**High School Number Sense Lesson 65: Converting Decimals in a Base other than 10 to Fractions in Base 10**

On Wednesday, we introduced converting a decimal to a fraction (all in a base other than 10). Today we will start with a decimal in a different base, but convert it to a fraction

**in base 10**. (If you haven't done so lately, I recommend you review

**BaseNtoBase10**before proceeding). This concept appeared

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

**question # 66**.

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For base 10 decimals, each digit to the right of the decimal has its specific place value, beginning with:

^{-1}),

2. hundredths (1/100 or 10

^{-2}),

3. thousandths (1/1000 or 10

^{-3}),

4. ten thousandths (1/1000 or 10

^{-4}), etc.

^{-1}),

2. sixteenths (1/16 or 4

^{-2}),

3. sixty-fourths (1/64 or 4

^{-3}),

4. two hundred fifty-sixths (1/256 or 4

^{-4}), etc.

**How to Solve:**

- Convert the digits (to the right of the decimal) to a base 10 number (using
**BaseNtoBase10**). This will be your numerator. - Count the number of digits in the decimal. Take the base to this power. This will be your denominator.
- Reduce the fraction if necessary.

**Example 1: Change 0.111, base 2, to a base 10 fraction. ___**

- Convert 111 (base 2) to base 10. 2 x 1 = 2, + 1 = 3, x 2 = 6, + 1 =
**7**. This is your numerator. - There are 3 digits. Take 2 to the 3rd power, which is
**8**. This is your denominator. - The answer is
**7/8**, which does not reduce.

**Example 2: Change 0.24 (base 5) to a base 10 fraction. ___**

- Convert 24 (base 5) to base 10. 5 x 2 = 10, + 4 =
**14**. This is your numerator. - There are 2 digits. Take 5 to the 2nd power, which is
**25**. This is your denominator. - The answer is
**14/25**, which does not reduce.

**Example 3: Change 0.43 (base 6) to a base 10 fraction. ___**

- Convert 43 (base 6) to base 10. 6 x 4 = 24, + 3 =
**27**. This is your numerator. - There are 2 digits. Take 6 to the 2nd power, which is
**36**. This is your denominator. - The answer is
**27/36**, which reduces to**3/4**.

**Example 4: Change .212 (base 4) to a base 10 fraction. ___**

- Convert 212 (base 4) to base 10. 4 x 2 = 8, + 1 = 9, x 4 = 36, + 2 =
**38**. This is your numerator. - There are 3 digits. Take 4 to the 3rd power, which is
**64**. This is your denominator. - The answer is
**38/****64**, which reduces to**19/32**.

**Example 5: Change .333... (base 8) to a base 10 fraction. ___**

- Notice the 1 repeating digit of 3. The numerator will be
**3**. - The denominator will be (the base minus 1), which is
**7**. - The answer is
**3/7**.

**Example 6: Change 0.3555.. (base 7) to a base 10 fraction. ___**

- Notice the 1 non-repeating digit of 3 and the 1 repeating digit of 5. The numerator will be (35 - 3) = 32 (base 7). Convert this to base 10. 7 x 3 = 21, + 2 =
**23**. - The denominator is 60 (base 7), which is
**42**base 10. - The answer is
**23/42**, which does not reduce.

**Example 7: Change 0.4232323... (base 5) to a base 10 fraction. ___**

- Notice the 1 non-repeating digit of 4 and the 2 repeating digits of 23. The numerator will be (423 - 4) = 414 (base 5). Convert this to base 10. 5 x 4 = 20, + 1 = 21, x 5 = 105, + 4 =
**109**. - The denominator is 440 (base 5), which is
**120**base 10. - The answer is
**109/120**, which does not reduce.

**Check back soon for a free worksheet to help you practice BaseNDecTo10F.**

**Up Next for High School: FunctionCompo**