Verified Answer :

  • The repeating decimal \(\overline{5}\) is equal to \(\frac{5}{9}\) because \(10x - x = 5.555\dots - 0.555\dots = 5\), so \(9x = 5\).
  • The fraction \(\frac{36}{45}\) simplifies to \(\frac{4}{5}\) by dividing both numerator and denominator by 9.
  • Since \(\frac{5}{9} eq \frac{4}{5}\) (as \(25 eq 36\)), the given equation is incorrect.
  • Final Answer: \(\boxed{\text{False}}\).

Explanation

  1. Problem Analysis

We are asked to determine if the equality \(\overline{5} = \frac{36}{45}\) is true, where \(\overline{5}\) represents the repeating decimal \(0.555\dots\). To check this, we need to express both sides of the equation as fractions in their simplest form and compare them.

  1. Converting the Repeating Decimal to a Fraction

Let \(x\) be the repeating decimal \(0.555\dots\). We can find its fractional representation using the following steps:

  1. Write the expression for \(x\): \(x = 0.555\dots\)
  2. Multiply by 10 to shift the decimal point: \(10x = 5.555\dots\)
  3. Subtract the first equation from the second: \(10x - x = 5.555\dots - 0.555\dots\)
  4. This yields \(9x = 5\), which simplifies to \(x = \frac{5}{9}\).

Thus, \(\overline{5} = \frac{5}{9}\).

  1. Simplifying the Fraction on the Right Side

Now, let us simplify the fraction \(\frac{36}{45}\) by dividing both the numerator and the denominator by their greatest common divisor. Both 36 and 45 are divisible by 9:
\(\frac{36}{45} = \frac{36 \div 9}{45 \div 9} = \frac{4}{5}\).
So, \(\frac{36}{45}\) is equivalent to \(\frac{4}{5}\).

  1. Conclusion

Finally, we compare the two simplified values:
Is \(\frac{5}{9} = \frac{4}{5}\)?
To compare these, we can cross-multiply:
\(5 \times 5 = 25\)
\(9 \times 4 = 36\)
Since \(25 eq 36\), the fractions are not equal. Therefore, the original statement \(\overline{5} = \frac{36}{45}\) is false. The final answer is \(\boxed{\text{False}}\).

Examples

In financial accounting, correctly converting repeating decimals into fractions is crucial for maintaining accuracy in interest rate calculations or proportional distributions. If an error like assuming \(\overline{5} = \frac{36}{45}\) were made in an automated system, it could lead to significant discrepancies in large-scale financial reporting. Understanding the exact fractional forms of decimals prevents such errors and ensures precision.

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