Math · Numbers, fractions & decimals
Fractions, Decimals, and Percents
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Start here: key ideas
- Fraction to decimal means numerator divided by denominator.
- Decimal to percent moves the decimal two places right.
- Percent to decimal moves it two places left.
- Write percent over 100, then simplify.
What you’ll be able to do
- Convert among fractions, decimals, and percents.
- Recall benchmark equivalents.
- Choose a useful representation for comparison.
Treat fraction, decimal, and percent as three names for one quantity.
Optional review
If you’d like to review the basics, visit Fractions: Operations and Mixed Numbers, Decimals and Place Value Operations. You can start this lesson without completing those first.
One amount, three names
A percent is a ratio per hundred. The statements 3/4, 0.75, and 75% are equivalent: they name the same point on the number line. A benchmark is a familiar equivalent used for quick comparison: 1/2=0.5=50%, 1/4=0.25=25%, and 3/4=0.75=75%.
Percent means divide by 100
12% means 12 out of 100. Dividing 12 by 100 gives 0.12. Think about the hundredths meaning rather than memorizing a direction alone.
Percent means divide by 100 — Denny Burzynski and Wade Ellis, Jr. / OpenStax CNX, via Mathematics LibreTexts. CC BY. Source image reproduced unchanged.
Convert in every direction
To change a fraction to a decimal, divide numerator by denominator. A terminating decimal ends, as 3/8=0.375. A repeating decimal continues a pattern, as 2/3=0.666…. To change a decimal to a percent, multiply by 100 and write %. Thus 0.42=42%. To change a percent to a decimal, divide by 100: 7%=0.07.
To make a fraction from a decimal, read its place value: 0.36=36/100=9/25. To make a fraction from a percent, place it over 100 and simplify: 35%=35/100=7/20. A percent can exceed 100%; 125%=1.25=5/4.
Choose the easiest form
Convert all values to the same form before comparing. For 5/8 and 60%, either compute 5÷8=0.625 and compare with 0.60, or recognize 60%=3/5. Keep exact values until the end; early rounding can reverse close comparisons.
Scroll the table sideways if needed.
| Fraction | Decimal | Percent |
|---|---|---|
| 1/4 | 0.25 | 25% |
| 1/2 | 0.50 | 50% |
| 3/4 | 0.75 | 75% |
Worked examples
Worked example 1: Convert 7/20. Divide 7÷20=0.35; multiply by 100 to obtain 35%.
Worked example 2: Order 0.58, 3/5, and 57%. Convert to decimals: 0.58, 0.60, 0.57. Therefore 57%<0.58<3/5.
A reliable way to reason
Conversion changes notation, not quantity. Picture one point on a number line wearing three labels. This prevents a common error: believing 0.4, 4%, and 4/10 are equivalent. In fact, 0.4=40%=4/10, while 4%=0.04=1/25. The percent sign supplies a hidden division by 100.
Use the form that exposes the comparison. To compare 7/12 and 58%, convert 7/12 to about 0.5833 and 58% to 0.58; the fraction is slightly larger. To calculate 25% of a quantity, the fraction 1/4 may be fastest. To show part of 100, percent is clearest. For a repeating result, preserve exactness when possible: 2/3 is exact, while 0.67 is a rounded approximation. If rounding is required, convert first, then round at the requested place. A final benchmark check protects scale: values below 1 correspond to percents below 100%; values above 1 correspond to percents above 100%. Thus 1.08 must be 108%, not 10.8%.
Guided reasoning and error checks
Check a conversion by reversing it: 62.5%=5/8 because 5÷8=0.625. For close values, carry enough digits. Comparing 5/6 with 83% requires 0.8333… versus 0.83; rounding too early would hide that the fraction is larger. Exact fractions are often the safest intermediate form.
Final self-check
Before finalizing a conversion, ask whether the scale is believable. A positive fraction smaller than 1 must become a decimal between 0 and 1 and a percent between 0% and 100%. For example, 11/20 becomes 0.55 and 55%. If a computation produced 5.5% or 550%, the digits may look familiar, but the scale test proves the decimal moved the wrong number of places.
Terms to remember
- percent
- A ratio measured per hundred.
- benchmark
- A familiar value used for comparison.
- equivalent
- Equal in value despite a different form.
- terminating decimal
- A decimal with finitely many digits.
- repeating decimal
- A decimal with a recurring digit pattern.
Your quick summary
- Use division for fraction-to-decimal conversion.
- Attach or remove the percent scale by factors of 100.
- Convert all quantities to one form before comparing.
- Benchmark equivalents such as 1/2 = 0.5 = 50% make conversions and comparisons easier to check.
Lesson quiz
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