MULTIPLICATION OF VULGAR FRACTIONS.

Rule. Multiply the numerators into one another for the nume rator of the product; and then do the same by the denominators, for the denominator of the product.

Example.

Multiply of a pound by of ditto: Say 3 times 5 is 15, the numerator; and 4 times 6 is 24, the denominator; So the answer is 12, or the lowest term §.

You are to note, That Multiplication in fractions lessens the product, though in whole numbers it augments it, as above:, or 12s. 6d. is less than , or 16s. 8d. and also less than the other fraction, or 15s. The reason of which I have not here room to insist on; but it is given in Fisher's Arithmetic, in Multiplication of Vulgar Fractions; to which book I refer the reader for that, and sundry enlargements in the several rules of the science of arithmetic.

2. To multiply a whole number by a fraction.

Rule. Multiply the integer by the numerator of the fraction, and place the product over the denominator.

Example.

Multiply 567. by 3-4+3=108

This improper fraction 128, reduced according to rule, makes but 421. which is less than 56, and confirms what was before asserted, viz. that Multiplication of Fractions lessen the product, &c.

To multiply a simple by a compound fraction.

Rule. Reduce the compound fraction to a simple one, as before taught, and work as above.

Example.

Multiply of a pound by

is 36, and 8 times 12 is 96. lowest terms; equal to 7s. 6d.

of of a pound: Say, 6 times 6 So that the answer is 36 or 3 in its

DIVISION OF VULGAR FRACTIONS.

Rule. Multiply the numerator of the divisor into the denominator of the dividend, and the product is the denominator of the quotient; and then multiply the denominator of the divisor into the numerator of the dividend, and the product will be the nume rator of the quotient.

45

Here 16, multiplied by 2, gives 32; and 15 by 3, gives 45, so that the quotient is, equal to 1, as in the work. Again, suppose was divided by, the quotient will be, equal to 12 And so for any other example.

OF DECIMAL FRACTIONS.

In Decimal Fractions the integer, or whole thing, as one pound, one yard, one gallon, &c, is supposed to be divided into ten equal parts, and those parts into tenths, and so on, without end.

So that the denominator of a decimal, being always known to consist of an unit, with as many ciphers as the numerator has places, is therefore never set down; the parts being only distinguished from the whole number, by a comma prefixed; thus, 5 which stands for,,25 for %,,123 for 1000⚫

But the different value of figures appears plainer by the following table.

Decimal

parts.

Whole numbers.

Millions.

Tens.
Hundreds.

C Thousands.

X Thousands.

Thousands.

-Units.

Parts of Tens.

Parts of Hundreds.
Parts of Thousands.
Parts of X Thousands.

Parts of C Thousands.
Parts of Millions.

From which it plainly appears that as whole numbers increase

in a ten-fold proportion to the left hand, decimal parts decrease in a ten-fold proportion to the right-hand; so that ciphers placed before decimal parts decrease their value, by removing them farther from the comma, or unit's place, thus,5 is 5 parts of 10, or,05 is 5 parts of 100, or ;,005, is 5 parts of 1000, or 100,0005, is 5 parts of 10000, or But ciphers after decimal parts do not alter their value, For,,5,50, ,500, &c. are each but of the unit.

Б

5

10000.

REDUCTION OF DECIMAL FRACTIONS.

Example 1.

Reduce of a pound sterling to a decimal.

4) 3,00)

that is, 75 hundreds, equal to 3 qrs. of any thing, whe,75) ther money, weight, measure, &c. as being of 100 ·

and so 25 hundreds is, in decimals, the quarter of any thing, as being the of a 100; and five-tenths expresses the half of any thing, as being the of 10.

In Reduction of Decimals sometimes it happens, that a cipher or ciphers must be placed to the left hand of the decimal, to supply the defect of the want of places in the quotient of Division. In this case always remember, that so many ciphers as you annex to the denominator of the vulgar fraction, so many places you must point off in the quotient towards the left hand; but if there be not so many places to point off, then you must supply the defect by placing a cipher or ciphers to the left of the decimal.

Example 2.

Reduce 9d. or to the decimal of a pound sterling, thus: 240) 9,000 (0375

The more ciphers you annex, the greater you bring your decimal to the truth: but in most cases, four ciphers annexed are sufficient. But when you are to reduce,,, as above, of an integer to a decimal, or any number of shillings to a decimal of a pound, two ciphers are sufficient. One example more.

Example 3.

Reduce 3 farthings to the decimal of a pound, that is, the vulgar fraction farthings being a pound.

960) 3,000000 (,003125. The work being performed according to Division, with two ciphers prefixed, quotes, 003125, or by 3125 ten hundred thousandth parts of a pound. By the same method, the vulgar fractions of weight, measure, &c. are reduced to decimals.

Example 4.

How is 12 pounds weight expressed in the decimal of 1 cwt. Avoirdupois, or 112lb. The vulgar fraction is, and the decimal,,1071, found as before, thus:

112) 12,0000 (,1072
112

80, &c.

The remainder, 43, is not worth notice, being less than the 100000 part of an unit or 1.

Example 5.

How is 73 days brought to the decimal of a year, vulgarly thus expressed?

365) 73,0 (,2 Answer, 2 tenths.

730

(0)

VALUATION OF DECIMALS.

To find the value of a decimal fraction, whether of corn, weigl.t, measure, &c.

Rule. Multiply the decimal given, by the units contained in the next inferior denomination, and point off as many places from the right hand as you have in your decimal; so those figures towards the left of the point are integers, or whole numbers; and those on the other side, towards the right hand, are parts of 1 or unity that is so many tenths, hundredths, thousandths, or ten thousandths, of one of those integers, whether a pound, a shilling, or a penny, &c. or of a ton, a hundred, a quarter, or a pound weight, &c. And so many of any other integer, of what quality

or kind soever.

In the example of money, I multiply the fraction by 20, and point off 520 for the three places in the decimal, &c. and the answer is 9s. 6d. nearly.

In the example of weight, I proceed as in that of money, (the fraction being the same,) but different with respect to the inferior denominations, and the answer is 9C. 2qrs. 2lb. 4 of a pound weight.

To find the true value of a decimal in money in a briefer method, viz.

Rule. Always account the double of the first figure (to the left hand) for shillings; and if the next to it be 5, reckon one shilling more; and whatever is above 5, call every one ten, and the next figure so many ones as it contains; which tens and ones call farthings; and for every 24, abate one as admit the last example of money, viz. 476; the double of 4 is 8, and there being one 5 in 7, (the next figure,) I reckon 1s. more, which makes 9s. and there being 2 (in the 7) above 5, they are to be accounted two tens, or 20; which with the next figure 6 being so many ones, make 26 farthings; and abating 1 for 24, they give 6d. and a farthing more.

ADDITION OF DECIMALS

Is the same in operation as in whole numbers; only in setting down, care must be taken that the decimal parts stand respectively under like parts: that is, primes under primes, seconds under seconds, thirds under thirds, &c. and the integers stand as in whole numbers.

Note. There must be as many places pointed off, as there are in that number which has most decimal places.

The casting up of the foregoing examples is the same with Addition of one denomination, in whole numbers. The total of

Fifths.