the mixed number by the given integer, and apnex the quotient to the integral quotient formerly found. bus, supposing 101 were to be divided by 5; I say 5)10(2, and 5); and so the complete quotient is 2 But if, in dividing the integral part, there happen to be a remainder, prefix this remainder to the fraction for a new mixed number, which reduce to an improper fraction; then divide the improper fraction by the given integer, and annex the quotient to the integral quotient formerly found. Thus-If it be required to divide 15 by 8, I say 8)15(1, and 7 remain ; which 7, prefixed to the fraction, gives 73 for a new mixed number; and this, reduced to an improper fraction, is, and 8)31(31: so the complete quotient is 131

5. If the factors of the denominator and numerator of the quotient, instead of being actually multiplied, be only connected with the sign of multiplication, it will be easy to cancel such factors, above and below, as happen to be the same. Thus: 2 of 3 ==1=3. Or a factor above and below may be divided by the same number, thus: === Or the factors of the numerator of the quotient may be exchanged, thus: 3)(

6. To divide an integer by a fraction, is to divide the product of the denominator and integer by the numerator, thus: 3)8(535x2=

10.

7. If the divisor and dividend have the same denominator, you have only to divide the numerator of the dividend by the numerator of the divisor; thus: ); for (.

8. If a dividend of two or more denominations be given to be divided by a fraction, reduce the higher part or parts of the dividend to the lowest species, and then divide. Thus, to divide £6 92s. by 2, I say £6-6×20=120; and 120+93-12938.519; and 4)519(1557 194s. £9 14s. 74d. Or, divide the given dividend by the numer ator of the fraction, and multiply the quotient by the denominator.

EXAMPLE.

Divide £276 16s. 8d. among 4 men, so that A, B, and C, may have equal shares, and D only two-thirds of one of their shares.

OPERATION.

1+1+1+3=3+3+3+3=4 of 1=1}.

£ S. d. £ S. d. £ S. d. Then 11)276 16 8(25 3 4X3=75 10

O A's part

1. Divide by 3. 2. Divide 24 by 33. 3. Divide 191 by 9. 4. Divide 18 by 53.

X3 75 10

O B's do.

X3 75.10

O C's do.

Ans..

Ane. 11340-£16 8s. 103d.

3. Divide of § by 71. 6. Divide £12 6s. 8d by 2. The reason of the rules laid down will appear evident, by considering that the method here used is nothing else but the reducing the divisor and dividend to a common denominator, and then dividing the one numerator by the other. Thus, for reducing the divisor and dividend to a common denominator, we have)1⁄2(—•

The truth of this rule may also be proved thus: assume two frac tions equal to two integers, such as § and 16, equal to 2 and 4, and the quotient of the fractions will be equal to the quotient of the integers. Thus (42, and 2)4(2.

QUESTIONS TO EXERCISE THE LEARNER ON THE PRECEDING RULES.

1. There are two numbers; the lesser is 363; their difference is 1125; what is the greater ? Ans. 487 2. The sum of two numbers is 12; the greater is 737; what is the Jesser? Ans. 47. 3. A has of a piece of land, and B has of the same land; which of them has the greater share, and what the difference? Ans. A has the greater share; and the difference of their shares is. 4. Three purses contain £30; in one purse is £55g; in another is £42; how much is in the third purse? 5. How much is the of 130 ?

6. What number multiplied by 3 will produce 253.

Ans. £32,
Ans. 813
Ans.427

7. A has share of a ship, of which he sells to B 3, and B sells out of his share to Cof his interest; the question is what part had each man in said ship? Ans. A ; B; and C. 8. What part of 3 pence is 3 of 2 pence? 9. A has of a ship, of which he sells is the value of the whole ship at that rate?

Ans. $.

thereof for £100; what Ans. £533 6s. 8d

13

DECIMAL FRACTIONS.

A decimal fraction (which is the most genuine and natural way of dividing unity, and perhaps the most ancient) always supposes the integer to be divided into 10, 100, or 1000 parts, &c. as you covet pre ciseness in your operation. Hence the denominator being known, needs not to be expressed, but you may place your fraction as an integer, by taking care to prefix its distinguishing point, or comma: so will be ex, ressed thus, .5, and thus, .05; thus, .05; 75% thus, .75, &c... The denominators need not be expressed, as the denominator of a decimal fraction is always 10, 100, or 1000, &c. For the numerator only may be made to express the true value thereof. For this purpose it is only required to write the numerator with a point prefixed to it, (or at the left hand) to distinguish it from a whole number, when it consists of so many figures as the denominator hath cyphers annexed to unity or 1, as above.

N. B. The point prefixed is called the separatrix; a cypher placed at the left hand of an integer, or at the right hand of a decimal, neither increases nor decreases the value; but placed at the right hand of an integer increases the value, and at the left hand of a decimal decreases it ten-fold. If the numerator has not so many places as the denominator has cyphers, put so many before it, viz. at the left hand, as will make up the defect; as, for example, write thus .05; and To thus .006, &c. and thus do these fractions receive the form of whole numbers.

The 1st, 2d, 2d, 4th, &c. places of decimals, counting from the left to the right hand, are called primes, seconds, thirds, fourths, &c. Unity may be considered as a fixed point, from whence whole numbers proceed, infinitely increasing toward the left hand, and decimals. infinitely decreasing toward the right hand to 0, as may following table, viz.

NUMERATION TABLE.

be seen in the

In this table you may see that as integers increase in a ten-fold pro portion to the left hand, so decimal fractions decrease in a ten-fold proportion to the right hand, each figure taking its value from its distance from unit's place; if it be in the first place after units (or the separating point) it signifies tenths; if in the second, hundredths, &c. decreasing in each place in a ten-fold proportion.

Consequently every single figure expressing a decimal, has for its denominator an unit, or 1, with so many cyphers as its place is distint from unit's place. Thus you may see in the decimal part of the table, 2=26 ; 3=180; 4 &c. and if a decimal be expressed by several figures, the denominator is 1, with so mary cyphers as the lows est figure is distaut from unit's place. So .248 signifies 348 aud -0029, &c.

Cyphers, placed at the right hand of a decimal fraction, do not alter its value, since every significant figure continues to possess the same place; so .5 .50 and .500 are all of the same value, and each equal to 1. But cyphers, placed at the left hand of a decimal, do alter its value, every cypher depressing it to of the value it had before, by re noving every significant figure one place further from the place of units. So .5, .05, .005, all express different decimals, viz .5, ; .05, t; .005, 100.

Hence may be observed the contrary effect of cyphers being annexed to whole numbers and decimals. It is likewise evident from the table, that since the places of decimals decrease in a ten-fold proportion from units downwards, so they consequently increase in a ten-fold proportion from the right towards the left hand, as the places of whole numbers do; for ten hundredth parts make one tenth ; ten tenths make 1; ten units, ten; ten tens, one hundred. &c. viz. = 1 =1, and 1×40=10, which proves that decimals are subject to the sane law of notation, and consequently of operation, as whole numbers are; and also that the same number may be differently expressed, according as the integer is chosen: thus, the time since our Saviour's birth may be written 1821; or thus. 182.1; or thus 18.21; or thus, 1.821; or thus, .1821, according as one year, a decade, a century, a chiliad, or myriad, is used as the integer. Hence arises the superiour excellency of decimal arithmetick, above every other sort of numerical computation; as will appear, with convincing evidence, in the following work.

ADDITION OF DECIMALS.

In addition of decimals, we observe the same method as in whole numbers, only in setting down, regard must be had that the fractional parts stand one under the other, viz. primes, or tenths, under primes, -seconds under seconds, thirds under thirds, &c. and if mixed numbers are to be added. then the fractions are to stand as before directed, and the whole numbers to stand, as in whole numbers; then find their sum as

in whole numbers, and point off so many places for decimals as are equal to the greatest number of places in any of the given numbers.

EXAMPLES.

1. What is the sum of ,45,,5,,095, ,225, when added together.

Here in the 3d example, the decimals are arranged from the separatrix towards the right hand, and the whole numbers from the same point towards the left hand, the greatest number of decimal places in any one of the numbers is three, consequently three figures in the sum must be pointed off for decimals.

If the decimals are to be run on to a great many places, it will be sufficient in most cases to use only four or five places, and observe to increase the figure at which you break off by an unit, if rejected figure on the right exceed 5; and in adding such approximates, omit the right hand figure of the sum, as uncertain, but take in the carriage. Here follows an example at large, and the same contracted.

Thus,