Question

Express the following as a vulgar fraction.

• A. $$4\frac{45}{11}$$
• B. $$4\frac{5}{11}$$
• C. $$4\frac{5}{99}$$
• D. $$4\frac{42}{11}$$

Answer 1

Answer:

$$4\frac{5}{11}$$

1. Understand the Notation:

• The overline (or vinculum) above the digits indicates a repeating decimal.

  • 0.23 means 0.232323... (the "23" repeats infinitely)

  • 0.2 means 0.22222... (the "2" repeats infinitely)

2. Convert the Repeating Decimals to Fractions:

• 0.23:

  • Let x = 0.232323...

  • 100x = 23.232323...

  • 100x - x = 23.232323... - 0.232323...

  • 99x = 23

  • x = 23/99

• 0.2:

  • Let y = 0.22222...

  • 10y = 2.22222...

  • 10y - y = 2.22222... - 0.22222...

  • 9y = 2

  • y = 2/9

3. Substitute the Fractions into the Expression:

• 23/99 + 2/9 + 4

4. Find a Common Denominator for the Fractions (99):

• 23/99 + (2/9) * (11/11) + 4

• 23/99 + 22/99 + 4

5. Add the Fractions:

• (23 + 22) / 99 + 4

• 45/99 + 4

6. Simplify the Fraction:

• 45/99 can be simplified by dividing both the numerator and denominator by 9.

• 45 / 9 = 5

• 99 / 9 = 11

• So, 45/99 = 5/11

7. Add the Whole Number:

• 5/11 + 4

• (5/11) + (44/11) (Converting 4 to a fraction with a denominator of 11)

• (5 + 44) / 11

• 49/11

8. Convert to a Mixed Number (if needed):

• 49 / 11 = 4 and 5/11

Therefore, the answer is $4\frac{5}{11}$