From a fraction to a percentage

Percentages are a simple topic, yet they are often used with uncertainty. Many books introduce them through a proportion such as a : b = n : 100. The concept of proportionality matters, but understanding percentages does not require a large set of rules about antecedents, consequents, extremes, and means. We can start directly from the fraction.

1. Fractions and ratios

The expressions 1/2, 18/19, and 1/20 are ratios:

• “1 out of 2” means 1 : 2 and is written 1/2;
• “18 out of 19” means 18 : 19 and is written 18/19;
• similarly, “1 out of 20” is written 1/20.

2. When a percentage appears

When a fraction has denominator 100, it is a percentage. There is no mathematical reason why a ratio must preferably be expressed as a percentage. This form is, however, familiar in everyday language and often makes a measure easier to perceive. For example, many people recognise 95 out of 100 faster than the equivalent fraction 19/20.

The percent sign means that the number before it is divided by 100:

1% = 1/100
n% = n/100

3. Denominators that divide 100

To convert 1/2 to a percentage, multiply numerator and denominator by 50. This gives an equivalent fraction:

1/2 = (1 · 50)/(2 · 50) = 50/100 = 50%

In the same way:

1/4   = (1 · 25)/(4 · 25) = 25/100 = 25%
19/20 = (19 · 5)/(20 · 5) = 95/100 = 95%

These conversions are immediate because 2, 4, and 20 divide 100.

4. The general method

What should we do with 5/6, since 6 does not divide 100? Make 100 appear in the denominator by multiplying by 100/100, which equals 1 and does not change the number:

5/6 = (5/6) · (100/100)
    = ((5/6) · 100) · (1/100)
    = ((5/6) · 100)%
    = 83.333...%

From this process we can first say: to convert a number to a percentage, multiply it by 100 and divide it by 100. Because division by 100 is already represented by %, the final rule is:

To convert a number to a percentage, multiply it by 100 and write the % sign.

Some checks:

1/2   = (1/2) · 100%   = 50%
3/4   = (3/4) · 100%   = 75%
19/20 = (19/20) · 100% = 95%

5. What the spreadsheet shows

The original document also checks the rule in Excel. If we enter 1 in a cell, open Format Cells, and select Percentage, the program visually multiplies the number by 100 and adds the percent sign.

1 = 1 · 100% = 100%

6. Conclusion

A percentage is a fraction with denominator 100, written using the % sign. It is nothing other than a fraction.

Prepared by Salvatore Mosaico.