fo many cyphers at the left hand as will make up the defect; fo is written thus, ,005, &c.

SUBTRACTION OF DECIMALS.

RULE.

"Place the numbers according to their value; then fubtract as in whole numbers, and point off the decimals as in addition."

All the operations in Decimal Fractons are extremely eafy; the only liability to error will be in placing the numbers and pointing off the decimals ; and here care will always be fecurity against mistakes.

MULTIPLICATION OF DECIMALS.

RULE.

"Whether they are mixed numbers, or pure decimals, place the factors and multiply them as in whole numbers."

"2. Point off fo many figures from the product as there are decimal places in both the factors; and if there be not fo many decimal places in the product, fupply the defect by prefixing cyphers."