Three ways to write the same thing
A fraction, a decimal and a percentage all describe a part of a whole. Three quarters is 3/4, 0.75 and 75 percent. Each form has a use: fractions are exact and easy to combine, decimals are easy to compare and calculate with, and percentages make sizes and changes easy to read.
Fluency means moving between them without thinking.
From a fraction to a decimal and a percentage
Divide the top number by the bottom to get the decimal, then multiply by 100 for the percentage. Five eighths is 5 divided by 8, which is 0.625, or 62.5 percent. Some fractions produce decimals that repeat forever, such as one third, which is 0.333 repeating.
Rounding is then a choice, and the exact value stays a fraction.
From a decimal back to a fraction
Write the digits over a power of ten and simplify. 0.6875 is 6875 over 10,000, which simplifies to 11/16. For a decimal that looks like a repeating pattern, an approximation with a limit on the denominator is often the useful answer.
A ruler reads sixteenths and a drill index reads sixty-fourths, so the limit should match the tool you are using.
Add after converting to a common denominator
For 3/4 + 5/6, use denominator 12: 9/12 + 10/12 = 19/12. Adding numerators and denominators separately gives 8/10, which is not the sum. This example assumes positive fractions; comparisons using a sum’s size need the signs specified.
Division uses the reciprocal
For (3/4) ÷ (5/6), multiply 3/4 by 6/5 to obtain 18/20 = 9/10. The divisor must be nonzero. A terminating decimal can be written exactly as a fraction using its stated digits; a rounded display of one-third is not exact one-third.
Keep the exact fraction through intermediate arithmetic when rounding would affect the final comparison.
Mixed numbers and improper fractions
A mixed number, such as 3 and 2/5, folds a whole number and a fraction together. To use it in arithmetic, convert it to an improper fraction by multiplying the whole number by the denominator and adding the numerator, giving 17/5. Convert back at the end if the answer should read naturally, such as in a recipe or a cut list.
Percent change and percent difference
Percent change measures the move from an old value to a new one, dividing the gap by the old value. From 80 to 100 is a 25 percent rise. Percent difference compares two values when neither is the starting point, dividing the gap by their average.
The same two numbers give 22.2 percent. Use the one that fits the question, and say which you used.
Proportions and scaling
A proportion says two ratios are equal, and cross-multiplying finds the missing part. If 3 parts of concentrate go with 4 of water, then 15 parts go with 20. The same step scales a recipe, a map, a photo or a dose.
A proportion equates two ratios with corresponding quantities in the numerators and denominators.
Rounding and the last check
Round at the end, not in the middle, because early rounding accumulates. After any calculation, ask whether the size is reasonable: three quarters plus five sixths must be a little under two. Estimating first catches most slips before they reach a final answer.
Assumed inputs and calculated outputs
Fraction Calculator: input sensitivity
| First numerator | Simplified numerator | Simplified denominator | Decimal value |
|---|---|---|---|
| 1 | 13 | 12 | 1.1 |
| 2 | 4 | 3 | 1.3 |
| 3 | 19 | 12 | 1.6 |
| 4 | 11 | 6 | 1.8 |
| 5 | 25 | 12 | 2.1 |
What goes wrong
Fractions, decimals and percentages explained: common errors
| The mistake | Why it misleads | Do this instead |
|---|---|---|
| Adding the tops and the bottoms of two fractions | The result is smaller than one of the inputs, which is impossible for a sum of positive fractions. | Find a common denominator, add the tops and keep the denominator. |
| Using percent difference for a before-and-after change | It treats both values as equals, so it understates a rise and overstates a fall. | Use percent change from the starting value. |
| Reading 0.45 hours as 45 minutes | A decimal of an hour is a fraction of 60 minutes, so 0.45 hours is 27 minutes. | Multiply the decimal part by 60 to get minutes. |
| Rounding partway through a calculation | Small early rounding errors grow when they are multiplied later. | Keep full precision until the last step, then round. |
Fractions, decimals and percentages explained: worked calculation
Using the Fraction Calculator worked example, moving First numerator from 1 to 5 changes the Simplified numerator from 4 to 25.
| Input | Value | Definition |
|---|---|---|
| First numerator | 3 | The top number of the first fraction. |
| First denominator | 4 | The bottom number of the first fraction. It cannot be zero. |
| Operation | Add (+) | Choose add, subtract, multiply or divide. |
| Second numerator | 5 | The top number of the second fraction. |
| Second denominator | 6 | The bottom number of the second fraction. It cannot be zero. |
| Simplified numerator | 19 |
|---|---|
| Simplified denominator | 12 |
| Decimal value | 1.6 |
| First numerator | Simplified numerator | Simplified denominator | Decimal value |
|---|---|---|---|
| 1 | 13 | 12 | 1.1 |
| 2 | 4 | 3 | 1.3 |
| 3 | 19 | 12 | 1.6 |
| 4 | 11 | 6 | 1.8 |
| 5 | 25 | 12 | 2.1 |
Which input moves the simplified numerator most in the fraction calculator?
Ranked by how far each moves the simplified numerator across the range tested: operation (10, 105%), second denominator (9, 47%), first denominator (32, 168%) and first numerator (7, 37%).
How much does first denominator matter in the fraction calculator?
The worked example uses 4. With the other inputs left at the worked example, moving first denominator from 3 to 5 takes the simplified numerator from 11 to 43, a swing of 168% of the worked-example figure.
Fractions, decimals and percentages explained: points to carry forward
- A fraction, a decimal and a percentage are three names for one value.
- Add fractions only over a common denominator, then simplify.
- Percent change uses the old value as the base, and percent difference uses the average.
- A proportion is solved by cross-multiplying, with matching quantities on top.
- Round at the end and estimate first to catch errors.
How do I convert a fraction to a percentage?
Divide the top by the bottom and multiply by 100. Five eighths is 0.625, or 62.5 percent.
How do I add fractions with different denominators?
Rewrite both with a common denominator, add the tops and simplify. The fraction calculator does the steps and reduces the answer.
How do I turn a decimal into a fraction?
Write it over a power of ten and simplify, or use a denominator limit to get the closest fraction for a ruler or a drill size.
What is the difference between percent change and percent difference?
Percent change divides by the starting value. Percent difference divides by the average of the two values and is used when neither is the start.
How do I convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, add the numerator and keep the denominator. 3 and 2/5 becomes 17/5.
How do I solve a proportion?
Cross-multiply. For a/b = c/x, x equals b times c divided by a. The rearrangement assumes the relevant denominator is nonzero.
Check the original equality after substituting the solved value into both ratios.
Why is my percentage answer over 100?
A part can exceed the whole when you compare an increase, such as a value that doubled, which is a 100 percent rise and 200 percent of the original.
How do I find a percentage of a number?
Multiply the number by the percentage divided by 100. Thirty percent of 250 is 250 times 0.30, or 75. The percentage calculator handles the reverse questions too.
How do I convert 3/8 to a decimal?
Divide 3 by 8, which is 0.375. Every fraction is a division, and the decimal either ends or repeats.
What does it mean when a decimal repeats?
The fraction has a denominator with factors other than 2 and 5, so the division never ends. One third is 0.333 repeating, and the exact value is best kept as a fraction.