CHAPTER VI PERCENTAGE 6 100 47. Explanation.-Percentage is merely a kind of fraction or, rather, a particular kind of decimal fraction, 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 18 or .06. These all mean the same thing; namely, 6 parts out of 100. 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 cts. an hour and gets an increase of 10%, this increase will be 10% (or 10% or .10) of 40 cts. and this is easily seen to be 4 cts., so his new rate is 44 cts. 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 X 40 44 cts., the new rate. = To find a certain per cent of anything, we multiply exactly as we would in finding a fractional part or decimal part of anything. The only difference is that before multiplying we must first change the per cent to its corresponding fraction or decimal. Unless the number of per cent can be reduced to a simple fraction like,,, it is generally easier and takes less time to convert it to a decimal instead of a fraction before making the calculation. For example, in figuring the interest on $1200 at the rate of 31%, we would first change 31% to .035 and then multiply $1200 by .035, which gives $42.00. $1200 .035 6000 3600 $42.000, Answer. Any decimal fraction may be easily changed to per cent. Here we first change the 1000 for a denominator. whose denominator is 100. decimal to a common fraction having Next we reduce this fraction to one 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 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% A common fraction is reduced to per cent by first changing it to a decimal, and then changing the decimal to per cent by moving the decimal point 2 places to the right. We reduce to per cent as follows: .25 25%. Thus, of anything is the same as 25% of it, because 25% .25%%. From this = = = = = it is clearly evident that fractions, decimals, and per cent are all one and the same thing but expressed differently. Thus: 25 100 25 100 = .25 (the decimal point indicates the denominator) = 25%(the % sign indicates the denominator) The following table gives a number of different per cents with the corresponding decimal fractions and common fractions: 48. 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 in terms of per cent, we would say "40% of those castings are bad," because "two out of five" is the same as 2 40 100 5 = 40%. In the second case, we would say, "90 per 90 cent of those cutters should be replaced." Here, “nine out of ten" = 10 = = 100 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 60% of the whole. 00 50 If a shop is running with 50% of its full force, it means that 10% or of the full force is working. If the full force of men is 1300, then the present force is 50% of 1300 = .50 X 1300 = 650. If the full force were 700 men, then the 50% would be 350. Suppose again a shop has 50 men working, which the foreman states is only 25% of his normal crew. If 50 is 25%, then 1% of his normal crew is 50 ÷ 25 2 men, and 100%, or the normal crew, is 100 X 2 = 200 men. This is readily checked by the fact that 50 is of 200, and 25%. = = 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. .65 X 8 = 5.20 lb., or 5 lb., the copper needed .35 X 8 = 2.80 lb., or 21% lb., the zinc needed = Sometimes, in dealing with very small per cents, we find a decimal per cent such as used 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 if we write this .04% as a decimal, it will be .0004. common mistake to misunderstand these decimal per the student should be very careful in reading them. Likewise, he should be careful in changing a decimal into per cent that the decimal point is moved two places to the right. 10,000, SO It is a very cents, and |