of 2446 above 24, and I scratch out the terms 4 and 24; I also find that 12 in the means will measure itself once, and 36 in the extremes 3 times; wherefore I scratch out the terms 12 and 36, and put the quotient 3 above 36; then I find that the 3 now found in the extremes will mcasure 21 in the means 7 times, which I scratch, as before, then there is only 6, and 7 remaining for the terms, which are both in the means; therefore I multiply the means for a dividend, and as there is no terms but units in the extremes to divide it, the product is the answer.

VULGAR FRACTIONS.

I.

A

DEFINITIONS.

FRACTION is some part or parts of an unit, or any whole number, and consists of two parts, the one called a numerator, and the other a denominator.

II. The denominator shows the number of parts the unit is divided into, and the numerator shows what number of these parts are to be taken.

Corollary 1. From this it is evident, if the number of parts expressed by the numerator, be equal to the number of parts in the denominator, then the fraction will be equal to an unit, and if the numerator is greater or less than the denominator, the value of the fraction will also be greater or less than an unit.

Corollary 2. Hence the value of any fraction is equal to the quotient arising, by dividing the numerator by its

denominator, together with the remaining parts, if any; therefore any vulgar fraction is properly expressed in this manner,, the number above the line being the numerator, and the number below the denominator.

III. When the value of a fraction is less than an unit, such as,, or, it is called a proper fraction.

IV. When the value of a fraction is greater than an unit, such as these,,,, &c. then the fraction is called improper.

V. If the numerator of an improper fraction be divided by its denominator, then the quotient, together with the fractional parts that is left, is called a mixed number; thus, 34, or 31, is a mixed number.

VI. A compound fraction is the fraction of a fraction; thus, of a is a compound fraction.

PROBLEM I.

To find the greatest common measure in two or more numbers.

I. If there are only two numbers, divide the greater by the less, and the divisor by the remainder, and proceed in this manner till nothing remains, then will the last divisor be the greatest common measure of the two numbers.

II. When there is more than two numbers, find the greatest common measure of any two of them, as before; also the greatest common measure of that common measure, and of the other numbers, and proceed in this manner through all the numbers to the last, then will the last common measure be the greatest that will measure all the given numbers.

EXAMPLE.

What is the greatest common measure of 108, 132, and 78?

And the greatest common measure of all the numbers is 6.

PROBLEM II.

To reduce a fraction to its lowest terms.

Find the greatest common measure by the last Problem, and divide both terms of the fractions by it, and the quotient will be the terms of the fraction required.

Divide the terms of the given fraction by any digit above 1, that will divide them without a remainder, and these quotients again in the same way; and proceed in this manner as often as you can, and the last quotient will be the terms of the fraction required.

EXAMPLE.

Reduce 1 to its lowest terms.

3) 4) 9)

314=1== Answer.

PROBLEM III.

To reduce a whole number to the form of a fraction, which shall have a given denomi

nator.

Multiply the whole number by the given denominator, and under that product place the said denominator.

EXAMPLE.

Reduce 7 to a fraction whose denominator shall be 3.

7x3=21, hence is the fraction required.

PROBLEM IV.

To reduce a fraction of different denominations to equivalent ones, having a common denominator.

Multiply each numerator into all the denominators but its own, for a numerator to the required fraction, and all the denominators continually, for a common denominator.

VOL. I.

P

EX

Therefore the fractions 12, 14, and 14, are the fractions required.

Another method.

I. Divide by any number that will divide two or more of the given denominators without a remainder, and set the quotients together, with the undivided numbers in a line. below them; divide the second line as before, and proceed in this manner, till there is no two numbers that can be divided; then multiply all the divisors and quotients together for the common denominator.

II. Divide the common denominator by the denominator of each fraction; then multiply the quotient by each numerator, and the products will be the numerators of the fractions required; by this method you will have the least common denominator.

EXAMPLE.

Reduce,, and, to fractions, having the least common denominator.

2)2 3 4

1 3 2.

2X1×3×2=12. The least common denominator.