1. The places of decimal parts in the divisor and quotient, counted together, must always be equal to those in the dividend; therefore, divide as in whole numbers, and, from the right hand of the quotient, point off so many places for decimals, as the decimal places in the dividend exceed those in the divisor.

2. If the places of the quotient be not so many as the rule requires, supply the defect by prefixing ciphers to the left hand.

3. If at any time there be a remainder, or the decimal places in the divisor be more than those in the dividend, ciphers may be annexed to the dividend or to the remainder, and the quotient carried on to any degree of exactness.

180. What is the rule for division of decimals?

When decimals or whole numbers are to be divided by 10, 100, 1000, &c. [viz. unity with ciphers] it is performed by removing the separatrix, in the dividend, so many places toward the left hand, as there are ciphers in the divisor.

*This character, placed here, shews that the quotient is not complete, and that it may be further continued by annexing ciphers to the remainder.

These questions are left unpointed in the quotient, to exercise the learner.

181 When decimals or whole numbers are to be divided by 100, 1000, &c., how do you proceed?

REDUCTION OF DECIMALS.

CASE I.

To reduce a vulgar fraction to its equivalent decimal.

RULE.

Divide the numerator by the denominator, as in Division of Decimals, and the quotient will be the decimal required.-Or, so many ciphers as you annex to the given numerator, so many places must be pointed off in the quotient, and if there be not so many places of figures in the quotient, the deficiency must be supplied by prefixing so many ciphers before the quotient figures.

NOTE. By annexing one, two, three, &c. ciphers to the numerator, the value of the fraction is increased ten, a hundred, &c. times. After dividing, the quotient will of course be ten, a hundred, &c. times too much; the quotient must therefore be divided by ten, a hundred, &c to give the true quotient or fraction. In the first example, is made 1900-125, which, divided by 1000, is 5,125, and is the rule.

3. Reduce 1, 1, 2, 3, 1, 3, and to decimals.

4. Reduce,,, and to decimals.

4 Ans. 25, 5, 75, ·333+, ·8, ·833+, ·875.

Ans. 263+, 692+, 025, 25.

CASE II.

To reduce numbers of different denominations, as of Money, Weight, and Measure, to their equivalent decimal fractions.

RULE.

1. Write the given numbers perpendicularly under each other, for dividends, proceeding orderly from the least to the greatest.

182. How do you proceed to reduce a vulgar fraction to its equivalent decimal ?183. What is your rule for reducing numbers of different denominations to their

equivalent decimal value?

2 Opposite to each dividend, on the left hand, place such a number, for a divisor, as will bring it to the next superior denomination, and draw a line perpendicularly between them.

3. Begin with the highest, and write the quotient of each division as decimal parts on the right hand of the dividend next below it, and so on, until they are all used, and the last quotient will be the decimal sought.

:

NOTE. The reason of the rule may be explained from the first example thus, three farthings are of a penny, which, brought to a decimal, is 75; consequently, 8d. may be expressed, 8-75d. but 8-75 is $75 of a penny-375 of a shilling, which, brought to a d'cimal, is 7291668.+ lo like manner, 17-729166s.+ are 17729168. 1773888886458+ as by the rule.

2000000

1000000

EXAMPLES.

1. Reduce 17s. 8d. to the decimal of a pound.

Here, in dividing 3 by 4, we suppose two ciphers to be annexed to the 3, which make it 3.00, and 75 is the quotient, which we write against 8 in the next line; this quotient, viz. 8.75, being pence and decimal parts of a penny, we divide them by 12, which brings them to shillings and decimal parts; we therefore divide by 20, and (there being no whole number) the quotient is decimal parts of a pound.

2. Reduce 1, 2, 3, 4, and so on to 19 shillings, to decimals.

Here, when the shillings are even, half the number, with a point prefixed, is their decimal expression; but if the number be odd, annex a cipher to the shillings, and then halving them, you will have their decimal expression.

3. Reduce 1, 2, 3, and so on to 11 pence, to the decimals of a

shilling.*

6

4. Reduce 1, 2, 3, &c. to 11 pence, to the decimals of a pound.

To find the decimal of any number of shillings, pence, and farthings,

by inspection.

RULE.

1. Write half the greatest even number of shillings for the first decimal figure.

2. Let the farthings in the given pence and farthings, possess the second and third places; observing to increase the second place, or place of hundredths, by 5, if the shillings be odd, and the third place by 1, when the farthings exceed 12, and by 2 when they exceed 36.

NOTE.-If there be no shillings, or only one shilling in the given sum, a cipher must be written in the first place, or place of tenths. If the farthings in the given pence and farthings, do not exceed 9, a cipher must be written in the second place, or place of hundredths. The invention of the rule is as follows:-As shillings are so many 20ths of a pound, half of them must be so many tenths, and consequently take the place of tenths in the decimals; but when they are odd, their half will always consist of two figures, the first of which will be half of the even number, next less, and the second a 5: Again, farthings are 960ths of a pound, and had it happened that 1000, instead of 960, had made a pound, it is plain any number of farthings would have made so many thousandths, and might have taken their place in the decimal accordingly. But 960 increased by part of

*The answers to this question are the same as the decimal parts of a foot.

184.

What is the rule for finding the decimal of any number of shillings, pence, &c. by inspection?

N