What should come in the place of question mark (?) in the following questions ?
If x% of `500 =y%` of 300 and x% of y% of 200=60, then x= ?

Options

A. `10sqrt2`
B. `20sqrt2`
C. `15 sqrt2`
D. `30 sqrt2`

Answer

To solve the problem step by step, we will follow the mathematical relationships given in the question.

### Step 1: Set up the equations
We start with the first equation from the problem statement:
\[ x\% \text{ of } 500 = y\% \text{ of } 300 \]

This can be rewritten as:
\[ \frac{x}{100} \times 500 = \frac{y}{100} \times 300 \]

### Step 2: Simplify the equation
Cancelling the common factor of 100 from both sides, we get:
\[ 5x = 3y \]

### Step 3: Express y in terms of x
From the equation \( 5x = 3y \), we can express \( y \) in terms of \( x \):
\[ y = \frac{5x}{3} \]

### Step 4: Set up the second equation
Now, we use the second part of the problem:
\[ x\% \text{ of } y\% \text{ of } 200 = 60 \]

This can be rewritten as:
\[ \frac{x}{100} \times \frac{y}{100} \times 200 = 60 \]

### Step 5: Simplify the second equation
Cancelling the common factors, we have:
\[ \frac{xy}{10000} \times 200 = 60 \]
\[ \frac{xy}{50} = 60 \]

### Step 6: Multiply both sides by 50
To eliminate the fraction, multiply both sides by 50:
\[ xy = 3000 \]

### Step 7: Substitute y in terms of x
Now, substitute \( y \) from Step 3 into this equation:
\[ x \left(\frac{5x}{3}\right) = 3000 \]

### Step 8: Simplify the equation
This simplifies to:
\[ \frac{5x^2}{3} = 3000 \]

### Step 9: Multiply both sides by 3
To eliminate the fraction, multiply both sides by 3:
\[ 5x^2 = 9000 \]

### Step 10: Divide by 5
Now, divide both sides by 5:
\[ x^2 = 1800 \]

### Step 11: Take the square root
Taking the square root of both sides gives:
\[ x = \sqrt{1800} \]
\[ x = \sqrt{36 \times 50} \]
\[ x = 6\sqrt{50} \]
\[ x = 6 \times 5\sqrt{2} \]
\[ x = 30\sqrt{2} \]

### Final Answer
Thus, the value of \( x \) is \( 30\sqrt{2} \).

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