The sum of $\overline{2}.75$ and $\overline{3}.78$ is

Concept & Logic When a bar is placed over the integer part of a decimal (the characteristic), it signifies that *only* the integer is negative, while the decimal part (the mantissa) remains positive. This notation is commonly used in logarithm calculations. $$ \overline{a}.bc = -a + 0.bc $$ Step-by-Step Solution Given expression: $\overline{2}.75 + \overline{3}.78$ 1. Break down each number into its negative integer and positive decimal components: $$ \overline{2}.75 = -2 + 0.75 $$ $$ \overline{3}.78 = -3 + 0.78 $$ 2. Add the equations together. Group the integer parts and the decimal parts separately: $$ (-2 - 3) + (0.75 + 0.78) $$ 3. Calculate the sum of the integers: $$ -2 - 3 = -5 $$ 4. Calculate the sum of the decimals: $$ 0.75 + 0.78 = 1.53 $$ 5. Combine the results: $$ -5 + 1.53 $$ Split the $1.53$ into its whole number and decimal parts: $$ -5 + 1 + 0.53 $$ $$ -4 + 0.53 $$ 6. Convert back into the bar notation format (negative integer, positive decimal): $$ \overline{4}.53 $$ Exam Strategy & Shortcut Add the decimal parts first: $0.75 + 0.78 = 1.53$. This generates a $+1$ carry-over. Now, sum the integers along with the carry-over: $-2 - 3 + 1 = -4$. The integer becomes $\overline{4}$ and the decimal remains $.53$, instantly giving $\overline{4}.53$. Common Pitfall A very common mistake is treating the entire number as negative (e.g., treating $\overline{2}.75$ as $-2.75$). This leads to $(-2.75) + (-3.78) = -6.53$, which might lead students to mistakenly guess an incorrect option. The bar on top is distinct from a negative sign in front. Final Answer Therefore, the correct answer is $\overline{4}.53$.