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.

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