Evaluate the following: 1.256 + 0.7 - 0.15

Step 1: Convert the repeating decimal $1.2\dot{5}\dot{6}$ to a fraction. Let $x = 1.2\dot{5}\dot{6}$. This means $x = 1.2565656...$ Multiply by $10$ to move the non-repeating digit to the left of the decimal point: $10x = 12.565656... \quad (*)$ Multiply by $1000$ to move one full cycle of the repeating part to the left of the decimal point: $1000x = 1256.565656... \quad (**)$ Subtract equation $(*)$ from equation $(**)$: $1000x - 10x = 1256.565656... - 12.565656...$ $990x = 1244$ $x = \frac{1244}{990}$ Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is $2$: $x = \frac{1244 \div 2}{990 \div 2} = \frac{622}{495}$

Step 2: Convert the other decimals to fractions. $0.7 = \frac{7}{10}$ $0.15 = \frac{15}{100} = \frac{3}{20}$

Step 3: Substitute the fractions into the original expression. The expression is $1.2\dot{5}\dot{6} + 0.7 - 0.15$. Substituting the fractional forms: $\frac{622}{495} + \frac{7}{10} - \frac{3}{20}$

Step 4: Find a common denominator for the fractions. The denominators are $495$, $10$, and $20$. Prime factorization of the denominators: $495 = 3^2 \times 5 \times 11$ $10 = 2 \times 5$ $20 = 2^2 \times 5$ The least common multiple (LCM) is $2^2 \times 3^2 \times 5 \times 11 = 4 \times 9 \times 5 \times 11 = 1980$.

Step 5: Convert each fraction to have the common denominator $1980$.