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31. The weights of a number of castings are: 412 lb., 270 lb., 1020 lb.,

75 lb., 68 lb. What is their total weight?

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32. Four studs are required: 2 in., 18 in., 216 in., and 132 in. long; how

3 long a piece of steel will be required from which to cut them allowing in. 4 altogether for cutting off and finishing their ends?

33. Monday morning an engineer bought 48 gallons of cylinder oil; 3 4

on Monday, Tuesday, and Wednesday he used gallon per day; on Thurs

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day he used gallon; and on Friday gallon. How much oil had he left on Saturday?

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34. Find the total length of the roll shown in the sketch in Fig. 3.

35. A piece of work on a lathe is 1 ft. in diameter; it is turned down in

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five cuts; in the first step the tool takes off 32

in. from the diameter; then

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36. How long must a machine shop be to accommodate the following 1 machines installed in a single line: lathe, 8 ft. long; planer, 14 ft. long;

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milling machine, 4 ft. long; engine, 77 ft. long; tool room, 12 ft. long?

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Allow 31 ft. between a wall and a machine, and 3 ft. between two machines. The tool room is to be placed at the end of the shop.

37. In doing a certain piece of work one man puts in 13 hours, a second

1 man hour, a third works 2 hours, and a fourth man works hours. 2

How long would it take one man to do the work?

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38. By mistake, the draftsman omitted the thickness of the flange on the drawing of a gas engine cylinder in Fig. 4. From the other dimensions given, calculate the thickness of the flange.

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39. A millwright has to rig up temporarily a 6 in. belt to be 3675 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 142 in. long, and one 1333 in. long.

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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 hours and 15 minutes lathe work, 3 hours and 10 minutes milling, 1 hour and 10 minutes planing, and 1 hour and 15 minutes bench work; what is the total number of hours to be charged to the job?

CHAPTER III

MULTIPLICATION AND DIVISION OF FRACTIONS

19. A Whole Number Times a Fraction.-In the study of multiplication, we learn that multiplying is only a short way of adding. 4×7 is the same as four 7's added together. Either 4X7, or 7+7+7+7 will give 28. If we apply this same principle to the multiplying of fractions, we see that 4×7 is the same as four of these fractions added together.

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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?

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20. "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 four equal parts and then taking three of these parts. These three parts will be of 5 in., and by actual measurement will be 3 in long, so we know that of 5-33. Now see what times 5 is

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Therefore, we see that the word

"of" in such a case signifies multiplication.

21. 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.

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Explanation: The numerator of the product is obtained from multiplying the numerators together: 7×2=14. The denominator of the product, in the same manner, is 8×3=24. This 7 12

14 24

gives the product which can be reduced to

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=32. Three of these parts will naturally contain three times as many thirty-seconds, or 3. Therefore:

= 32

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22. 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.

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To multiply a mixed number by a whole number, we can reduce the mixed number to an improper fraction and then multiply it; or we can multiply the fractional part and the whole number part separately by the number and then add the products.

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