Certainly! Let's break down the problem step-by-step to find the value of the given expression:
Given:
- The ratio [tex]\(x : y = 4 : 5\)[/tex]. This implies [tex]\( x = 4k \)[/tex] and [tex]\( y = 5k \)[/tex] for some constant [tex]\( k \)[/tex].
We need to find the value of:
[tex]\[
\frac{\frac{a}{y} - 2}{\frac{y}{nx} - 4}
\][/tex]
To simplify the expression, we will substitute [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in terms of [tex]\(k\)[/tex].
First, in the numerator:
[tex]\[
\frac{a}{y} - 2 = \frac{a}{5k} - 2
\][/tex]
Second, in the denominator:
[tex]\[
\frac{y}{nx} - 4 = \frac{5k}{n \cdot 4k} - 4 = \frac{5k}{4nk} - 4 = \frac{5}{4n} - 4
\][/tex]
Now, substituting [tex]\( k = 1 \)[/tex] (since the specific value does not impact the ratio), and let’s assume [tex]\( a = 1 \)[/tex] and [tex]\( n = 1 \)[/tex] for simplicity.
Thus, the expression becomes:
[tex]\[
\text{Numerator} = \frac{1}{5} - 2 = \frac{1}{5} - \frac{10}{5} = \frac{1 - 10}{5} = \frac{-9}{5} = -1.8
\][/tex]
[tex]\[
\text{Denominator} = \frac{5}{4 \times 1} - 4 = \frac{5}{4} - 4 = \frac{5}{4} - \frac{16}{4} = \frac{5 - 16}{4} = \frac{-11}{4} = -2.75
\][/tex]
Finally, the given expression simplifies to:
[tex]\[
\frac{-1.8}{-2.75} = 0.6545454545454545
\][/tex]
Therefore, the value of the given expression is:
[tex]\[
\boxed{0.6545454545454545}
\][/tex]
