Percentage and Its Business Applications
The expression “per cent,” which is an abbreviation of the Latin words “per centum,” means “for each hundred.”
The symbol % is often used to denote “per cent.” Thus, 7 per cent, or 7%, means 7 parts out of every 100 parts, i.e., 7⁄100 of the whole.
Since per cent means hundredths, we may write any fraction whose denominator is 100 as so many per cent. In some cases the corresponding common fractions are so simple that it is advisable to remember them. For example:
| 25% | = | 25100 | = | 14, | 50% | = | 50100 | = | 12, |
| 75% | = | 75100 | = | 34, | 331⁄3% | = | 331⁄3100 | = | 13, |
| 662⁄3% | = | 662⁄3100 | = | 23, | 5% | = | 5100 | = | 120, |
| 21⁄2% | = | 21⁄2100 | = | 140, | 121⁄2% | = | 121⁄2100 | = | 18, |
and so on.
The number per cent is called the rate per cent.
Table of Additional Values
| Symbol | Decimal | Common Fraction | |||||
|---|---|---|---|---|---|---|---|
| 1% | = | .01 | = | 1⁄100 | |||
| 2% | = | .02 | = | 2⁄100 | = | 1⁄50 | |
| 3% | = | .03 | = | 3⁄100 | |||
| 4% | = | .04 | = | 4⁄100 | = | 1⁄25 | |
| 5% | = | .05 | = | 5⁄100 | = | 1⁄20 | |
| 6% | = | .06 | = | 6⁄100 | = | 3⁄50 | |
| 7% | = | .07 | = | 7⁄100 | |||
| 8% | = | .08 | = | 8⁄100 | = | 2⁄25 | |
| 9% | = | .09 | = | 9⁄100 | |||
| 10% | = | .10 | = | 10⁄100 | = | 1⁄10 | |
| 20% | = | .20 | = | 20⁄100 | = | 1⁄5 | |
| 25% | = | .25 | = | 25⁄100 | = | 1⁄4 | |
| 50% | = | .50 | = | 50⁄100 | = | 1⁄2 | |
| 100% | = | 1 | .00 | = | 100⁄100 | = | 1 |
Here are a few others that should be learned:—
| 61⁄4% | = | 1⁄16 of 100% | 162⁄3% | = | 1⁄6 of 100% |
| 81⁄2% | = | 1⁄12 of 100% | 331⁄3% | = | 1⁄3 of 100% |
| 121⁄2% | = | 1⁄8 of 100% | 662⁄3% | = | 2⁄3 of 100% |
A Decimal as Per Cent
Write the decimal as hundredths, and the number expressing the number of hundredths is the per cent.
Examples:
| .4 | = | .40 | = | 40100 | = | 40% |
| .8 | = | .80 | = | 80100 | = | 80% |
| .25 | = | 25100 | = | 25% | |
| .33 | 1⁄3 | = | 331⁄3100 | = | 331⁄3% |
| .50 | = | 50100 | = | 50% | |
| .87 | 1⁄2 | = | 871⁄2100 | = | 871⁄2% |
If the decimal has more than two decimal places, the figures after the second one are written as a fraction of a per cent, as,—
| .255 | = | 251⁄2100 | = | 251⁄2%. |
| .163 | = | 163⁄10100 | = | 163⁄10%. |
To change a common fraction to per cent:
1. Change the fraction to a decimal.
2. Express the decimal as hundredths.
3. The result is the per cent desired.
Examples:
| 1⁄2 | = | .5 | = | .50 | = | 50% |
| 3⁄4 | = | .75 | = | 75% | ||
| 2⁄3 | = | .662⁄3 | = | 662⁄3% | ||
| 9⁄10 | = | .90 | = | 90% | ||
| 8⁄9 | = | .888⁄9 | = | 882⁄9% | ||
| 7⁄8 | = | .871⁄2 | = | 871⁄2% | ||
| 25⁄26 | = | .962⁄13 | = | 962⁄13% | ||
Or, they may be written this way:
| 3⁄4 | = | 3⁄4 of | 100100 | = | 75100 | = | 75% |
| 2⁄3 | = | 2⁄3 of | 100100 | = | 662⁄3100 | = | 662⁄3% |
| 1⁄2 | = | 1⁄2 of | 100100 | = | 50100 | = | 50% |
Terms Used in Percentage
In Percentage, there are five terms or quantities considered; namely, the Base, Rate per cent, Percentage, Amount and Proceeds or Difference; any two being given, a third one may be found.
The base and rate given, to find the percentage.
Rule.—Multiply the base by the rate per cent expressed decimally.
Example: How many dollars is 6% of $50?
| $50, | the Base, or number on which percentage is computed. |
| .06, | the Rate, or term denoting number of hundredths taken. |
| $3.00, | the Percentage, or the product of the base and rate per cent. |
| $53.00, | the Amount, or the base increased by the percentage. |
| $47.00, | the Proceeds, or Difference, the base less the percentage. |
| Ans. $3.00. | |
When the rate per cent is an aliquot part of 100, the percentage is readily found by taking such a part of the base as the rate per cent is part of 100. Thus, at 10%, take 1⁄10 of base; at 121⁄2%, 1⁄8; at 162⁄3%, 1⁄6, etc.
The base and percentage given, to find the rate.
Rule.—Divide the percentage by 1% of the base
Example: Bought a watch for $15 and sold it for $18; what per cent did I make?
.15 ) 3.00
20
Ans. 20%
Here, $15.00 is the base, and ($18 - $15) $3.00, the gain or percentage. Now, as 1% of 15.00 is.15, it is evident that 3.00 is as many per cent of 15.00, as.15 is contained times is 3.00, which is 20.
Proof: 20% or 1⁄5 of $15 = $3.
The percentage and rate given, to find the base.
Rule.—Divide the percentage by the rate per cent expressed decimally.
Example: Received $6.40, percentage or interest, for money loaned at 4%, what was the base or principal?
.04 ) 6.40
Ans. $160
If $1 produces.04 (4 cents) in a certain time, $6.40 must be the percentage of as many dollars as.04 is contained times in $6.40, which is 160.
Proof: 4% of $160 (160 ×.04) = $6.40.
The amount and rate given, to find the base.
Rule.—Divide the given amount by 1.00 plus the rate per cent.
Example: Bought a horse at a certain price, and sold him for $84, making 12% on cost; what did he cost?
1.12 ) 84.00
Ans. $75
If I made 12% on cost, every dollar invested gained 12 cents; hence, the horse cost as many dollars as 1.12 is contained times in 84.00, which is 75.
Proof: 12% of $75 (75 ×.12) = $9; $75 + $9 = $84.
The proceeds and the rate given, to find the base.
Rule.—Divide the given proceeds by 1.00 minus the rate per cent.
Example: Sold a wagon for $51, which is 40% less than it cost; what did it cost?
.60 ) 51.00
Ans. $85
If I lost 40%, or 40 cents on the dollar, I received only 60 cents for every dollar the wagon cost; hence, it cost as many dollars as.60 is contained times in 51.00, which is 85.
Proof: 40% of $85 (85 ×.40) = $34; $85 - $34 = $51.
Note.—The principles of percentage, in one form or another, enter into nearly all commercial calculations, besides many others. It is therefore of the utmost importance to business men, clerks, accountants, bookkeepers, and others, to become expert in percentage, and to adopt the easiest, simplest and shortest methods in computing interest, partial payments, trade discount, profit and loss, commission, insurance, stocks, bonds, taxes, exchange, etc.