**High School Number Sense Lesson 64: Converting Decimals to Fractions in a Base other than 10**

Today we will leave behind the safety and familiarity of base 10 and live in the marvelous world of other bases. Specifically, we will be converting repeating decimals to fractions, but all in a base other than base 10. We'll be combining the ideas found in several other topics we've already covered:

**ConvDeciFrac**: Converting Decimals to Fractions**RepDec.aaaa**: Repeating Decimals (with One Repeating Digit)**RepDec.abab**: Repeating Decimals (with Two Repeating Digits)**RepDec.abbb**: Repeating Decimals (with One Repeating & One Nonrepeating Digit)

**10 times**last year, with a median placement at

**question # 66**.

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For the past few years, each of these problems has been in

**a base between 3 and 8**, inclusive:

- Base 3: 3 times
- Base 4: 6 times
- Base 5: 9 times
- Base 6: 6 times
- Base 7: 3 times
- Base 8: 10 times

Also, each of these problems has included a

**repeating decimal**.

- One repeating digit: 7 times
- Two repeating digits: 11 times
- One repeating and One nonrepeating digit: 17 times
- Two repeating and One nonrepeating digit: 2 times

We need to remember the

**rules of repeating decimals**(in base 10):

- For each digit that repeats, the denominator of the fraction will begin with a 9. (For bases other than 10, this digit will be the base minus one).
- For each digit that does not repeat, the denominator will end with a 0. (This is true in any base).

**Example 1: Change 0.333... (base 6) to a base 6 fraction. ___ (base 6)**

- There is only one repeating digit:
**3**. This will be the numerator. - The denominator will have one
**5**(think base 6, minus one). - The answer is
**3/5**.

**Example 2: Change 0.444... (base 8) to a base 8 fraction. ___ (base 8)**

- There is only one repeating digit:
**4**. This will be the numerator. - The denominator will have one
**7**(think base 8, minus one). - The answer is
**4/7**.

**Example 3: Change 0.242424... (base 5) to a base 5 fraction. ___ (base 5)**

- There are two repeating digits:
**24**. This will be the numerator. - The denominator will have two
**4**s (think base 5, minus one). - The answer is
**24/44**, but this looks like it may reduce... - In base 10, this is equal to
**14/24**, which reduces to**7/12**. Back in base 5, this is**12/22**.

**Example 4: Change 0.1333... (base 4) to a base 4 fraction. ___ (base 4)**

- There is one repeating digit (3) and one nonrepeating digit (1). The numerator will be 13 - 1 =
**12**. - The denominator will start with one
**3**(think base 4, minus one) and end in one**0**:**30**. - The answer is
**12/30**, which may reduce... - In base 10, this is equal to
**6/12**, which reduces to**1/2**. Back in base 4, this is still**1/2**.

**Example 5: Change 0.2353535... (base 6) to a base 6 fraction. ___ (base 6)**

- There are two repeating digits (35) and one nonrepeating digit (2). The numerator will be 235 - 2 =
**233**. - The denominator will start with two
**5**s (think base 6, minus one) and end in one**0**:**550**. - The answer is
**233/550**, which may reduce... - In base 10, this is equal to
**93/210**, which reduces to**31/70**. Back in base 6, this is**51/154**.

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

**Up Next for High School: BaseNDecTo10F**