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88. A 28-tooth 7-pitch gear has an outside diameter of 4.286 in. The diameter at the bottom of the teeth is 3.67 in. How deep are the teeth cut?

89. A 2 in. pipe has an actual inside diameter of 2.067 in. The metal of the pipe is .154 in. thick. What is the outside diameter of the pipe? 90. Read the micrometer shown below in Fig. 7.

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44. Explanation.-Percentage is merely another kind of fractions or, rather, a particular kind of decimal fractions, of which the denominator is always 100. Instead of writing the denominator, we use the term "per cent" to indicate that the denominator is 100. When we speak of "6 per cent" we mean Tor.06. These all mean the same thing; namely, six parts out of one hundred. Instead of writing out the words "per cent" we more often use the sign % after the number, as, for instance, 6%, which means "6 per cent." Since per cent means hundredths of a thing, then the whole of anything is 100% of itself, meaning 188, or the whole. If a man is getting 40 cents an hour and gets an increase of 10%, this increase will be 10% (or .10) of 40 cents and this is easily seen to be 4 cents, so his new rate is 44 cents. Another way of working this would be to say that his old rate is 100% of itself and his increase is 10% of the old rate, so that altogether he is to get 110% of the old rate. Now 110% is the same as 1.10 and 1.10×40=44 cents, the new rate.

or

Any decimal fraction may be easily changed to per cent.

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Here we first change the decimal to a common fraction having 100 for a denominator. Then we drop this denominator and use, instead, the per cent sign (%) written after the numerator. This sign indicates, in this case, 87.5 parts out of 100, or

87.5
100

The change from a decimal to percentage can be made without changing to a common fraction as was just done. Having a decimal, move the decimal point two places to the right and write per cent after the new number.

.625=62.5% .06=6%

1.10=110%

If it is desired to use a certain number of per cent in calculations, it is usually expressed as a decimal first and then the calculations are made. For example, when figuring the interest on $1250 at the rate of 6%, we would first change 6% to .06 and multiply $1250 by .06 which gives $75.00.

$1250 .06 $75.00

A common fraction is reduced to per cent by first reducing it to a decimal and then changing the decimal to per cent.

Example:

The force in a shop is cut down from 85 men to 62. What per cent of the original number of men are retained?

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Therefore, the number of men retained is 72.9% or nearly 73% of the original number of men.

If we want to reduce the fraction to per cent, we first get .125 and then, changing this decimal to per cent, we have .125 12.5%. Then of anything is the same as 12% of it, because 121%=125=}.

100

The following table gives a number of different per cents with the corresponding decimals and common fractions:

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45. The Uses of Percentage. In shop work, the chief use of percentage is to express loss or gain in certain quantities or to state portions or quantities that are used or unused, good or bad, finished or unfinished, etc. Very often we hear expressions like: "two out of five of those castings are bad;" or "nine out of ten of those cutters should be replaced." If, in the first illustration, we wanted to talk on the basis of a hundred castings instead of five, we would say "40 per cent of those castings are bad," because “two out of five" is the same as 3,10,=40%. And in the second case: "90 per cent of those cutters should be replaced." Here, "nine out of ten" == 10%, 90%. If a piece of work is said to be 60% completed, it means that, if we divide the whole work on the job into 100 equal parts, we have already done 60 of these parts or 0% of the whole.

100

60

100

90

100'

If a shop is running with 50% of its full force, it means that 50% or of the full force is working. If the full force of men is 1300, then the present force is 50% of 1300.50X1300=650. If the full force were 700 men, then the 50% would be 350.

Another very common use of percentage is in stating the portions or quantities of the ingredients going to make up a whole. We often see formulas for brasses, bronzes, and other alloys in

which the proportions of the different metals used are indicated by per cents. For example, brass usually contains about 65% copper and 35% zinc. Then, in 100 lb. of brass, there would be 65 lb. of copper and 35 lb. of zinc. Suppose, however, that instead of 100 lb. we wanted to mix a smaller amount, say 8 lb. The amount of copper needed would be 65% or .65 of 8 lb.

.65x8 5.20 lb., or 5 lb., the copper needed.

8

.35X8=2.80 lb., or 2 lb., the zinc needed.

10

Sometimes, in dealing with very small per cents, we see a decimal per cent such as found in the specifications for boiler steel, where it is stated that the sulphur in the steel shall not exceed .04%. Now this is not 4%; neither is it .04; but it is .04%, meaning four one-hundredths per cent, or four onehundredths of one one-hundredth. This is 10 of 100=10000, So if we write this .04% as a decimal, it will be .0004. It is a very common mistake to misunderstand these decimal per cents, and the student should be very careful in reading them. Likewise, be careful in changing a decimal into per cent that the decimal point is shifted two places to the right.

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46. Efficiencies.-Another common use of percentage is in stating the efficiencies of engines or machinery. The efficiency of a machine is that part of the power supplied to it, that the machine delivers up. This is generally stated in per cent, meaning so many out of each hundred units. If it requires 100 horse-power to drive a dynamo and the dynamo only generates 92 horse-power of electricity, then the efficiency of the dynamo is 10% or 92%. If the engine driving a machine shop delivers 250 horse-power to the lineshaft, but the lineshaft only delivers 200 horse-power to the machines, then the efficiency of the lineshaft is 288.80 =80%. The other 50 horse-power, or 20%, is lost in the friction of the shaft in its bearings and in the slipping of the belts. The efficiencies of all machinery should be kept as high as possible because the difference between 100% and the efficiency means money lost. The large amount of power that is often lost in line shafting can be readily appreciated when we try to turn a shaft by hand and try to imagine the power that would be required to turn it two or three hundred times a minute.

47. Discount.-In selling bolts, screws, rivets, and a great many other similar articles, the manufacturers have a standard list of prices for the different sizes and lengths and they give their

customers discounts from these list prices. These discounts or reductions in price are always given in per cent. Sometimes they are very complicated, containing several per cents to be deducted one after another. Each discount, in such a case, is figured on the basis of what is left after the preceding per cents have been deducted.

Example:

The list price of in. by 1 in. stove bolts is $1 per hundred. If a firm gets a quotation of 75, 10 and 10% discount from list price, what would they pay for the bolts per hundred?

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Explanation: 75, 10 and 10% discount means 75% deducted from the list price, then 10% deducted from that remainder, then 10% taken from the second remainder.

Starting with 100 cents, the list price, we deduct the first discount of 75%. This leaves 25 cents. The next discount of 10% means 10% off from this balance. Deducting this leaves 223 cents. Next, we take 10% from this, leaving 20 cents per hundred as the actual cost of these stove bolts.

48. Classes of Problems.-Nearly all problems in percentage can be divided into three classes on the same basis as explained in Article 26. There are three items in almost any percentage problem: namely, the whole, the part, and the per cent. For example, suppose we have a question like this: "If 35% of the belts in a shop are worn out and need replacing, and there are 220 belts altogether, how many belts are worn out?" In this case, the whole is the number of belts in the shop, 220. The part is the number of belts to be replaced, which is the number to be calculated. The per cent is given as 35%.

Any two of these items may be given and we can calculate the missing one. We thus have the three cases:

1. Given the whole and the per cent, to find the part.

2. Given the part and the per cent that it is of the whole, to find the whole.

3. Given the whole and the part, to find what per cent the part is of the whole.

The principles taught under common fractions will apply equally well in working problems under these cases, the only difference being that here per cent is used instead of a common fraction. In working problems, the per cent should always be changed to a decimal.

One difficulty in working percentage problems is in deciding

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