It will be noted that if each of the last quotients are multiplied by 12, the greatest common divisor, the products thus found will be the respective numbers from which they were derived. This is a proof of the operation:

As 2 × 12=24, 4 × 12=48, etc.

Multiples: A multiple of a number is the product of that number multiplied by any other whole number.

A common multiple of two or more numbers is a product found by using those numbers as factors alone, or with other factors.

The least common multiple of two or more numbers is the smallest number which can be evenly divided by each of those numbers.

In solving various problems involving whole numbers and fractions, it is often necessary to find the least common multiple of two or more numbers.

Rule for Finding the Least Common Multiple: Write the numbers in a column and draw a vertical line on each side, divide by any small number that will evenly divide two or more of them, place this divisor to the left and the quotients to the right of their respective numbers, the numbers which are not divisible at this operation are carried in full to the right of the line, continue till no number can be found which will divide two or more of the numbers, the product then of the divisors, the final quotients and the undivided numbers is the least. common multiple.

3×2×2×2×3×5×2=720 least common multiple.

FRACTIONS.

A fraction is one or more of the equal parts of a unit or whole number. "Fraction" is derived from the Latin word

"fractus" meaning broken or divided, broken up into separate parts.

A fractional unit is one of the parts into which the unit is equally divided. A fraction is expressed by two numbers, one written above a line and the other below; this arrangement indicates that the number below the line is the number of parts into which the whole unit was divided, and is called the denominator, and the number above the line is the number of equal parts taken away from the others to make the fraction, and is called the numerator, as ; here the unit has been divided into 4 denominate parts, 3 of which are taken or enumerated to make this fraction, therefore 3 of these parts make of the whole.

When the numerator is less than the denominator it is called a proper fraction, as 4, 3, 2, are proper fractions.

When the numerator is greater than the denominator it is called an improper fraction, as,., are improper fractions. A compound fraction is a fraction of a fraction, as of §, of, are compound fractions.

A complex fraction is one containing a fraction in its 321 numerator or denominator, as 5'1'6

are compl x fractions. A number consisting of a whole number and a fraction is called a mixed number, as 31, 21, 121, are mixed numbers.

To multiply the numerator by a number multiplies the value of the fraction by that number; to multiply the denominator by a number is equivalent to dividing the value of the fraction by that number.

To divide the numerator by a number divides the value of the fraction; to divide the denominator multiplies the value of the fraction by that number.

To multiply both the numerator and denominator by the same number does not change the value of the fraction; multiplying raises it to higher terms, and dividing reduces it to lower terms.

To reduce a fraction to its lowest terms, divide both the numerator and denominator by their greatest common divisor. To add and subtract fractions, it is necessary that they have the same or a common denominator.

To reduce two or more fractions to a common denominator, find the least common multiple of the denominators, this will be the least common denominator, and multiply each numerator, by the quotient of the common denominator divided by its respective denominator for a new numerator.

Example: Reduce 4, §, 2, to a common denominator.

2, 1, 24÷4=6, 6x3=1=1

2x2 4, 2, 24÷8-3, 3x5=1=8

24

2×2×3×2×1=24 common denominator.

If a fraction is an improper fraction, it may be reduced to a whole or mixed number by dividing the numerator by the denominator, if there is a remainder write it over the denominator as a fraction.

Example: Reduce 33 to a whole or mixed number.

33÷8=41.

A whole or mixed number is reduced to an improper fraction by multiplying the whole number by the denominator and adding to this product the numerator, if there be any, which is written over the denominator as the numerator of the required fraction.

33

Addition of Fractions.

As only units or things of the same nature, kind or denomination can be added, so it is that only fractions having the same or a common denominator can be added.

Two dogs and 3 cats cannot be added together as dogs or cats, but when a denomination or name common to both is used they may then be added, as 2 dogs and 3 cats make 5 animals. It is therefore necessary before adding fractions to reduce them to a common denominator, after which add the numerators, the sum of which, used as a numerator of the common denominator, is the sum of the fractions added, expressed as a fraction.

Rule for Addition: Reduce the fractions to be added to the least common denominator, add the numerators and place the sum as a numerator over the common denominator. If this produces a proper fraction, reduce it to its lowest terms, if an improper fraction, reduced it to a whole or mixed number, which, as shown above, is done by dividing the numerator by the denominator; where mixed numbers are to be added, add the whole numbers separately, after which add the sums of the fractions and whole numbers.

To subtract fractions it is necessary that they should be reduced to a common denominator.

Rule: Write the subtrahend under the minuend, reduce the fractions to a common denominator, subtract the numerator of

the subtrahend from the numerator of the minuend, write the result as a numerator over the common denominator, which is the remainder expressed as a fraction.

If the operation be composed of mixed numbers, and the fraction of the minuend is less than that of the subtrahend, borrow one unit from the whole number of the minuend, reduce it to a fraction having the common denominator, add it to the fraction in the minuend and proceed to subtract; reduce the remainder to its lowest terms and write it under the column of fractions, the unit borrowed from the whole number of the minuend is returned to the whole number of the subtrahend before the whole numbers are subtracted. 1-3=1. Example:

66

Multiplication of Fractions.

Multiplication of fractions is indicated by writing the word of" or the sign × of multiplication between them, as of or 1×1.

Rule: To multiply a fraction by a fraction, multiply the numerators together for a numerator and the denominators together for a denominator, the fraction thus obtained is the product.

Example: =3 or by analysis of

.

To multiply a fraction by a whole number, multiply the numerator by the whole number and divide the product by the denominator, the quotient thus formed is the product. In practice, it is customary to multiply whole numbers by fractions, that is, to multiply the whole number by the numerator and divide the product by the denominator.