A whole or integer number may be expressed like a fraction, by writing 1 below it, as a denominator; so 3 is, or 4 is, &c.

A fraction denotes division; and its value is equal to the quotient obtained by dividing the numerator by the denominator; so 12 is equal to 3, and 20 is equal to 4.

Hence then, if the numerator be less than the denominator, the value of the fraction is less than 1. But if the numerator be the same as the denominator, the fraction is just equal to 1. And if the numerator be greater than the denominator, the fraction is greater than 1.

REDUCTION OF VULGAR FRACTIONS.

REDUCTION of Vulgar Fractions, is the bringing them out of one form or denomination into another; commonly to prepare them for the operations of Addition, Subtraction, &c. of which there are several cases.

PROBLEM.

To find the Greatest Common Measure of Two or more Numbers.

THE Common Measure of two or more numbers, is that number which will divide them both without remainder; so, 3 is a common measure of 18 and 24; the quotient of the former being 6, and of the latter 8. And the greatest number that will do this, is the greatest common measure: so 6 is the greatest common measure of 18 and 24; the quotient of the former being 3, and of the latter 4, which will not both divide further.

RULE.

If there be two numbers only; divide the greater by the less; then divide the divisor by the remainder; and so on, dividing always the last divisor by the last remainder, till nothing remains; so shall the last divisor of all be the greatest common measure sought.

When there are more than two numbers, find the greatest common measure of two of them, as before; then do the same for that common measure and another of the numbers;

and

and so on, through all the numbers; so will the greatest common measure last found be the answer.

If it happen that the common measure thus found is 1; then the numbers are said to be incommensurable, or not having any common measure.

EXAMPLES.

1. To find the greatest common measure of 1908, 936,

and 630.

936) 1908 (2

1872

So that 36 is the greatest common measure of 1908 and 936.

Hence then 18 is the answer required.

2. What is the greatest common measure of 246

3. What is the greatest common measure of 324,

1032?

CASE I.

To Abbreviate or Reduce Fractions to their Lowest Terms.

* DIVIDE the terms of the given fraction by any number that will divide them without a remainder; then divide these quotients

That dividing both the terms of the fraction by the same number, whatever it be, will give another fraction equal to the former, is evident. And when these divisions are performed as often as can be done, or when the common divisor is the greatest possible, the terms of the resulting fraction must be the least possible.

Note 1. Any number ending with an even number, or a cipher, is divisible, or can be divided, by 2.

2. Any number ending with 5, or 0, is divisible by 5.

CASE II.

To find the Value of a Decimal in terms of the Inferior Denominations.

MULTIPLY the decimal by the number of parts in the next lower denomination; and cut off as many places for a remainder to the right-hand, as there are places in the given decimal.

Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before.

Proceed in the same manner through all the parts of the integer; then the several denominations separated on the lefthand, will make up the answer.

Note, This operation is the same as Reduction Descending in whole numbers.

EXAMPLES.

1. Required to find the value of 775 pounds sterling.

CASE III.

To reduce Integers or Decimals to Equivalent Decimals of Higher Denominations.

DIVIDE by the number of parts in the next higher denomination; continuing the operation to as many higher denominations as may be necessary, the same as in Reduction Ascending of whole numbers.

EXAMPLES.

1. Reduce 1 dwt to the decimal of a pound troy.

2. Reduce 9d to the decimal of a pound.

Ans. 03751.

3. Reduce 7 drams to the decimal of a pound avoird.

4. Reduce 26d to the decimal of a l.
5. Reduce 2.15 lb to the decimal of cwt.

Ans. 02734375lb. Ans. 0010833 &c. l.

Ans. 019196+cwt.`

6. Reduce 24 yards to the decimal of a mile.

Ans. 013636 &c. mile.

7. Reduce 056 pole to the decimal of an acre.

Ans. 00035 ac.

8. Reduce 1.2 pint of wine to the decimal of a hhd.

Ans. 00238+hhd.

9. Reduce 14 minutes to the decimal of a day.

Ans. 009722 &c. da.

10. Reduce 21 pint to the decimal of a peck.

Ans. 013125 pec.

11. Reduce 28" 12" to the decimal of a minute.

NOTE, When there are several numbers, to be reduced all to the decimal of the highest :

Set the given numbers directly under each other, for dividends, proceeding orderly from the lowest denomination to the highest.

Opposite to each dividend, on the left-hand, set such a number for a divisor as will bring it to the next higher name; drawing a perpendicular line between all the divisors and dividends.

Begin at the uppermost, and perform all the divisions : enly observing to set the quotient of each division, as decimal

CASE III.

To Reduce an Improper Fraction to its Equivalent Whole or
Mixed Number.

* DIVIDE the numerator by the denominator, and the quotient will be the whole or mixed number sought.

EXAMPLES.

1. Reduce
Here

2. Reduce
Here

to its equivalent number.

or 12÷ 3 = 4, the Answer.
to its equivalent number.

or 15724, the Answer.

3. Reduce 749 to its equivalent number.
Thus, 17) 749 (447

68

69 68

So that 749447, the Answer.

To Reduce a Whole Number to an Equivalent Fraction, having a Given denominator.

MULTIPLY the whole number by the given denominator: then set the product over the said denominator, and it will form the fraction required.

*This rule is evidently the reverse of the former; and the reason of it is manifest from the nature of Common Division.

† Multiplication and Division being here equally used, the result must be the same as the quantity first proposed.

EXAMPLES.