Find the value of 0.7̅ + 0.4899̅ - 0.5322̅.

Concept

A repeating (recurring) decimal is a rational number, and every repeating decimal can be converted into an exact fraction: multiply the decimal by a power of 10 that shifts one full repeating block to the left of the decimal point, subtract a similarly-shifted copy so the infinite repeating tail cancels exactly, and solve the resulting whole-number equation for the fraction.

Application

1. Convert 0.7̅: let x = 0.777... ; then 10x = 7.777... ; subtracting gives 9x = 7, so x = 7/9.

2. Convert 0.4899̅ (only the last 9 repeats): let x = 0.48999... ; then 10000x = 4899.999... and 1000x = 489.999... ; subtracting gives 9000x = 4410, so x = 4410/9000 = 49/100 (a decimal ending in an endless string of 9s equals the value rounded up, so 0.48999... = 0.49).

3. Convert 0.5322̅ (only the last 2 repeats): let x = 0.53222... ; then 10000x = 5322.222... and 1000x = 532.222... ; subtracting gives 9000x = 4790, so x = 4790/9000 = 479/900.

4. Bring all three fractions to a common denominator of 900: 7/9 = 700/900, 49/100 = 441/900, and 479/900 stays as it is.

5. Add and subtract the numerators: 700 + 441 - 479 = 662, so the sum is 662/900, which reduces (dividing by 2) to 331/450.

6. Convert 331/450 back to a decimal by long division: 331/450 = 0.7355555..., i.e. 0.73 followed by an endlessly repeating 5, written 0.735̅.

Cross-check

Adding the three decimals directly to ten places gives the same result: 0.7777777778 + 0.4900000000 - 0.5322222222 = 0.7355555556, which agrees with 331/450 to full precision, confirming the fraction-based computation.

So the value of the expression is 0.735̅ (0.7355555...), matching the option that reads 0.735̅.