If `(a)/(b) = (1)/(2),` find the value of the expression `((2a - 5b))/((5a + 3b)):`

Options

A. `-32 `
B. 11
C. `- (8)/(11)`
D. 17

Answer

To solve the problem, we start with the given ratio:

\[ \frac{a}{b} = \frac{1}{2} \]

From this, we can express \( a \) in terms of \( b \):

\[ a = \frac{1}{2}b \]

Now, we need to find the value of the expression:

\[ \frac{2a - 5b}{5a + 3b} \]

### Step 1: Substitute \( a \) in the expression

Substituting \( a = \frac{1}{2}b \) into the expression:

\[ 2a = 2 \left(\frac{1}{2}b\right) = b \]

Now, substituting \( a \) into the expression:

\[ 5a = 5 \left(\frac{1}{2}b\right) = \frac{5}{2}b \]

So, the expression becomes:

\[ \frac{b - 5b}{\frac{5}{2}b + 3b} \]

### Step 2: Simplify the numerator and denominator

Now simplify the numerator:

\[ b - 5b = -4b \]

And simplify the denominator:

\[ \frac{5}{2}b + 3b = \frac{5}{2}b + \frac{6}{2}b = \frac{11}{2}b \]

So the expression now looks like:

\[ \frac{-4b}{\frac{11}{2}b} \]

### Step 3: Cancel \( b \) from numerator and denominator

Since \( b \neq 0 \), we can cancel \( b \):

\[ \frac{-4}{\frac{11}{2}} = -4 \times \frac{2}{11} = \frac{-8}{11} \]

### Final Answer

Thus, the value of the expression is:

\[ \frac{-8}{11} \]

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