This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Assumption: The notation 5.165 is interpreted as 5.165, meaning the digits 165 repeat. Step 1: Set up the equation for the repeating decimal. Let m be the repeating decimal: m = 5.165165165... (1) Step 2: Multiply the equation to shift the repeating part to the left of the decimal. Since there are three repeating digits (165), multiply equation (1) by 1000: 1000m = 5165.165165... (2) Step 3: Subtract the original equation from the multiplied equation. Subtract equation (1) from equation (2): 1000m - m = 5165.165165... - 5.165165165... 999m = 5160 Step 4: Solve for m. Divide both sides by 999: m = (5160)/(999) Step 5: Simplify the fraction. Both the numerator (5160) and the denominator (999) are divisible by 3. 5160 ÷ 3 = 1720 999 ÷ 3 = 333 So, the simplified improper fraction is: m = (1720)/(333) Step 6: Convert the improper fraction to a mixed number. Divide 1720 by 333: 1720 = 5 × 333 + 55 So, the mixed number is: m = 5 (55)/(333) The final answer is $5