69. A man rated at 25 cents an hour is working under the premium system. In one day of 9 hours he performs 3 operations the limits of which are 4 hours each. If he gets paid a premium of half the time saved, how much will he make for the day?

70. The following figure represents a page from a time book of a shop working entirely on day work. Calculate the total time of each man, the amount of money due him, and the total pay roll for the shop.

CHAPTER V

DECIMAL FRACTIONS

36. What are Decimals?—In the old days, when no machinist pretended to work much closer than in. and the micrometer was unknown, the mechanic had little use for decimals except in figuring his pay. Now, however, we find that micrometer measurements are used so generally that a knowledge of decimal fractions is essential.

A Decimal Fraction is merely a fraction having a denominator of 10, 100, 1000, or some similar multiple of 10. The denominator is never written, however, but a system similar to that used in writing U. S. money is used. A decimal fraction is written by first putting down a period or "decimal point" and then writing the numerator of the fraction after the decimal point in such a manner that the denominator can be understood. Everything that comes after the decimal point (to the right of it) is a fraction, or part of a unit.

In writing sums of money, the first figure after the decimal point indicates dimes or tenths of a dollar; the second figure indicates cents, or hundredths of a dollar; the third figure, if any, indicates mills or thousandths of a dollar. This system has proved so handy that it has been extended to representing fractions of any sort of a unit (not necessarily dollars).

Let us take a decimal, say .253, and find out its meaning. We said that the first figure was tenths; the second, hundredths; the third, thousandths, and so on. Then .253 would be + 180 +1000. This is not a very handy system unless there is

some easier way to read it. If we reduce these to a common denominator and add them, we get:

This shows that one place to the right of the decimal point indicates a denominator of 10, two places a denominator of 100, three places a denominator of 1000, and so on. The number at the right of the decimal point can, therefore, be taken as the numerator, and the denominator obtained as follows: Put down a 1 below the decimal point and a cipher (0) after it for each figure in the numerator. This will give the denominator. In the case just given, we would have 5 showing that the denominator is 1000..

In the same manner:

1000

The last case (.042) presents an interesting problem. Here we have a numerator so small in respect to the denominator that it is necessary to have a cipher, or zero (0) between it and the decimal point, in order that the denominator can be indicated correctly. Let us see how we would go about writing such a common fraction into a decimal. Take To. If we merely wrote .5 that would be and would, therefore, not be right. From the rule for finding denominators of decimals we see that

there must be as many figures after the decimal point as there are ciphers in the denominator. In this case the denominator (1000) has 3 ciphers, so we must have three figures in our decimal. We, therefore, put two ciphers to the left of the 5 and then put down the decimal point. We now have .005, which can be easily seen to be To

One thing that must be carefully borne in mind is that adding ciphers after a decimal does not change the value of the fraction. .5 is the same in value as .50 or .500 because is the same in value as 5 or 100%. On the other hand, ciphers immediately following the decimal point do affect the value of the fraction, as has just been shown.

Mixed numbers are especially easy to handle by decimals, because the whole number and the fraction can be written out in a horizontal line with the decimal point between them. We read mixed decimals just as we would any mixed number-first the whole number, then the numerator, and lastly, the denominator.

Example:

42137.24697

In this example, 42137 is the whole number, and .24697 is the fraction. The number reads "forty-two thousand one hundred thirty-seven and twenty-four thousand six hundred ninety-seven hundred-thousandths.

The names and places to the right and left of the decimal point are as follows:

1 2 3

37. Addition and Subtraction. Knowing that all figures to the right of the decimal point are decimal parts of 1 thing and that all figures to the left are whole numbers and represent whole things, it will be seen readily that in addition and subtraction the figures must be so placed that the decimal points come under

each other. As was shown under U. S. Money, the operations can then be carried out just as if we were dealing with whole numbers.

Pay no attention to the number of figures in the decimal. Place the decimal points in line vertically. You can, if you desire, add ciphers to make the number of decimal places equal in the two numbers. Remember, however, that the ciphers must be added to the right of the figures in the decimal. Proceed as in ordinary addition and subtraction, carrying the tens forward in addition and borrowing, where necessary, in subtraction just as with whole numbers.

38. Multiplication. In multiplication forget all about the decimal point until the work is finished; multiply as usual with whole numbers. Then point off in the product as many decimal places, counting from the right, as there are decimal places in the multiplier and multiplicand together.

Since there are three decimal places in one number, and one in the other, we count off in the product four (3+1) places from the right and place the point between the 7 and the 4. The last 0 can be dropped after pointing off the product, giving the result 34.762 (or 3478). The reason for this can be seen from the following: The whole numbers are 6 and 5. The result must be a little more than 6×5-30, and less than 7×6=42, since the numbers are more than 6 and 5, and less than 7 and 6. The actual result is 34.762.

The position of the decimal point can be reasoned out in this way for any example, but the quickest way is to point off from the right a number of decimal places equal to the sum of the numbers of decimal places in the multiplier and multiplicand.