Analyze the manufacturing dataset below and answer the following question.

The ratio of total laptops produced by Factory X to Factory Y is 5:3, and standard-grade laptops account for 40% of the total units manufactured at Factory Y. The quantity of premium-grade laptops produced at Factory X is $11\frac{1}{9}$ % more than those produced at Factory Y, and the average quantity of premium-grade laptops across both factories is 19.

Calculate the ratio of premium-grade laptops produced at Factory X to standard-grade laptops produced at Factory Y.

The correct option is 5:3.

Step 1: Define variables for total production at each factory.

Let the total number of laptops produced by Factory X be $T_X$ and by Factory Y be $T_Y$.
We are given that the ratio of total laptops produced by Factory X to Factory Y is 5:3.

So, we can write:

$T_X=5k$

$T_Y=3k$

where $k$ is a common multiplier.

Step 2: Find standard-grade laptops produced at Factory Y.

Standard-grade laptops account for 40% of the total units manufactured at Factory Y.
Therefore, the quantity of standard-grade laptops produced at Factory Y ($S_Y$) is:

$S_Y=40\%\times T_Y=0.4\times 3k=1.2k$

Step 3: Find premium-grade laptops produced at Factory Y.

Since laptops are categorized into standard-grade and premium-grade, the remaining 60% of Factory Y's production consists of premium-grade laptops ($P_Y$):

$P_Y=60\%\times T_Y=0.6\times 3k=1.8k$

Step 4: Find premium-grade laptops produced at Factory X.

The quantity of premium-grade laptops produced at Factory X ($P_X$) is $11\frac{1}{9}\%$ more than those produced at Factory Y.
Convert the mixed fraction to an improper fraction: $11\frac{1}{9}\%=\frac{100}{9}\%=\frac{1}{9}$.

So, $P_X$ is greater than $P_Y$ by $\frac{1}{9}$ of $P_Y$:

$P_X=P_Y+\frac{1}{9}{P}_Y=\frac{10}{9}{P}_Y$

Substitute $P_Y=1.8k$:

$P_X=\frac{10}{9}\times 1.8k=2k$

Step 5: Calculate the required ratio.

We need to find the ratio of premium-grade laptops produced at Factory X ($P_X$) to standard-grade laptops produced at Factory Y ($S_Y$):

$\text{Ratio}=\frac{P_X}{S_Y}=\frac{2k}{1.2k}=\frac{2}{1.2}=\frac{20}{12}=\frac{5}{3}$

Thus, the required ratio is 5:3.