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.

Refer to the manufacturing dataset provided to answer the question.

The premium-grade laptops produced at Factory Y represent what percentage of the standard-grade laptops produced at Factory X?

The correct option is 60%.

Let us break down the problem step-by-step to find the required percentage.

Step 1: Understand the factory production ratio and percentage components.

Let the total number of laptops produced by Factory X be $T_X$ and by Factory Y be $T_Y$.

Given the ratio of total laptops produced by Factory X to Factory Y is 5 : 3:

$\frac{T_X}{T_Y}=\frac{5}{3}$

Let $T_X=50k$ and $T_Y=30k$ for some constant $k$.

At Factory Y, standard-grade laptops account for 40% of the total units manufactured.

Therefore, premium-grade laptops at Factory Y account for the remaining 60% of Factory Y's total production:

$P_Y=60\%\text{ of }{T}_Y=0.60\times 30k=18k$

Step 2: Determine 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 ($P_Y$).

Note that:

$11\frac{1}{9}\%=\frac{100}{9}\%=\frac{1}{9}$

Thus,

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

Substituting $P_Y=18k$:

$P_X=\frac{10}{9}\times 18k=20k$

Step 3: Determine standard-grade laptops produced at Factory X.

Since total laptops at Factory X ($T_X$) consist of premium-grade ($P_X$) and standard-grade ($S_X$) laptops:

$S_X=T_X-P_X=50k-20k=30k$

Step 4: Calculate the required percentage.

We need to find what percentage $P_Y$ (premium-grade laptops at Factory Y) is of $S_X$ (standard-grade laptops at Factory X):

$\text{Required Percentage}=\left(\frac{P_Y}{S_X}\right)\times 100\%$

Substituting $P_Y=18k$ and $S_X=30k$:

$\text{Required Percentage}=\left(\frac{18k}{30k}\right)\times 100\%=\frac{3}{5}\times 100\%=60\%$

Thus, the premium-grade laptops produced at Factory Y represent 60% of the standard-grade laptops produced at Factory X.