What is $$ \frac{\frac{7}{8} \times \frac{7}{8} + \frac{5}{6} \times \frac{5}{6} + \frac{7}{8} \times \frac{5}{3}}{\frac{7}{8} \times \frac{7}{8} - \frac{5}{6} \times \frac{5}{6}} $$ equal to?

### Concept & Formula This problem masks fundamental algebraic identities inside complex-looking fraction multiplication. The two governing formulas to recognize here are: $$ (a + b)^2 = a^2 + b^2 + 2ab $$ $$ a^2 - b^2 = (a - b)(a + b) $$ ### Step-by-Step Solution Let's substitute the repeating fractions with variables to spot the pattern more easily: * Let $a = \frac{7}{8}$ * Let $b = \frac{5}{6}$ Look closely at the numerator: $$ a \times a + b \times b + a \times \frac{5}{3} $$ The trick lies in the fraction $ \frac{5}{3} $. Notice that $ \frac{5}{3} $ is exactly double $ \frac{5}{6} $. Therefore, $ \frac{5}{3} = 2 \times b $. Substitute this back into the numerator: $$ \text{Numerator} = a^2 + b^2 + a(2b) = a^2 + b^2 + 2ab $$ Which simplifies to $ (a + b)^2 $. Now look at the denominator: $$ \text{Denominator} = a \times a - b \times b = a^2 - b^2 $$ Which expands to $ (a - b)(a + b) $. Combine the simplified numerator and denominator: $$ \text{Expression} = \frac{(a + b)^2}{(a - b)(a + b)} $$ Cancel out one $(a + b)$ term from top and bottom: $$ \text{Expression} = \frac{a + b}{a - b} $$ Now, evaluate $ (a + b) $ and $ (a - b) $ by substituting the fractions back. Find a common denominator for 8 and 6, which is 24. * $a = \frac{7}{8} = \frac{21}{24}$ * $b = \frac{5}{6} = \frac{20}{24}$ Calculate the sum and difference: * $a + b = \frac{21}{24} + \frac{20}{24} = \frac{41}{24}$ * $a - b = \frac{21}{24} - \frac{20}{24} = \frac{1}{24}$ Finally, divide the sum by the difference: $$ \frac{41/24}{1/24} = 41 \times \frac{24}{1} = 41 $$ ### Exam Strategy & Shortcut Whenever you see a $a^2 + b^2 + \dots$ structure, actively look for how the remaining term represents $2ab$. Here, seeing $5/3$ alongside $5/6$ immediately signals $2b$. You can then instantly simplify the whole block of math down to $ (a+b)/(a-b) $, requiring only one basic fraction addition and subtraction. ### Common Pitfall Failing to recognize $5/3$ as $2 \times 5/6$ causes students to abandon the algebraic approach entirely and attempt brute-force arithmetic, which under exam conditions almost always results in a miscalculation given the large mixed denominators. ### Final Answer **Therefore, the correct answer is 41.**