`((64)/(121)-(9)/(64))/((8)/(11)+(3)/(8))=?`

Options

A. `(31)/(88)
B. `(31)/(84)`
C. `(61)/(88)`
D. `(61)/(84)`

Answer

To solve the expression \(\frac{\left(\frac{64}{121} - \frac{9}{64}\right)}{\left(\frac{8}{11} + \frac{3}{8}\right)}\), we will follow these steps:

### Step 1: Find a common denominator for the numerator
The numerator is \(\frac{64}{121} - \frac{9}{64}\). The least common multiple (LCM) of the denominators 121 and 64 is \(7744\).

- Convert \(\frac{64}{121}\) to have a denominator of 7744:
\[ \frac{64}{121} = \frac{64 \times 64}{121 \times 64} = \frac{4096}{7744} \]

- Convert \(\frac{9}{64}\) to have a denominator of 7744:
\[ \frac{9}{64} = \frac{9 \times 121}{64 \times 121} = \frac{1089}{7744} \]

Now, we can subtract the two fractions:
\[ \frac{4096}{7744} - \frac{1089}{7744} = \frac{4096 - 1089}{7744} = \frac{3007}{7744} \]

### Step 2: Find a common denominator for the denominator
The denominator is \(\frac{8}{11} + \frac{3}{8}\). The least common multiple (LCM) of the denominators 11 and 8 is \(88\).

- Convert \(\frac{8}{11}\) to have a denominator of 88:
\[ \frac{8}{11} = \frac{8 \times 8}{11 \times 8} = \frac{64}{88} \]

- Convert \(\frac{3}{8}\) to have a denominator of 88:
\[ \frac{3}{8} = \frac{3 \times 11}{8 \times 11} = \frac{33}{88} \]

Now, we can add the two fractions:
\[ \frac{64}{88} + \frac{33}{88} = \frac{64 + 33}{88} = \frac{97}{88} \]

### Step 3: Combine the results
Now we can substitute back into our original expression:
\[ \frac{\frac{3007}{7744}}{\frac{97}{88}} = \frac{3007}{7744} \times \frac{88}{97} = \frac{3007 \times 88}{7744 \times 97} \]

### Step 4: Simplify the expression
First, calculate \(7744\):
\[ 7744 = 88 \times 88 \]
So, we can rewrite:
\[ \frac{3007 \times 88}{7744 \times 97} = \frac{3007 \times 88}{88 \times 88 \times 97} = \frac{3007}{88 \times 97} \]

### Step 5: Calculate the final answer
Now we need to calculate \(88 \times 97\):
\[ 88 \times 97 = 8536 \]

Thus, the final answer is:
\[ \frac{3007}{8536} \]

### Final Answer
The simplified result of the expression is:
\[ \frac{31}{88} \]