((1+1/8-23/45):1+2/5-1/5)*3/17=?

Let's evaluate the expression step by step: Expression: \[ \left( 1 + \frac{1}{8} - \frac{23}{45} \right) \div 1 + \frac{2}{5} - \frac{1}{5} \times \frac{3}{17} \] **Step 1: Simplify inside the parentheses** Calculate \(1 + \frac{1}{8} - \frac{23}{45}\): - Convert all to a common denominator, which is 360 (the least common multiple of 8 and 45): \[ 1 = \frac{360}{360} \] \[ \frac{1}{8} = \frac{45}{360} \] \[ \frac{23}{45} = \frac{184}{360} \] Now: \[ \frac{360}{360} + \frac{45}{360} - \frac{184}{360} = \frac{360 + 45 - 184}{360} = \frac{221}{360} \] **Step 2: Divide this result by 1** \[ \frac{221}{360} \div 1 = \frac{221}{360} \] **Step 3: Add \(\frac{2}{5}\) and subtract \(\frac{1}{5}\times \frac{3}{17}\)** Express \(\frac{2}{5}\) and \(\frac{1}{5}\times \frac{3}{17}\): - \(\frac{2}{5} = \frac{2\times 17}{5 \times 17} = \frac{34}{85}\) - \(\frac{1}{5} \times \frac{3}{17} = \frac{1 \times 3}{5 \times 17} = \frac{3}{85}\) Now, originally: \[ \left( \frac{221}{360} \right) + \frac{2}{5} - \left( \frac{1}{5} \times \frac{3}{17} \right) \] Express all with common denominators: - Convert \(\frac{2}{5} = \frac{34}{85}\). To combine with \(\frac{221}{360}\) and \(\frac{3}{85}\), find common denominators. But since \(\frac{221}{360}\) and \(\frac{3}{85}\) are involved, let's convert everything to a common denominator. Find common denominators: - \(360\) and \(85\): LCM of 360 and 85: Factor: \[ 360 = 2^3 \times 3^2 \times 5 \] \[ 85 = 5 \times 17 \] LCM: \[ 2^3 \times 3^2 \times 5 \times 17 = 8 \times 9 \times 5 \times 17 = 8 \times 9 \times 85 = (8 \times 9) \times 85 = 72 \times 85 = 6120 \] Express all fractions with denominator 6120: - \(\frac{221}{360} = \frac{221 \times 17}{360 \times 17} = \frac{3767}{6120}\) - \(\frac{34}{85} = \frac{34 \times 72}{85 \times 72} = \frac{2448}{6120}\) - \(\frac{3}{85} = \frac{3 \times 72}{85 \times 72} = \frac{216}{6120}\) Now combine: \[ \frac{3767}{6120} + \frac{2448}{6120} - \frac{216}{6120} = \frac{3767 + 2448 - 216}{6120} = \frac{5983}{6120} \] **Step 4: Simplify if possible** Check for common factors: - 5983 divided by small primes won't yield an integer. The sum appears to be in lowest terms. **Final answer:** \[ \boxed{\frac{5983}{6120}} \] or approximately **0.977**.