((5)/(1)+(5)/(9)-(5)/(11))-((5)/(9)-(5)/(11)-5)

To solve the expression \((\frac{5}{1} + \frac{5}{9} - \frac{5}{11}) - (\frac{5}{9} - \frac{5}{11} - 5)\), we need to follow these steps:<br /><br />1. Simplify the terms inside each set of parentheses.<br />2. Subtract the second set of parentheses from the first set.<br /><br />Let's start with the first set of parentheses:<br />\[<br />\frac{5}{1} + \frac{5}{9} - \frac{5}{11}<br />\]<br /><br />Convert \(\frac{5}{1}\) to \(5\):<br />\[<br />5 + \frac{5}{9} - \frac{5}{11}<br />\]<br /><br />To combine these fractions, find a common denominator. The least common multiple (LCM) of 9 and 11 is 99.<br /><br />Convert \(\frac{5}{9}\) and \(\frac{5}{11}\) to have a denominator of 99:<br />\[<br />\frac{5}{9} = \frac{5 \times 11}{9 \times 11} = \frac{55}{99}<br />\]<br />\[<br />\frac{5}{11} = \frac{5 \times 9}{11 \times 9} = \frac{45}{99}<br />\]<br /><br />Now, add and subtract these fractions:<br />\[<br />5 + \frac{55}{99} - \frac{45}{99} = 5 + \frac{55 - 45}{99} = 5 + \frac{10}{99}<br />\]<br /><br />So, the first set of parentheses simplifies to:<br />\[<br />5 + \frac{10}{99}<br />\]<br /><br />Next, simplify the second set of parentheses:<br />\[<br />\frac{5}{9} - \frac{5}{11} - 5<br />\]<br /><br />Again, find a common denominator for \(\frac{5}{9}\) and \(\frac{5}{11}\):<br />\[<br />\frac{5}{9} = \frac{55}{99}<br />\]<br />\[<br />\frac{5}{11} = \frac{45}{99}<br />\]<br /><br />Subtract these fractions:<br />\[<br />\frac{55}{99} - \frac{45}{99} = \frac{10}{99}<br />\]<br /><br />So, the second set of parentheses simplifies to:<br />\[<br />\frac{10}{99} - 5<br />\]<br /><br />Now, subtract the second set of parentheses from the first set:<br />\[<br />(5 + \frac{10}{99}) - (\frac{10}{99} - 5)<br />\]<br /><br />Distribute the negative sign:<br />\[<br />5 + \frac{10}{99} - \frac{10}{99} + 5<br />\]<br /><br />Combine like terms:<br />\[<br />5 + 5 = 10<br />\]<br /><br />Thus, the final answer is:<br />\[<br />\boxed{10}<br />\]