2. Definition of a Fraction.-A Fraction is one or more of the equal parts into which anything may be divided. Every fraction must contain two numbers, a denominator and a numerator. These are called the terms of a fraction.

3. The Denominator.-The Denominator tells into how many equal parts the unit is divided. In the case shown in Fig. 1, 1 in. was the unit, and it was divided into 8 equal parts. The denominator in this case was 8.

4. The Numerator. The Numerator shows how many of these parts are taken. In giving the length of the piece of steel in Fig. 1, we divided the inch into 8 parts and took 5 of them for the length. Five (5) is the numerator, and 8 is the denominator.

5. Writing and Reading Fractions. In writing fractions, the numerator is placed over the denominator and either a slanting line, as in 5/8, or a horizontal line, as in §, drawn between them. The horizontal line is the better form to use, as mistakes are easily made when a whole number and a fraction with a slanting line are written close together.

We can have fractions of many things. An hour of time is divided into 60 equal parts called minutes. A minute is thus of an hour. Likewise, 20 minutes is 8 of an hour. In the same way, 1 second is of a minute.

In the early days, before the unit called the inch was in use, the foot was the common unit for measuring lengths. When it was necessary to measure lengths less than 1 ft., fractions of a foot were used. This became too troublesome, so of a foot was given the name of inch to avoid using so many fractions. For instance, where formerly the term of a foot was used, we now use 5 in. This shows how the use of a smaller unit reduces the use of fractions. Another small unit, the one-thousandth of an inch, is used by electricians, and is called the mil. The diameter

2

of wire, for example, is expressed more conveniently in mils than in thousandths of an inch.

6. Proper Fractions. If the numerator and denominator of a fraction are equal, the value of the fraction is 1, because there are just as many parts taken as there are parts in 1 unit.

In each of these cases, the numerator shows that we have taken the full number of parts into which the unit has been divided. Consequently, each of the fractions equals a full unit, or 1.

A Proper Fraction is one whose numerator is less than the denominator. The value of a proper fraction, therefore, is always less than 1.

7. Improper Fractions.-An Improper Fraction is one whose numerator is equal to or larger than the denominator. Therefore, an improper fraction is equal to or more than 1.

8. Mixed Numbers.—A Mixed Number is a whole number and a fraction written together: for example, 4 is a mixed number; 4 is read four and one-half and means 4 whole units and a unit more.

9. Reduction of Fractions.-Quite often we find it desirable to change the form of a fraction in order to make certain calcula

FIG. 2.-Scale divided to illustrate reduction of fractions.

tions; but, of course, the real value of the fraction must not be changed. The operation of changing a fraction from one form to another without changing its value is called Reduction.

By referring to the scale in Fig. 2 it will be seen that if we take the first inch and divide it into 8 parts, each in. will contain 4 of these parts. Hence, in. in. In this case, we make the denominator of the fraction 4 times as large, by making

=

4 times as many parts in the whole. It then takes a numerator 4 times as large to represent the same fractional part of an inch. This relation holds whether we are dealing with inches or with any other thing as a unit.

10. Changing to Higher Terms. When we change a fraction to higher terms, we increase the number of parts in the whole, as just shown, and this likewise increases the number of parts taken. Therefore, the numerator and denominator both become larger numbers.

A fraction is raised to higher terms by multiplying both numerator and denominator by the same number.

3 16

Suppose we want to change of an inch to 64ths. To get 64 for the

denominator, we must multiply 16 by 4 and, therefore, must multiply 3 by the same number.

3 3 X 4 12
16
16 X 4 64

=

=

11. Reduction to Lower Terms.-When we reduce a fraction to lower terms, we reduce the number of parts into which the whole unit is divided. This likewise reduces the number of parts which are taken.

A fraction is reduced to lower terms by dividing both numerator and denominator by the same number. When there is no number which will exactly divide both numerator and denominator, the fraction is already in its lowest terms.

There is no number that will evenly divide both 9 and 32. Therefore, the fraction is reduced to its lowest terms.

12. Reduction of Improper Fractions. When the numerator of a fraction is just equal to the denominator, we know that the value of the fraction is 1 (see Art. 6).

In each of these cases we have taken the full number of parts into which we have divided the unit. Consequently, each of these fractions is one whole unit, or 1.

When the numerator is greater than the denominator, the value of the fraction is one or more units, plus a proper fraction, or a whole plus some part of a whole.

From these examples we may see that to reduce an improper fraction to a whole or mixed number the simplest way is as follows:

Divide the numerator by the denominator. The quotient will be the number of whole units. If there is anything left over, or a remainder, write this remainder over the denominator, since it represents the number of parts left in addition to the whole units. We now have a mixed number, or an exact whole number, in place of the improper fraction.

These show that a fraction represents unperformed division. In fact, division is often indicated in the form of a fraction. The numerator is the dividend and the denominator is the divisor.

13. Reduction of Mixed Numbers.-It is often necessary or desirable to change mixed numbers to improper fractions. The method of doing this may be seen from the following examples.

were to be reduced to an improper fraction, we would say: "Since there are 4 fourths in 1, in 7 there are 4 X 7, or 28, fourths. 28 fourths plus 1 fourth equals 29 fourths."

The rule which this gives us is very simple: Multiply the whole number by the denominator of the fraction and write the product over the denominator. This reduces the whole number to a fraction. Add to this the fractional part of the mixed number. The sum is the desired improper fraction.

In working problems like the above, the work should be arranged as in the following example.

14. Use of Signs. Throughout the preceding paragraphs and in all the following text material, five "signs of operation" are used repeatedly: the plus sign (+); the minus sign (−); the equals sign (=); the multiplication sign (X); and the division sign (÷).

The plus sign (+) is a signal, or indication, of addition and when placed between two numbers means that they are to be added. The minus sign (-) when placed in this manner is a signal of subtraction. When the minus sign is placed between two numbers, it must be remembered that the second number is to be subtracted from the first; thus in the subtraction 7 - 6 = 1, the 6 is subtracted from the 7. The multiplication sign (X) signifies that the numbers are to be multiplied; and the division sign (÷) that the first number is divided by the second.

The most misused of all these signs is the equals sign (=). This sign placed between two sets of numbers simply means that