First-degree equation involving percentages

In mathematics, a percentage (from Latin per centum "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number (pure number); it has no unit of measurement.

Calculations

The percent value is computed by multiplying the numeric value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, one first computes the ratio ${\frac {50}{1250}}$ = 0.04, and then multiplies by 100 to obtain 4%. The percent value can also be found by multiplying first instead of later, so in this example, the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1250 to give 4%.

To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them. For example, 50% of 40% is: ${\frac {50}{100}}\times {\frac {40}{100}}$ = 0.50 x 0.40 = 0.20 = ${\frac {20}{100}}$ = 20%

It is not correct to divide by 100 and use the percent sign at the same time; it would literally imply division by 10,000. For example, 25% = ${\frac {25}{100}}$ = 0.25, not ${\frac {25\%}{100}}$, which actually is ${\frac {25/100}{100}}$ = 0.0025. A term such as ${\frac {100}{100}}$ % would also be incorrect, since it would be read as 1 percent, even if the intent was to say 100%.

Whenever communicating about a percentage, it is important to specify what it is relative to (i.e., what is the total that corresponds to 100%). The following problem illustrates this point.

In a certain college 60% of all students are female, and 10% of all students are computer science majors. If 5% of female students are computer science majors, what percentage of computer science majors are female?

We are asked to compute the ratio of female computer science majors to all computer science majors. We know that 60% of all students are female, and among these 5% are computer science majors, so we conclude that ${\frac {60}{100}}\times {\frac {5}{100}}={\frac {3}{100}}$ or 3% of all students are female computer science majors. Dividing this by the 10% of all students that are computer science majors, we arrive at the answer: ${\frac {3\%}{100\%}}={\frac {30}{100}}$ or 30% of all computer science majors are female.

This example is closely related to the concept of conditional probability.

Percentage increase and decrease

Due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the initial value of that quantity. For example, if an item is initially priced at $200 and the price rises 10% (an increase of $20), the new price will be $220. Note that this final price is 110% of the initial price (100% + 10% = 110%).

Some other examples of percent changes:

• An increase of 100% in a quantity means that the final amount is 200% of the initial amount (100% of initial + 100% of increase = 200% of initial). In other words, the quantity has doubled.
• An increase of 800% means the final amount is 9 times the original (100% + 800% = 900% = 9 times as large).
• A decrease of 60% means the final amount is 40% of the original (100% – 60% = 40%).
• A decrease of 100% means the final amount is zero (100% – 100% = 0%).

In general, a change of x percent in a quantity results in a final amount that is 100 + x percent of the original amount (equivalently, (1 + 0.01x) times the original amount).

Resources

• Solving Percent Problems, Khan Academy
• Taking Percentages, Khan Academy
• Part, Whole, & Percent Proportion Word Problems, The Organic Chemistry Tutor

Licensing

Content obtained and/or adapted from:

• Percentage, Wikipedia under a CC BY-SA license