Fraction‑Decimal‑Percent Converter

Relationships Among Fractions, Decimals, and Percents

Fractions, decimals, and percents are three different ways to describe the same underlying idea: a ratio between two quantities. Being able to move fluently between these formats is essential in school mathematics, everyday budgeting, cooking, statistics, and many other areas. This page explains how the conversions work and what this calculator does for you.

A fraction such as a/b represents the division of the numerator a by the denominator b, with the condition that b ≠ 0. A decimal writes the result of that division in base ten, like 0.75. A percent is a decimal scaled so that the whole is 100, for example 75% means 75 out of 100.

These relationships can be summarized with simple formulas:

• Fraction to decimal: divide numerator by denominator.
• Decimal to percent: multiply by 100 and add the percent sign.
• Percent to decimal: divide by 100 and remove the percent sign.

In algebraic form, if the fraction is a/b and the decimal is d and the percent is p%, then we have:

and, going the other direction, if you start from a percent value p (without the percent sign), then:

From Fraction to Decimal and Percent

To convert a fraction to a decimal, you simply carry out the indicated division. For example, the fraction 3/4 means three divided by four:

3 ÷ 4 = 0.75

To convert this decimal to a percent, multiply by 100:

0.75 × 100 = 75%

So you can write:

3/4 = 0.75 = 75%

For many fractions, such as those with denominators of 2, 4, 5, 8, or 10, the decimal form terminates. Examples include 1/2 = 0.5, 1/4 = 0.25, and 3/5 = 0.6.

Other fractions lead to repeating decimals. A classic example is 1/3:

1 ÷ 3 = 0.3333… (the 3 repeats forever)

When decimals repeat, no finite string of digits can show the exact value. The converter displays a rounded decimal approximation while internally treating the value as a fraction so that the relationship between forms stays mathematically consistent.

From Decimal to Fraction

Converting a decimal to a fraction usually involves placing the digits after the decimal point over the correct power of ten and then simplifying. A systematic manual approach is:

1. Count how many digits appear after the decimal point.
2. Write the number without the decimal point as the numerator.
3. Use 1 followed by as many zeros as there are decimal digits as the denominator.
4. Simplify the fraction by dividing numerator and denominator by their greatest common divisor.

Example with a terminating decimal:

• Start with 0.125.
• There are three digits after the decimal point: 1, 2, and 5.
• Write the numerator as 125 and the denominator as 1000: 125/1000.
• Divide both numerator and denominator by 125, giving 1/8.

The converter automates this process for you. For inputs such as 0.2, it produces the simplified fraction 1/5 instead of a larger but equivalent fraction like 2/10.

For non-terminating decimals, the calculator uses an internal tolerance to find a nearby simple fraction. For instance, a decimal like 0.333333 (entered by a user) will be detected as very close to 1/3, and the result is shown in that simplified fractional form.

From Percent to Decimal and Fraction

A percent always means “out of 100”. To convert a percent to a decimal, divide by 100. To convert a percent to a fraction, place the number over 100 and then simplify.

Example with 75%:

• Percent to decimal: 75% ÷ 100 = 0.75.
• Percent to fraction (first step): 75/100.
• Simplify by dividing numerator and denominator by 25: 3/4.

You again arrive at 3/4 = 0.75 = 75%.

Example with 12.5%:

• Percent to decimal: 12.5% ÷ 100 = 0.125.
• Decimal to fraction: 0.125 = 1/8 (as shown earlier).

The calculator performs these chained conversions automatically. When you select “Percent” as the input type, it first converts to decimal, then to fraction, and finally returns all three forms.

Interpreting the Results

When you use the converter, you will usually see three key outputs for any valid input:

• Fraction form: given as a simplified rational number, typically in the form a/b with no common factors between a and b other than 1.
• Decimal form: a decimal representation, rounded to a practical number of digits for display.
• Percent form: a number out of 100, generally rounded consistently with the decimal display.

Here is how to interpret each form:

• If the fraction has a small denominator (such as 2, 3, 4, 5, 8, or 10), it is often easiest to visualize as slices of a whole or as portions of a set.
• The decimal is well suited for measurement, science, and calculator work; it makes ordering and comparing numbers straightforward, especially when they share the same number of decimal places.
• The percent is usually the most intuitive when dealing with discounts, tax rates, test scores, and probability, because it directly answers “How many out of 100?”

The calculator preserves these relationships so that whatever format you start with, you can quickly see and compare the equivalent values in the other two formats.

Worked Example

Suppose you enter the value 0.375 and mark it as a decimal. The converter will determine the fraction and percent forms as follows.

1. Decimal to fraction:
   • There are three digits after the decimal: 3, 7, and 5.
   • Write 375/1000.
   • The greatest common divisor of 375 and 1000 is 125.
   • Divide numerator and denominator by 125 to get 3/8.
2. Decimal to percent:
   • Multiply by 100: 0.375 × 100 = 37.5.
   • Add the percent sign: 37.5%.

The final combined result is:

3/8 = 0.375 = 37.5%

In practice, you only need to enter 0.375 and specify that it is a decimal. The calculator performs all of these steps instantly and shows the outcomes in each form.

Common Fraction, Decimal, and Percent Equivalents

The table below lists several frequently used numbers and their equivalent forms. This can be helpful for quick reference or to check that you understand the patterns.

Comparison of Formats

Each representation has its own strengths depending on the situation. The table below summarizes some of the typical advantages.

Assumptions, Notes, and Limitations

This converter is designed to be practical and easy to use, but it does make a few assumptions and applies some limits so that results stay readable:

• Input format: The tool expects a single number in one of three forms: a fraction like a/b, a decimal like 0.375, or a percent like 12.5 or 12.5%, depending on how the page is configured. Mixed numbers (such as 1 3/4) may not be accepted unless you rewrite them as improper fractions (for example, 7/4).
• Denominator not zero: Fractions with a denominator of zero are mathematically undefined. If you attempt to enter something like 5/0, the calculator will treat it as invalid and will not produce a result.
• Rounding of decimals and percents: When a fraction corresponds to a repeating decimal, the converter rounds the decimal and percent forms to a limited number of digits. This keeps the output compact but means that the decimal and percent values are approximations, not exact values.
• Size of numbers: Extremely large numerators or denominators, or decimals with very many digits, may be simplified to an approximate fraction in order to keep the output within reasonable size and to avoid overflow in your browser.
• Invalid characters: Non-numeric input (letters, symbols other than the decimal point, slash, or percent sign) will typically be rejected. If you do not see an answer, check that you have entered a valid numeric format.
• Educational use: This tool is meant for learning, checking homework, and general-purpose calculations. It should not be the sole basis for critical engineering, financial, or scientific decisions where exact specifications and professional review are required.

If you are studying related topics, you may also find tools such as a percent change calculator, a ratio simplifier, or a general percentage calculator useful. Linking ideas between these tools can help strengthen your understanding of proportional reasoning.