$5 \frac{5}{6} - 3 \frac{8}{9} - x = 1$

### Concept & Logic When adding or subtracting mixed fractions, it is mathematically sound and computationally faster to separate the whole integer parts from the fractional parts rather than converting everything into large improper fractions. ### Step-by-Step Solution * The given mathematical equation is: $5 \frac{5}{6} - 3 \frac{8}{9} - x = 1$ * Rearrange the equation to isolate the unknown variable $x$ on one side: $x = 5 \frac{5}{6} - 3 \frac{8}{9} - 1$ * Group the integer components and the fractional components independently: $x = (5 - 3 - 1) + (\frac{5}{6} - \frac{8}{9})$ * Calculate the result of the integer part: $5 - 3 - 1 = 1$ * Calculate the fractional part by finding the LCM of $6$ and $9$, which is $18$: $\frac{5}{6} = \frac{5 \times 3}{18} = \frac{15}{18}$ $\frac{8}{9} = \frac{8 \times 2}{18} = \frac{16}{18}$ * Subtract the fractions: $\frac{15}{18} - \frac{16}{18} = -\frac{1}{18}$ * Combine the processed integer and fractional parts back together: $x = 1 - \frac{1}{18} = \frac{17}{18}$ ### Exam Strategy & Shortcut Split the integers and fractions immediately in your head. $5 - 3 - 1 = 1$. The fraction part is $\frac{5}{6} - \frac{8}{9}$. Multiply cross-wise mentally for rapid subtraction: $(45 - 48) / 54 = -3/54 = -1/18$. Deducting this from the integer yields $1 - 1/18 = 17/18$. Skipping the improper fraction conversion saves massive amounts of time and eliminates arithmetic errors on large numbers. ### Common Pitfall Converting $5 \frac{5}{6}$ to $\frac{35}{6}$ and $3 \frac{8}{9}$ to $\frac{35}{9}$ creates unnecessarily large numerators. This significantly increases the likelihood of calculation errors when finding a common denominator and performing subtraction under time pressure. ### Final Answer **Therefore, the correct answer is $\frac{17}{18}$.**