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.

Correct Answer: The correct option is 5:3.

Step-by-step Solution:

Let us analyze the given information step-by-step to find the required ratio.

1. Expressing the Total Production of Laptops:

The ratio of total laptops produced by Factory X to Factory Y is 5:3.

Let the total number of laptops produced at Factory X be $5k$ and at Factory Y be $3k$, where $k$ is a constant multiplier.

2. Categorizing Factory Y's Production:

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

Standard-grade laptops at Factory Y = $40\%\text{of}3k=0.40\times 3k=1.2k$

Therefore, the remaining laptops at Factory Y must be premium-grade:

Premium-grade laptops at Factory Y = $3k-1.2k=1.8k$

3. Determining Premium-Grade Production at Factory X:

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

Note that $11\frac{1}{9}\%=\frac{100}{9}\%=\frac{1}{9}$.

Thus, Premium-grade laptops at Factory X = $\left(1+\frac{1}{9}\right)\times \text{Premium-grade laptops at Factory Y}$

Premium-grade laptops at Factory X = $\frac{10}{9}\times 1.8k=2k$

4. Calculating the Required Ratio:

We need to find the ratio of premium-grade laptops produced at Factory X to standard-grade laptops produced at Factory Y:

Required Ratio = $\frac{\text{Premium-grade laptops at Factory X}}{\text{Standard-grade laptops at Factory Y}}$

Required Ratio = $\frac{2k}{1.2k}=\frac{2}{1.2}=\frac{20}{12}=\frac{5}{3}$

Hence, the ratio is 5:3.