Converting decimals to fractions

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This may be a statement of the obvious, but converting a decimal to a fraction is the opposite of converting a fraction to a decimal which you do by dividing the numerator by the denominator. For example the fraction 1/2 is the decimal 0.5 which you get by dividing 1 by 2.

The strategy you use depends on the type of decimal you have. Let's do these from easiest to hardest.

Non-repeating decimals
These are also called terminating decimals because they don't go on forever.
Example 1: Convert 0.45 to a fraction.
Step 1: Let x = 0.45.
Step 2: Count how many numbers there are after the decimal point. In this case, there are 2.
Step 3: Multiply both sides by 100, because 100 has 2 zeroes. We get 100x = 45.
Step 4: Solve for x. In this case x = 45/100. Using the Euclidean Algorithm to reduce the fraction, we get x = 9/20.

Simple repeating decimals
You can tell if it is a simple repeating decimal number if the repeating part starts with the first number after the decimal point.
Example 2: Convert 4.372372372... to a fraction.
Step 1: Let x = 4.372372. Call this equation #1.
Step 2: Count how many numbers there are in the repeating part. In this example, the repeating part is 372. So there are 3 numbers in the repeating part.
Step 3: Multiply both sides by 1000, because 1000 has 3 zeroes. We get 1000x = 4372.372372... Call this equation #2.
Step 4: Subtract equation #1 from equation #2:

Solving for x, we get x = 4368/999. Using the Euclidean Algorithm to reduce the fraction, we get x = 1456/333.

Ugly repeating decimals
These are the ones where there are some numbers after the decimal point before the repeating part. About as much fun as watching a documentary with your parents.
Example 3: Convert 2.173333... to a fraction.
Step 1: Let x = 2.173333. Call this equation #1.
Step 2: Count how many non-repeating numbers there are after the decimal point. In this case, there are 2.
Step 3: Multiply both sides by 100, because 100 has 2 zeroes. We get 100x = 217.3333... Call this equation #2.
Step 4: Now that we have equation #2 as a simple repeating decimal number, we can carry out the same strategy as in Example 2, starting with step 2:

Solving for x, we get x = 1956/900. Using the Euclidean Algorithm to reduce the fraction, we get x = 163/75.