31. A boiler maker and his helper renew of the total number of tubes in a boiler the first day on the job; the second day they renew of the total number of tubes. What fraction of the total number remains to be completed after the second day?

32. The weights of a number of castings are; 412 lb., 270 lb., 1020 lb., 75 lb., and 68 lb. What is their total weight?

33. In a certain boiler test about of the total heating value of the coal was unavoidably lost up the chimney, about was lost through unburned coal in the ash, and the remainder was absorbed by the

FIG. 6. Casting for gas engine cylinder.

boiler. Compute the fraction of the total heat absorbed by the boiler.

34. Four studs are required; 23 in., 13 in., 25 in., and 133 in. long; how long a piece of steel will be required from which to cut them, allowing in. altogether for cutting off and finishing their ends? 35. Monday morning an engineer bought 48 gal. of cylinder oil; on Monday, Tuesday, and Wednesday he used gal. per day; on Thursday he used gal.; and on Friday gal. How much oil had he left on Saturday?

36. A piece of work on a lathe is 1 ft. in diameter; it is turned down in five cuts: in the first step the tool takes off 3 in. from the diameter; then in.; then in.; then 2 in.; and the fifth time, in. What

is the diameter of the finished piece?

37. A small foundry uses 15 tons of coke the first week of a month, 143 tons the second week, 17 tons the third week, and 192 tons during the remainder of the month. What was the total amount of coke burned during the month?

38. By mistake, the draftsman omitted the thickness of the flange on the drawing of the gas engine cylinder shown in Fig. 6. From the other dimensions given, calculate the thickness of the flange.

39. A millwright has to rig up temporarily a 6-in. belt to be 367 in. long. In looking over the stock of old belting he finds the following pieces of the right width: one piece 126 in. long, one 1423 in. long, and one 1333 in. long. How many inches must be cut from one of the pieces so that these pieces can be laced together to give the right length?

40. The time cards for a certain piece of work show 2 hr. and 15 min. lathe work, 3 hr. and 10 min. milling, 1 hr. and 10 min. planing, and 1 hr. and 15 min. bench work; what is the total number of hours to be charged to the job?

CHAPTER III

MULTIPLICATION AND DIVISION OF FRACTIONS

21. A Whole Number Times a Fraction. In the study of multiplication, we learn that multiplying is only a short way of adding. For example, 4 X 7 is the same as four 7's added together. Either 4 X 7, or 7+7+7+ 7 will give 28. If we apply this same principle to the multiplying of fractions, we see that 4 X is the same as four of these fractions added together. 7 7 7 7 7 28

4 X

=

+ 8

=

+ +
8 8 8 8 8

This shows that multiplying a fraction by a whole number is performed by multiplying the numerator by the whole number and placing the product over the denominator of the fraction.

In other words, the size of the parts is not changed, but the number of parts is increased by the multiplication. After multiplying, the product should be reduced to lowest terms; and, if an improper fraction, should be reduced to a whole or mixed number. Example:

What would be the total weight of 12 brass castings each weighing of a pound?

22. "Of" Means "Times."-The word "of" is often seen in problems in fractions, as, for instance, "What is of 5 in.?" In such a case, we work the problem by multiplying, so we say that "of" means "times." You can see that this is so by taking a piece of wood 5 in. long and cutting it into 4 equal parts and then taking 3 of these parts. These 3 parts will be of 5 in., and 2 by actual measurement will be 32 in. long, so we know that of 5 3. Now see what

=

times 5 is:

23. A Fraction Times a Fraction. To multiply two or more fractions together, multiply the numerators together for the numerator of the product, and multiply the denominators together for the denominator of the product.

Multiply.

8

Explanation: The numerator of the product is obtained from multiplying the numerators together: 7 X 2 = 14. The denominator of the product, in the same manner, is 8 X 3 = 24. This gives the product which can be reduced

14

24

Let us see what multiplication of fractions really means, and why the work is done as just shown. Suppose we are to find of in. This means that of an inch is to be divided into 4 equal parts and 3 of these parts are wanted. If we divide in. into 4 equal parts, each part will be one-fourth as large as in. and, therefore, can be considered as being made up of 7 parts, each one-fourth as large as in. Then of

=

Three (3) of these parts will naturally contain 3 times as many thirty

24. Multiplying Mixed Numbers. This is one of the most difficult operations in the study of fractions, unless one adopts a fixed rule and follows it in all cases. The student will have no trouble if he will first reduce the mixed numbers to improper fractions, and then multiply these like any other fractions.

To multiply a mixed number by a whole number, we can either reduce the mixed number to an improper fraction and then perform the multiplication, or we can multiply the fractional

part and the whole number part separately by the number and then add the products.

Example:

cts. apiece?

What would be the cost of ten 4-in. by 6-in. machine bolts at 13

10

132 cts., Answer.

30

8

Explanation: First multiply 10 by This gives

or

6

8

3 or 33 Set this down. Then multiply

10 by 1. This gives 10, and we add this to the 3 giving a total of 133 cts.

25. Cancellation.-Very often the work of multiplying fractions may be lessened by cancellation, as it avoids the necessity of reducing the product to lowest terms. To get an idea of cancellation we must first understand what a "factor" is. A factor of a number is a number which will divide it evenly. Thus, 2 is a factor of 8, 3 is a factor of 27, 5 is a factor of 35, etc. When the same number will exactly divide two or more numbers, it is called a common factor of those numbers. Thus, 2 is a common factor of 8 and 12, because it will divide into both 8 and 12 without leaving a remainder. Four (4) is also a common factor of 8 and 12. Similarly, 7 is a common factor of 14 and 21. This idea of common factors we have already used in reducing fractions to lowest terms. Thus, when we have we divide both 8 and 12 by 4 and get

8

12

Cancellation is a process of shortening the work of reduction by removing or canceling the equal factors from the numerator and denominator.

Suppose we wish to multiply by