If `a /b = c/d = 3/2,` then ` (13 a - 111c)/(13 b - 111d ) = ?`

Options

A. `3/2`
B. `1/2`
C. `5/2`
D. `3/5`

Answer

To solve the problem step by step, we start with the given ratios and then manipulate the expressions accordingly.

Given:
\[ \frac{a}{b} = \frac{c}{d} = \frac{3}{2} \]

### Step 1: Express \(a\) and \(c\) in terms of \(b\) and \(d\)
From the ratio \(\frac{a}{b} = \frac{3}{2}\), we can express \(a\) as:
\[ a = \frac{3}{2}b \]

Similarly, from the ratio \(\frac{c}{d} = \frac{3}{2}\), we can express \(c\) as:
\[ c = \frac{3}{2}d \]

### Step 2: Substitute \(a\) and \(c\) into the expression
We need to evaluate:
\[ \frac{13a - 111c}{13b - 111d} \]

Substituting the expressions for \(a\) and \(c\):
\[ = \frac{13\left(\frac{3}{2}b\right) - 111\left(\frac{3}{2}d\right)}{13b - 111d} \]

### Step 3: Simplify the numerator
Now, simplify the numerator:
\[ = \frac{\frac{39}{2}b - \frac{333}{2}d}{13b - 111d} \]

### Step 4: Factor out \(\frac{1}{2}\) from the numerator
Factoring out \(\frac{1}{2}\) from the numerator:
\[ = \frac{\frac{1}{2}(39b - 333d)}{13b - 111d} \]

### Step 5: Simplify the expression
Now, we can cancel \(\frac{1}{2}\) from the numerator and denominator:
\[ = \frac{39b - 333d}{2(13b - 111d)} \]

### Step 6: Factor out common terms
Notice that \(39b - 333d\) can be factored:
\[ = \frac{3(13b - 111d)}{2(13b - 111d)} \]

### Step 7: Cancel out the common terms
Now, we can cancel \(13b - 111d\) from the numerator and denominator:
\[ = \frac{3}{2} \]

### Final Answer
Thus, the value of \(\frac{13a - 111c}{13b - 111d}\) is:
\[ \frac{3}{2} \]