**High School Number Sense Lesson 63: Converting from a Fraction to a Repeating Decimal**

Back in August we introduced repeating decimals in the form of .aaaa... (

**RepDec.aaaa**) , .abab... (

**RepDec.abab**), and .abbb... (

**RepDec.abbb**). I would recommend reviewing each of these lessons before proceeding. Today we will go the other direction--from the fraction to the repeating decimal. This concept appeared a massive

**17 times**last year on high school tests, with a median placement at

**question # 69**.

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These questions require a little bit of logic and a little bit of calculation. The fractions with denominator 11, 33, 99, 999, etc., are pretty straightforward:

**Example 1: The first 4 digits of the decimal of 31/99 is 0.___**

- We know that any fraction with denominator 99 simply repeats the numerator (as a 2 digit number).
- The answer is
**3131**.

**Example 2: The first 4 digits of the decimal of 253/999 is 0.___**

- We know that any fraction with denominator 999 simply repeats the numerator (as a 3 digit number).
- The answer is
**2532**.

**Example 3: The first four digits of the decimal for 14/33 is 0.___**

- In this example, multiply both the numerator and denominator by 3 to get
**42/99**. - The answer is
**4242**.

**Non-Repeating Digits**

Denominators that end in 0 require a little more calculation (and a little less logic?). But there is a pretty cool pattern for denominator 90:

**Example 4: The first 4 digits of the decimal of 29/90 is 0.___**

- Divide 29 by 9 to get
**3 remainder 2**. The whole number portion of the answer is the first digit of the decimal, and the remainder is the repeating digit. - The answer is
**3222**.

**Example 5: The first 4 digits of the decimal of 37/45 is 0.___**

- In this example, multiply both the numerator and denominator by 2 to get
**74/90**. - Now divide 74 by 9 to get
**8 remainder 2**. The whole number portion of the answer is the first digit of the decimal, and the remainder is the repeating digit. - The answer is
**8222**.

**Example 6: The first 4 digits of the decimal of 221/900 is 0.___**

- In this example, notice that the denominator is 900. We will do the same thing we did with 90, but then divide by 10 (move the decimal over once to the left).
- Divide 221 by 9 to get
**24 remainder 5**. The whole number portion of the answer becomes the two non-repeating digits, and the remainder is the repeating digit. - The answer is
**2455**.

__: Any of these problems can be reverse-engineered, if we know the patterns and rules for converting the repeating decimal to the fraction.__

**Note****Example 7: The first 4 digits of the decimal of 245/990 is 0.___**

- In this case, we need to think of a 3-digit number in the form of
**abc**, where**(abc - a) = 245**--our numerator. - Logically,
**a**would have to be**2**--it's already given.**abc**would have to be**247**. In other words,**247 - 2 = 245**. - There is only 1 non-repeating digit in the decimal, so the answer is
**2474**.

**Example 8: The first four digits of the decimal for 71/330 is 0.___**

- In this example, multiply both the numerator and denominator by 3 to get
**213/990**. - We need to think of a 3-digit number in the form of
**abc**, where**(abc - a) = 213**--our numerator. - Logically,
**a**would have to be**2**--it's already given. So**abc**would have to be**215**. In other words,**215 - 2 = 213**. - There is only 1 non-repeating digit in the decimal, so the answer is
**2151**.

**Example 9: The first 3 digits of the decimal for 5/7 is 0.___**

- This recent test question requires a good knowledge of the pattern of repeating decimals for fractions with denominator 7. Regardless of the numerator (as long as it's not a multiple of 7), the decimal will repeat the digits 142857, in that order. But the starting number will be different, depending on the numerator.
- The answer is
**714**. The pattern goes like this:

**1/7 = .142857...**

2/7 = .285714...

3/7 = .428571...

4/7 = .571428...

5/7 = .714285...

6/7 = .857142...

2/7 = .285714...

3/7 = .428571...

4/7 = .571428...

5/7 = .714285...

6/7 = .857142...

**Check back soon for a free worksheet to help you practice RepDecFracDec.**

**Up Next for High School: BaseNDeciFrac**