Question

32/40
which rational number is equivalent to 6.2\overline{7}
\frac{2090}{333} \quad \frac{69}{11} \quad \frac{113}{18} \quad \frac{627}{100}

Explanation:

Step1: Recall repeating decimal to fraction

Let \( x = 6.2\overline{7} \). The bar is over 7, so we multiply by 10 to move the decimal past the non - repeating part: \( 10x=62.\overline{7} \). Then multiply by 10 again to move past the repeating part: \( 100x = 627.\overline{7} \).

Step2: Subtract to eliminate the repeating part

Subtract \( 10x\) from \( 100x \): \( 100x-10x=627.\overline{7}-62.\overline{7} \). This gives \( 90x = 565 \), so \( x=\frac{565}{90}=\frac{113}{18}\) (simplify by dividing numerator and denominator by 5: \( \frac{565\div5}{90\div5}=\frac{113}{18} \)). We can also check the other options:

• For \( \frac{2090}{333}\approx6.276\) (not equal to \( 6.2\overline{7}\approx6.277\cdots\) as closely as \( \frac{113}{18}\)).
• \( \frac{69}{11}\approx6.2727\) (the repeating part is 27, not 7).
• \( \frac{627}{100} = 6.27\) (no repeating part).

Answer:

\(\frac{113}{18}\) (the third option)