If `(a)/(b)=(4)/(5) and (b)/(c)=(15)/(16)` then `(18c^(2)-7a^(2))/(45c^(2)+20a^(2))` is equal to
यदि `(a)/(b)=(4)/(5)` और `(b)/(c)=(15)/(16)` तो `(18c^(2)-7a^(2))/(45c^(2)+20a^(2))` का मान है -

Options

A. `(1)/(3)`
B. `(2)/(5)`
C. `(3)/(4)`
D. `(1)/(4)`

Explanation

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

1. \(\frac{a}{b} = \frac{4}{5}\)
2. \(\frac{b}{c} = \frac{15}{16}\)

### Step 1: Express \(a\) in terms of \(b\)
From the first ratio, we can express \(a\) as:
\[ a = \frac{4}{5}b \]

### Step 2: Express \(b\) in terms of \(c\)
From the second ratio, we can express \(b\) as:
\[ b = \frac{15}{16}c \]

### Step 3: Substitute \(b\) in the expression for \(a\)
Now, we substitute the expression for \(b\) into the expression for \(a\):
\[ a = \frac{4}{5} \left(\frac{15}{16}c\right) = \frac{4 \times 15}{5 \times 16}c = \frac{60}{80}c = \frac{3}{4}c \]

### Step 4: Substitute \(a\) and \(c\) into the expression
We need to evaluate the expression:
\[ \frac{18c^2 - 7a^2}{45c^2 + 20a^2} \]
Substituting \(a = \frac{3}{4}c\):
\[ a^2 = \left(\frac{3}{4}c\right)^2 = \frac{9}{16}c^2 \]

### Step 5: Substitute \(a^2\) into the expression
Now we substitute \(a^2\) into the expression:
\[ \frac{18c^2 - 7\left(\frac{9}{16}c^2\right)}{45c^2 + 20\left(\frac{9}{16}c^2\right)} \]
This simplifies to:
\[ \frac{18c^2 - \frac{63}{16}c^2}{45c^2 + \frac{180}{16}c^2} \]

### Step 6: Find a common denominator
The common denominator for the terms in the numerator and denominator is 16:
\[ = \frac{\frac{288}{16}c^2 - \frac{63}{16}c^2}{\frac{720}{16}c^2 + \frac{180}{16}c^2} \]
This simplifies to:
\[ = \frac{\frac{225}{16}c^2}{\frac{900}{16}c^2} \]

### Step 7: Simplify the expression
The \(c^2\) cancels out:
\[ = \frac{225}{900} \]
This simplifies to:
\[ = \frac{1}{4} \]

### Final Answer
Thus, the value of the expression \(\frac{18c^2 - 7a^2}{45c^2 + 20a^2}\) is:
\[ \frac{1}{4} \]