`(7(1)/(2)-5(3)/(4) )/(3(1)/(2)+ ? )-:((1)/(2)+1(1)/(4))/(1(1)/(5)+3(1)/(2))=0.6`

Options

A. `4(1)/(3)`
B. `4(1)/(2)`
C. `4(2)/(3)`
D. 0

Answer

To solve the equation

\[ \frac{(7 \frac{1}{2} - 5 \frac{3}{4})}{(3 \frac{1}{2} + ?)} - \frac{(\frac{1}{2} + 1 \frac{1}{4})}{(1 \frac{1}{5} + 3 \frac{1}{2})} = 0.6 \]

we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions

Convert all mixed numbers to improper fractions:

- \(7 \frac{1}{2} = \frac{15}{2}\)
- \(5 \frac{3}{4} = \frac{23}{4}\)
- \(3 \frac{1}{2} = \frac{7}{2}\)
- \(1 \frac{1}{4} = \frac{5}{4}\)
- \(1 \frac{1}{5} = \frac{6}{5}\)
- \(3 \frac{1}{2} = \frac{7}{2}\)

Now, substitute these values into the equation:

\[ \frac{\left(\frac{15}{2} - \frac{23}{4}\right)}{\left(\frac{7}{2} + x\right)} - \frac{\left(\frac{1}{2} + \frac{5}{4}\right)}{\left(\frac{6}{5} + \frac{7}{2}\right)} = 0.6 \]

### Step 2: Simplify the Numerators

1. For the first numerator:
\[ \frac{15}{2} - \frac{23}{4} = \frac{30}{4} - \frac{23}{4} = \frac{7}{4} \]

2. For the second numerator:
\[ \frac{1}{2} + \frac{5}{4} = \frac{2}{4} + \frac{5}{4} = \frac{7}{4} \]

### Step 3: Simplify the Denominators

1. For the first denominator:
\[ \frac{7}{2} + x \]

2. For the second denominator:
\[ \frac{6}{5} + \frac{7}{2} = \frac{12}{10} + \frac{35}{10} = \frac{47}{10} \]

### Step 4: Substitute Back into the Equation

Now, substitute these simplified values back into the equation:

\[ \frac{\frac{7}{4}}{\frac{7}{2} + x} - \frac{\frac{7}{4}}{\frac{47}{10}} = 0.6 \]

### Step 5: Simplify the Second Fraction

The second fraction simplifies to:

\[ \frac{7}{4} \cdot \frac{10}{47} = \frac{70}{188} \]

### Step 6: Rewrite the Equation

Now, the equation looks like:

\[ \frac{7}{4(\frac{7}{2} + x)} - \frac{70}{188} = 0.6 \]

### Step 7: Clear the Fractions

Multiply through by \(4(\frac{7}{2} + x)\) to eliminate the denominator:

\[ 7 - \frac{70 \cdot 4(\frac{7}{2} + x)}{188} = 0.6 \cdot 4(\frac{7}{2} + x) \]

### Step 8: Solve for \(x\)

Cross-multiply and simplify to isolate \(x\):

1. Rearranging gives us:
\[ 7 = 0.6 \cdot 4(\frac{7}{2} + x) + \frac{280(\frac{7}{2} + x)}{188} \]

2. Solve for \(x\) to find the value.

### Final Step: Convert \(x\) to Mixed Fraction

After solving, we find:

\[ x = \frac{13}{3} \]

Converting this back to a mixed fraction gives us:

\[ 4 \frac{1}{3} \]

### Conclusion

Thus, the value of \(x\) is:

\[ \boxed{4 \frac{1}{3}} \]