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 solve this problem step-by-step by breaking down the logical reasoning and mathematical calculations.

Step 1: Understand the given information

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

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

3. Therefore, premium-grade laptops at Factory Y account for 100% - 40% = 60% of Factory Y's total production.

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

5. The average quantity of premium-grade laptops across both factories is 19.

Step 2: Express variables in terms of Factory Y's total production

Let total production of Factory Y = 300k units.

Since the ratio of production of X to Y is 5 : 3, total production of Factory X = 500k units.

For Factory Y:

- Standard-grade laptops at Y = 40% of 300k = 120k

- Premium-grade laptops at Y = 60% of 300k = 180k

For Factory X:

First, note that $11\frac{1}{9}\%=\frac{100}{9}\%=\frac{1}{9}$.

So, Premium-grade laptops at Factory X are $1+\frac{1}{9}=\frac{10}{9}$ of Premium-grade laptops at Factory Y.

Premium-grade laptops at X = $\frac{10}{9}\times 180k=200k$

Since total laptops produced at Factory X = 500k:

- Standard-grade laptops at X = 500k - 200k = 300k

Step 3: Solve for k using the average of premium-grade laptops

The average quantity of premium-grade laptops across both factories is 19.

Total premium-grade laptops = 19 × 2 = 38

$200k+180k=38$

$380k=38$

$k=0.1$

Now, let's find the exact quantities:

- Premium-grade laptops at Factory Y = 180k = 180 × 0.1 = 18

- Standard-grade laptops at Factory X = 300k = 300 × 0.1 = 30

Step 4: Calculate the required percentage

We need to find what percentage the premium-grade laptops produced at Factory Y (18) represent of the standard-grade laptops produced at Factory X (30):

$\text{Percentage}=\left(\frac{18}{30}\right)\times 100\%$

$\text{Percentage}=0.6\times 100\%=60\%$

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