Meaning of percentage and fraction–decimal–percentage conversion

Learning outcome

Interpret a percentage as a comparison with a reference quantity. Convert fractions, decimals and percentages while distinguishing exact values from rounded approximations.

Meaning and conversion

“Per cent” means “per hundred”: 1% = 1/100. A percentage expresses a comparison on a common scale of 100, even when the actual collection does not contain 100 objects. It describes relative size rather than an independent amount.

The reference quantity is the amount treated as the whole. It must be nonzero; throughout these lessons, reference quantities are positive. Without knowing the reference, a percentage alone cannot tell us a mass, price or count.

For any positive number p, p% = p/100. Consequently, convert a fraction or decimal into a percentage by multiplying its numerical value by 100 and attaching %. Convert back by dividing the percentage number by 100. These operations preserve the underlying value; they merely change its notation.

For instance, 0.35 = 35%, whereas 0.35% = 0.0035. The percent sign already means division by 100. A percentage may exceed 100% when the compared quantity exceeds the reference; percentages are not always shares within a fixed collection.

Reduce percentage fractions using common factors. Keep recurring decimals exact as fractions when possible. Use ≈, not =, when replacing an exact value with a rounded decimal.

Worked example 1

Convert 7/20 into a decimal and a percentage.

Multiply numerator and denominator by 5: 7/20 = 35/100. Therefore its decimal form is 0.35 and its percentage form is 35%. The denominator has become 100 without changing the fraction because both terms received the same multiplier.

Worked example 2

Convert 0.045 into a percentage, then into a reduced fraction.

The percentage number is 0.045 × 100 = 4.5, giving 4.5%. As a fraction, 4.5% = 4.5/100 = 45/1000 = 9/200. Converting back gives the same original decimal.

Worked example 3

Express 5/6 as an exact percentage and approximately to two decimal places.

Multiplying by 100 gives the exact percentage (250/3)% = 83⅓%. Its decimal percentage is recurring, so write 83⅓% ≈ 83.33%. The rounded value is useful for reporting but is not exactly equal to the original fraction.

Common traps

Do not attach % without changing the numerical value. Do not confuse a decimal with an identically written percentage. A rounded recurring percentage should not silently become an exact input in later calculations.

Practice questions

1. Convert 11/25 into a decimal and a percentage.
2. Convert 0.0075 into a percentage and a reduced fraction.
3. Convert 132% into a decimal and a reduced fraction.
4. Express 7/12 as an exact percentage and approximately to two decimal places.

Worked answers

1. Multiply both fraction terms by 4: 11/25 = 44/100. Thus the decimal is 0.44 and the percentage is 44%; all three forms have the same value.
2. Multiplying 0.0075 by 100 gives 0.75%. The fraction is 75/10000 = 3/400, obtained by dividing both terms by 25.
3. Divide by 100: 132% = 132/100 = 1.32. Reducing the fraction by 4 gives 33/25; being greater than the whole is valid here.
4. Calculate (7/12) × 100 = 175/3. Thus the exact percentage is (175/3)% = 58⅓%, and its two-decimal approximation is 58.33%.