ARTICLE VIII.

VULGAR FRACTIONS.

We have seen that Division naturally produces a species of numbers DIFFERING from an integer, i. e., they are BROKEN PARTs of an integer, hence they are called Fractions. The word Vulgar means Common, and is very properly applied to this kind of numbers, because they are so common in practical arithmetic. It is also used to distinguish them from Denominate and Decimal Fractions; hence,

1. A Vulgar Fraction is a BROKEN PART of a unit, and consists of two parts, called the Numerator and Denominator.

2. The numerator shows how many parts the fraction contains.

3. The denominator shows how many equal parts the UNIT is divided INTO.

When the numerator and denominator are GIVEN, the frac tion is formed by drawing a line and placing the former ABOVE, and the latter BELOW it; thus,, which is called three-fourths, because it signifies that the UNIT is divided into 4 equal parts, called fourths, and that the fraction contains 3 of them.

In the same manner any two numbers of which the less is above, and the greater below the line, or vice versa, will form a vulgar fraction.

4. An improper fraction has its numerator greater than the denominator, and is always equivalent to a mixed number; as, 9=12.

5. A mixed number consists of a whole number and a proper fraction; as, 5.

6. A compound fraction is a fraction or a fraction, &c. ; asof of 1, &c.

7. A fractional unit consists of one of those parts into which the unit is divided, and a fraction is said to be reduced to a fractional unit when the numerator is one; as, ; but it is said to be reduced to an integral unit, or unity, when both the numerator and denominator are units; as, †, which is equal to 1.

Since fractions are a NECESSARY part of any system of numbers, it follows that the addition, subtraction, multiplication, and division of fractions is also NECESSARY. In the addition of integers, we have seen that numbers cannot be added unless they are of the same denomination, or kind; the same is true with regard to fractions, i. e., cannot be added to, because the denominator (denomination) of the first (1), is not like the denominator of the second, (3), i. e., thirds and halves are not the same; but may be added to, or to, because the denominators are alike.

We are now to show how fractions of different denominators may be reduced to EQUIVALENT fractions, having the SAME, or a COMMON denominator. We have seen in division, that when the dividend and divisor are both multiplied by the same quantity, the quotient remains the same, (page 44); therefore, if the NUMERATOR and DENOMINATOR of a fraction be multiplied by the SAME NUMBER, the VALUE of the fraction will remain unaltered, since there is no difference in the quantities. On this principle we may multiply by 3, and we shall have, and by 2, and we shall have Again, suppose 1,, and, are to be reduced to a COMMON denominator, on the SAME principle, we have 1×15=15, and 1818, and 9; hence,,, and, reduced to a coMMON denominator, equal 15, 10, and. Now, it 38 is plain, that 15 is the product of 3 and 5, 10 is the product of 2 and 5, and 6 is the product of 2 and 3; hence, for reducing fractions to a common denominator, without altering their value, we have the following

General RULE.

Multiply the NUMERATOR and DENOMINATOR of EACH FRACTION Successively, by the DENOMINATORS of all the rest.

EXAMPLES.

3670

4725

1. Reduce,,,, and, to a common denominator. Thus: ××××8=3788; 2d. ××××3=230; 3d. ××××3=4538; 4th. ××××}=}}}; 5th. ××××8848; hence, 378, 2430, 4818, 1778, and 5040, have common denominators, and are respectively equivalent to,,,, and

5040

8.

670 4536 4725

2. Reduce,,, and, to equivalent fractions having a

common denominator.

In the same manner reduce the following, viz.

5. 1, 1, 3, and 18. 6. 19, 13, 13, 18.

18

321

468'

10. 12, 18, 11, 13, to equivalent fractions having common denominators.

We have seen that fractions may be reduced to a HIGHER denomination, without altering their value; hence, vice versa, (Latin, versa, changed; vice, in turn; i. e., changed in turn, or taken contrarily,) a fraction may be reduced to a LOWER denomination without altering its value; this may always be done, except when the numerator and denominator are prime to each other; i. e., when both of them cannot be divided by the same number without a remainder. It has already been shown that division is the reverse of multiplication; therefore, it follows, that, if the denomination of a fraction may be increased, without altering its value, by multiplication, a fraction may (on the other hand,) be decreased in denomination, without altering its value, by DIVISION; hence, for reducing fractions to their lowest terms, we have the following

General RULE.

Divide the numerator and denominator of the fraction by any number which is contained in them both without a remainder till they become PRIME to each other; i. e., till they have no COMMON DIVISOR but unity; and the work will be done.

8. A prime number is one which has no divisor but unity; and can, therefore, only be divided into as many integral parts, as it contains units; as, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, &c.

9. A composite number is the product of two or more factors, and may, therefore, be divided into two or more integral parts greater than unity. A composite number, from its nature, may very properly be called a common multiple; hence,

10. The GREATEST common multiple of two or more factors is their product.

11. The LEAST common multiple of two or more factors is their product when reduced to their lowest relative value, i. e., when they are so reduced as to become prime to each other, by the use of common divisors.

12. A COMMON divisor is a number greater than a unit, that will measure two or more numbers, any number of times without a remainder; it is sometimes called a common measure; as, 4, which is a common divisor of 8 and 12, i. e., =2 and 12=3; hence, when 8 and 12 are reduced to their lowest relative value, they become 2 and 3. Again, 12×8=96, which is the greatest common multiple of 8 and 12; but 2×3=6, which is the least common multiple of 8 and 12, there is, therefore, a great difference between the least and greatest common multiples of two or more composite numbers, and since the common denominator of fractions is (by the Rule,) the GREATEST common multiple of all the denominators, it follows that their LEAST common multiple, if they have one, would be much more convenient for such purposes, if it were possible to use it as a substitute for the former; this, we say, is possible, and will be explained after the learner has made himself practically familiar with the following

RULE for finding the Least Common Multiple of Numbers.

First. Place the numbers on a line from left to right, and separate them by a (·).

Second. Place the common divisor of two or more of the numbers on the left of the line.

Third. Divide as many of the numbers as possible by the common measure, and place the respective quotients of cach number immediately under IT, below the line, and set those which are prime to the common divisor, below the line also, immediately under themselves, i. e., immediately below their former situation.

Fourth. Repeat this operation by taking convenient divisors, and proceeding as before, till all the numbers are so reduced as to have become a UNIT, or prime to each other; and, lastly, multiply the COMMON MEASURES and prime numbers (if any,) together, their product will be the least common multiple required.

EXAMPLES.

Find the least common multiple of the following series of numbers.

1. (6, 12, 15, 9, 18.) Thus: 6) 6.12.15.9.18

Hence, 6 and 3 are common measures; 2, 3, and 5, prime numbers, therefore, 6 × 3 × 2 × 3 × 5=540, the least common measure required.

P

Instead of 6 and 3, if we had taken 2, 3, and 3, as common measures, the result would have been the same.

which are sufficient to illustrate the Rule, and if this process is well understood, the learner will have little difficulty in applying it to the process of REDUCING fractions, to their LEAST common denominator, as directed by the following

RULE.

1st. Reduce the given fractions to the lowest terms.