**High School Number Sense Lesson 26: Repeating Decimals (with One Repeating & One Nonrepeating Digit)**

This idea showed up

**10 times**this year in high school, with the median spot at

**question 28**. If you missed the last lesson on repeating decimals, you may want to review it here.

**Number Dojo Level: 199**

In this concept, you will be asked to convert a repeating decimal (such as .2777...) to a fraction. Once again, here are the rules of determining the denominator:

- For every unique
**digit that repeats**, begin your denominator with**that many 9's**. - For every unique
**digit that does not repeat**, end your denominator with**that many 0's**.

**To determine the numerator for RepDec.abbb:**

- Put the 2 digits together (1 nonrepeating & 1 repeating). For .2777..., this would be
**27**. - Subtract the nonrepeating digit (in this case, 2).
**27 - 2 = 25**.

Then fully reduce the fraction before you write down your answer! 25/90 reduces to

**5/18**.

**Example 1: .3222... = ___ (fraction)**

- Think of the nonrepeating & repeating digits together as a 2-digit number:
**32**. - Subtract the nonrepeating digit (3):
**32 - 3 = 29**. - Your denominator begins with one 9 (because of the 1 repeating digit) and ends with one 0 (because of the 1 nonrepeating digit), so:
**90**. **29/90**does not reduce.

**Example 2: .6444... = ___ (fraction)**

- Think of the nonrepeating & repeating digits together as a 2-digit number:
**64**. - Subtract the nonrepeating digit (6):
**64 - 6 = 58**. - Your denominator begins with one 9 (because of the 1 repeating digit) and ends with one 0 (because of the 1 nonrepeating digit), so:
**90**. - 58/90 reduces to
**29/45**.

**Example 3: .1333... = ___ (fraction)**

- Think of the nonrepeating & repeating digits together as a 2-digit number:
**13**. - Subtract the nonrepeating digit (1):
**13 - 1 = 12**. - Your denominator begins with one 9 (because of the 1 repeating digit) and ends with one 0 (because of the 1 nonrepeating digit), so:
**90**. - 12/90 reduces to
**2/15**.

**Example 4: 0.4333... = ___ (fraction)**

- Think of the nonrepeating & repeating digits together as a 2-digit number:
**43**. - Subtract the nonrepeating digit (4):
**43 - 4 = 39**. - Your denominator begins with one 9 (because of the 1 repeating digit) and ends with one 0 (because of the 1 nonrepeating digit), so:
**90**. - 39/90 reduces to
**13/30**.

**Example 5: 0.7222... = ___ (fraction)**

- Think of the nonrepeating & repeating digits together as a 2-digit number:
**72**. - Subtract the nonrepeating digit (7):
**72 - 7 = 65**. - Your denominator begins with one 9 (because of the 1 repeating digit) and ends with one 0 (because of the 1 nonrepeating digit), so:
**90**. - 65/90 reduces to
**13/18**.

**Example 6: 1.8666... = ___ (mixed number)**

- Ignore the whole number in front of the decimal for now. Think of the nonrepeating & repeating digits together as a 2-digit number:
**86**. - Subtract the nonrepeating digit (8):
**86 - 8 = 78**. - Your denominator begins with one 9 (because of the 1 repeating digit) and ends with one 0 (because of the 1 nonrepeating digit), so:
**90**. - 78/90 reduces to
**13/15**. - Place the 1 back in front to get
**1 13/15**.

**Here is a free worksheet to help you practice RepDec.abbb:**

repdec.abbb.pdf |

**Up Next for High School: MoneyRatio**