SHOP ARITHMETIC

CHAPTER I

COMMON FRACTIONS

1. Why We Use Fractions.-When we find it necessary to deal with things that are less than one unit, we must use fractions. A machinist cannot do all his work in full inches because it is generally impossible to have all measurements in exact inches. Consequently, for measurements less than 1 in., he uses fractions of an inch; he also makes use of fractions for measurements between one whole number of inches and the next whole number. If a bolt is wanted longer than 4 in. but shorter than 5 in., it would be 4 in. and a fraction of an inch. This fraction of an inch might be nearly a whole inch or it might be a very small part of an inch. The system used to designate parts of a unit is

easily seen by looking at a machinist's scale or at a foot-rule of any sort. Each inch on the scale is divided into a number of equal parts. A wooden foot-rule usually has eight or sixteen parts to each inch, while a machinist's steel scale has much finer divisions. Now, if we want to measure a piece of steel which is not an inch long, we hold a scale against it, as in Fig. 1, and find out how many of these divisions of an inch it takes to equal the length of the piece. The scale in Fig. 1 is 3 in. long and each inch is divided into eight parts. We see that this piece is as

long as five of these eight parts of an inch, or we say that it is "five-eighths" of an inch long.

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 numerator and a denominator. 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 eight equal parts. The denominator in this case was eight.

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 eight parts and took five of them for the length. Five is the numerator and eight 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 all sorts of things besides inches. An hour of time is divided into sixty equal parts called minutes. A minute is merely of an hour. Likewise, 20 minutes is 28 of an hour. In the same way, 1 second is of a minute.

In the early days, before we had the unit called the inch, the foot was the common unit for measuring lengths. When it was necessary to measure lengths less than 1 ft., fractions of a foot

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were used. This got to be too troublesome, so one-twelfth of a foot was given the name of inch to avoid using so many fractions. For instance, where formerly one said of a foot, we can now say 5 in. This shows how the use of a smaller unit reduces the use of fractions. In Europe, a unit called the millimeter is used in nearly all shop work. This is so small, being only about of an inch, that it is seldom necessary in shop work to use fractions of a millimeter.

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 one 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 four whole units and one-half 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 calculations; 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 con