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.

Find the total number of standard-grade laptops produced by both factories combined.

The correct option is 42.

Step 1: Define the variables and express total laptop production.

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, we can represent total production in terms of a variable $k$:

$T_X=5k$

$T_Y=3k$

Step 2: Calculate standard-grade and premium-grade laptops for Factory Y.

Standard-grade laptops account for 40% of the total units manufactured at Factory Y.

Standard-grade laptops at Factory Y = $0.40\times 3k=1.2k$

Since each factory produces standard-grade and premium-grade laptops, the remaining 60% at Factory Y are premium-grade laptops:

Premium-grade laptops at Factory Y ($P_Y$) = $0.60\times 3k=1.8k$

Step 3: Calculate premium-grade laptops for Factory X.

The quantity of premium-grade laptops produced at Factory X is $11\frac{1}{9}\%$ more than those produced at Factory Y.

Converting the percentage to a fraction:

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

Therefore, premium-grade laptops at Factory X ($P_X$) = $P_Y+\frac{1}{9}{P}_Y=\frac{10}{9}{P}_Y$

Substituting $P_Y=1.8k$:

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

Step 4: Determine the value of $k$ using the average premium-grade quantity.

The average quantity of premium-grade laptops across both factories is given as 19:

$\frac{P_X+P_Y}{2}=19$

$P_X+P_Y=38$

Substituting the expressions for $P_X$ and $P_Y$:

$2k+1.8k=38$

$3.8k=38$

$k=10$

Step 5: Find the total number of standard-grade laptops produced by both factories combined.

First, find the total laptops produced at each factory:

Total laptops at Factory X ($T_X$) = $5\times 10=50$

Total laptops at Factory Y ($T_Y$) = $3\times 10=30$

Next, find the standard-grade laptops at each factory:

Standard-grade laptops at Factory X = $T_X-P_X=50-(2\times 10)=50-20=30$

Standard-grade laptops at Factory Y = $T_Y-P_Y=30-(1.8\times 10)=30-18=12$

Combining the standard-grade laptops from both factories:

Total standard-grade laptops = $30+12=42$

Thus, the total number of standard-grade laptops produced by both factories combined is 42.