**High School Number Sense Lesson 25: Repeating Decimals (with One Repeating Digit and Two Repeating Digits)**

Today you get 2 concepts for the price of 1. (Technically, since the price is $0, you get 2 for the price of 2 as well). Combined, these two ideas were tested

**12 times**this year (

**6 times apiece**). RepDec.aaaa showed up, on average, at

**question # 25**, and

**RepDec.abab**showed up at around

**question # 32**.

**Number Dojo Levels: 166, 168**

For these question types, you are given a repeating decimal (such as .555...), and you are asked to convert it to a fraction. Or you are given more than one repeating decimal, and you need to perform operations (usually addition or subtraction) with them.

Repeating decimals are easier than they look. Here are the rules of 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**. (We will cover this in detail in Lesson 26).

If you have no non-repeating digits, your numerator will simply be the

**unique repeating digit(s)**. Once you figure out what your fraction is, make sure to

**reduce it completely**. This is number sense, after all! Here are some examples to make this a little easier to see:

**Example 1: .777... = ___ (fraction)**

- 7 is the only unique digit that repeats, so the denominator has 1 digit:
**9**. **7**is the numerator.**7/9**is the fraction, which does not reduce (because 7 is prime).

**Example 2: .333 = ___ (fraction)**

- 3 is the only unique digit that repeats, so the denominator has 1 digit:
**9**. **3**is the numerator.**3/9**is the fraction, which reduces to**1/3**.

**Example 3: .252525... = ___ (fraction)**

- Two digits repeat (2 & 5), so the denominator has 2 digits:
**99**. **25**is the numerator.**25/99**is the fraction, which does not reduce.

**Example 4: .424242... = ___ (fraction)**

- Two digits repeat, so the denominator has 2 digits:
**99**. **42**is the numerator.**42/99**is the fraction, which reduces to**14/33**.

**Example 5: .636363.... = ___ (fraction)**

- Two digits repeat, so the denominator has 2 digits:
**99**. **63**is the numerator.**63/99**is the fraction, which reduces to**7/11**.

**Example 6: 1.666... = ___ (mixed number)**

- Don't let the digit in front of the decimal throw you off. One digit repeats, so the denominator has 1 digit:
**9**. **6**is the numerator.**6/9**is the fraction, which reduces to**2/3**. Place the**1**back in front to get**1 2/3**.

**Example 7: 4.090909... + 3.454545... = ___**

- The first number has 2 repeating digits, so it is
**4 09/99**, which reduces to**4 1/11**. - The second number has 2 repeating digits, so it is
**3 45/99**, which reduces to**3 5/11**. - Add them together.
**4 1/11 + 3 5/11 = 7 6/11**.

**Example 8: 88 x .181818... = ___**

- The repeating decimal becomes
**18/99**, which reduces to**2/11**. **88 x 2/11**is the same as**88 x 2 ÷ 11**. Divide first, then multiply!**88 ÷ 11 = 8**, and**8 x 2**=**16**.

**Here are 2 free worksheets to help you practice RepDec.aaaa and RepDec.abab:**

repdec.aaaa.pdf |

repdec.abab.pdf |

**Up Next for High School: RepDec.abbb**