if `P:Q:R=2:3:4 and P^(2)+Q^(2)+R^(2)=11600`, then find P+Q-R, where P,Q and R are whole numbers.

Options

A. 15
B. 16
C. 18
D. 20

Explanation

To solve the problem step by step, we start with the given ratios and the equation involving the squares of P, Q, and R.

### Step 1: Express P, Q, and R in terms of a variable
Given the ratio \( P:Q:R = 2:3:4 \), we can express P, Q, and R in terms of a common variable \( x \):
- \( P = 2x \)
- \( Q = 3x \)
- \( R = 4x \)

### Step 2: Substitute P, Q, and R into the equation
We are given that:
\[ P^2 + Q^2 + R^2 = 11600 \]
Substituting the expressions for P, Q, and R:
\[ (2x)^2 + (3x)^2 + (4x)^2 = 11600 \]

### Step 3: Simplify the equation
Calculating the squares:
- \( (2x)^2 = 4x^2 \)
- \( (3x)^2 = 9x^2 \)
- \( (4x)^2 = 16x^2 \)

Now, substituting these back into the equation:
\[ 4x^2 + 9x^2 + 16x^2 = 11600 \]

### Step 4: Combine like terms
Combining the terms on the left side:
\[ (4 + 9 + 16)x^2 = 11600 \]
\[ 29x^2 = 11600 \]

### Step 5: Solve for x^2
Now, divide both sides by 29:
\[ x^2 = \frac{11600}{29} \]
Calculating the right side:
\[ x^2 = 400 \]

### Step 6: Solve for x
Taking the square root of both sides:
\[ x = \sqrt{400} \]
\[ x = 20 \]

### Step 7: Find P, Q, and R
Now, substitute \( x \) back to find P, Q, and R:
- \( P = 2x = 2 \times 20 = 40 \)
- \( Q = 3x = 3 \times 20 = 60 \)
- \( R = 4x = 4 \times 20 = 80 \)

### Step 8: Calculate P + Q - R
Now we can find \( P + Q - R \):
\[ P + Q - R = 40 + 60 - 80 \]
\[ P + Q - R = 100 - 80 \]
\[ P + Q - R = 20 \]

### Final Answer
Thus, the value of \( P + Q - R \) is **20**.

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