Finding Repeating Decimals Worksheet Download - Free Printable

Problem Overview:

The task is to determine whether each given fraction or division problem, when converted to a decimal, will result in a repeating decimal (R) or a terminating decimal (T). The key rule provided is:

- A fraction will result in a terminating decimal if the prime factors of the simplified denominator contain only 2s or 5s (or both).
- A fraction will result in a repeating decimal if the prime factors of the simplified denominator contain any prime factor other than 2 or 5.

Solution Approach:

1. Simplify Each Fraction or Division Problem: Convert each division problem into a fraction and simplify it.
2. Factorize the Denominator: Determine the prime factors of the denominator.
3. Apply the Rule:
- If the denominator's prime factors are only 2s and/or 5s, the decimal is terminating (T).
- If the denominator contains any prime factor other than 2 or 5, the decimal is repeating (R).

Step-by-Step Analysis:

#### 1. \( 10 \div 3 \)
- Simplified form: \( \frac{10}{3} \)
- Denominator: \( 3 \)
- Prime factors of 3: \( 3 \) (not 2 or 5)
- Result: Repeating decimal (R)

#### 2. \( \frac{1}{8} \)
- Denominator: \( 8 \)
- Prime factors of 8: \( 2 \times 2 \times 2 \) (only 2s)
- Result: Terminating decimal (T)

#### 3. \( \frac{16}{20} \)
- Simplify: \( \frac{16}{20} = \frac{4}{5} \)
- Denominator: \( 5 \)
- Prime factors of 5: \( 5 \) (only 5)
- Result: Terminating decimal (T)

#### 4. \( 102 \div 19 \)
- Simplified form: \( \frac{102}{19} \)
- Denominator: \( 19 \)
- Prime factors of 19: \( 19 \) (not 2 or 5)
- Result: Repeating decimal (R)

#### 5. \( \frac{2}{17} \)
- Denominator: \( 17 \)
- Prime factors of 17: \( 17 \) (not 2 or 5)
- Result: Repeating decimal (R)

#### 6. \( 288 \div 27 \)
- Simplified form: \( \frac{288}{27} = \frac{32}{3} \)
- Denominator: \( 3 \)
- Prime factors of 3: \( 3 \) (not 2 or 5)
- Result: Repeating decimal (R)

#### 7. \( \frac{11}{13} \)
- Denominator: \( 13 \)
- Prime factors of 13: \( 13 \) (not 2 or 5)
- Result: Repeating decimal (R)

#### 8. \( \frac{6}{16} \)
- Simplify: \( \frac{6}{16} = \frac{3}{8} \)
- Denominator: \( 8 \)
- Prime factors of 8: \( 2 \times 2 \times 2 \) (only 2s)
- Result: Terminating decimal (T)

#### 9. \( 196 \div 30 \)
- Simplified form: \( \frac{196}{30} = \frac{98}{15} \)
- Denominator: \( 15 \)
- Prime factors of 15: \( 3 \times 5 \) (contains 3, not just 2 or 5)
- Result: Repeating decimal (R)

#### 10. \( \frac{21}{24} \)
- Simplify: \( \frac{21}{24} = \frac{7}{8} \)
- Denominator: \( 8 \)
- Prime factors of 8: \( 2 \times 2 \times 2 \) (only 2s)
- Result: Terminating decimal (T)

#### 11. \( 101 \div 15 \)
- Simplified form: \( \frac{101}{15} \)
- Denominator: \( 15 \)
- Prime factors of 15: \( 3 \times 5 \) (contains 3, not just 2 or 5)
- Result: Repeating decimal (R)

#### 12. \( 243 \div 26 \)
- Simplified form: \( \frac{243}{26} \)
- Denominator: \( 26 \)
- Prime factors of 26: \( 2 \times 13 \) (contains 13, not just 2 or 5)
- Result: Repeating decimal (R)

#### 13. \( 45 \div 18 \)
- Simplified form: \( \frac{45}{18} = \frac{5}{2} \)
- Denominator: \( 2 \)
- Prime factors of 2: \( 2 \) (only 2)
- Result: Terminating decimal (T)

#### 14. \( 84 \div 22 \)
- Simplified form: \( \frac{84}{22} = \frac{42}{11} \)
- Denominator: \( 11 \)
- Prime factors of 11: \( 11 \) (not 2 or 5)
- Result: Repeating decimal (R)

#### 15. \( 144 \div 14 \)
- Simplified form: \( \frac{144}{14} = \frac{72}{7} \)
- Denominator: \( 7 \)
- Prime factors of 7: \( 7 \) (not 2 or 5)
- Result: Repeating decimal (R)

Final Answer:

\[
\boxed{R, T, T, R, R, R, R, T, R, T, R, R, T, R, R}
\]