How to Convert a Decimal to a Fraction? (Terminating Decimals)

Converting a terminating decimal (a decimal that ends) to a fraction involves these steps:

1. Write down the decimal as the numerator of the fraction (without the decimal point).
2. The denominator is a power of 10. The number of zeros in the denominator is equal to the number of digits after the decimal point in the original decimal.
   • If there's 1 digit after the decimal, the denominator is 10.
   • If there are 2 digits after the decimal, the denominator is 100.
   • If there are 3 digits after the decimal, the denominator is 1000, and so on.
3. Simplify the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD).

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"How to convert decimal to mixed fraction" is shown in Example 3 above.

How to Convert a Repeating Decimal to a Fraction? (Infinite Decimals)

Converting a repeating (or recurring) decimal to a fraction requires a bit more algebra:

For a simple repeating decimal (e.g., 0.333... or 0.777...):

1. Let x equal the repeating decimal.
2. Multiply x by a power of 10 so that the repeating block of digits shifts to the left of the decimal point. If one digit repeats, multiply by 10. If two digits repeat, multiply by 100, etc.
3. Subtract the original equation (x = ...) from the new equation. This should eliminate the repeating decimal part.
4. Solve for x and simplify the resulting fraction.

For a mixed repeating decimal (e.g., 0.1232323... where '1' doesn't repeat but '23' does):

1. Let x equal the repeating decimal.
2. Multiply x by a power of 10 to move the non-repeating part to the left of the decimal point.
3. Multiply x by another (larger) power of 10 to move one full repeating block to the left of the decimal point.
4. Subtract the smaller equation from the larger equation.
5. Solve for x and simplify.

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How to Convert a Fraction to a Decimal?

To convert a fraction to a decimal, you simply divide the numerator by the denominator.

1. Perform long division, with the numerator inside the division symbol (the dividend) and the denominator outside (the divisor).
2. Add a decimal point and zeros to the numerator as needed to continue the division until it either terminates (ends with a remainder of 0) or a repeating pattern of digits emerges.

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How to Convert a Mixed Fraction to a Decimal?

There are two main ways:

1. Convert the fractional part to a decimal and add it to the whole number.
   • Example: Convert 2 and 3/4 to a decimal.
     • Convert 3/4 to a decimal: 3 ÷ 4 = 0.75.
     • Add the whole number: 2 + 0.75 = 2.75.
     • So, 2 and 3/4 = 2.75.
2. Convert the mixed fraction to an improper fraction first, then divide.
   • Example: Convert 2 and 3/4 to an improper fraction: (2 × 4 + 3) / 4 = (8 + 3) / 4 = 11/4.
   • Convert the improper fraction to a decimal: 11 ÷ 4 = 2.75.
   • So, 2 and 3/4 = 2.75.

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How to Convert Fractions to Decimals Without a Calculator?

This involves performing long division manually, as described in the "How to Convert a Fraction to a Decimal?" section. You place the numerator inside the division bracket and the denominator outside, then carry out the division, adding decimal points and zeros as needed.

          0.625
        _______
      8 | 5.000
          -0
          ---
           5 0  (Bring down 0)
          -4 8  (8 x 6 = 48)
          ----
             20 (Bring down 0)
            -16 (8 x 2 = 16)
            ----
              40 (Bring down 0)
             -40 (8 x 5 = 40)
             ----
               0 (Remainder is 0, so it terminates)
        

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How to Use a Calculator for Conversions?

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General Combined Question:

"How to convert decimals to fractions and fractions to decimals" is answered by combining the methods described above for each individual conversion type.