The value of \(2.\overline{2}+1.\overline{18}-1.\overline{2}\) is:

Concept

A repeating decimal is a rational number. If a block of k digits repeats, the repeating fractional part is obtained by placing that block over \(10^k-1\).

Separate the whole-number part first, convert each repeating fractional part into a fraction, and then combine the fractions using ordinary arithmetic.

Application

1. Convert \(2.\overline{2}\): \(2+\frac{2}{9}=\frac{20}{9}\).

2. Convert \(1.\overline{18}\): \(1+\frac{18}{99}=1+\frac{2}{11}=\frac{13}{11}\).

3. Convert \(1.\overline{2}\): \(1+\frac{2}{9}=\frac{11}{9}\).

4. Combine the values: \(\frac{20}{9}+\frac{13}{11}-\frac{11}{9}=1+\frac{13}{11}=\frac{24}{11}\).

Cross-check

Since \(\frac{24}{11}=2.181818\ldots\), it agrees with \(2.2222\ldots+1.181818\ldots-1.2222\ldots=2.181818\ldots\).

Therefore, the value is \(\frac{24}{11}\).