If `|{:(a,5x,p),(b,10y,5),(c,15z,15):}|` = 125, then find the value of `|{:(3a,3b,c),(x,2y,z),(p,5,5):}|`

Options

A. 25
B. 125
C. 5
D. 10

Answer

To solve the problem, we need to find the value of the determinant \( |{:(3a, 3b, c), (x, 2y, z), (p, 5, 5):}| \) given that \( |{:(a, 5x, p), (b, 10y, 5), (c, 15z, 15):}| = 125 \).

### Step 1: Understanding the Determinant Properties
We know that if we multiply a row of a determinant by a scalar \( k \), the value of the determinant is also multiplied by \( k \).

### Step 2: Applying Row Multiplication
In our case, we can express the new determinant in terms of the original determinant:
\[ |{:(3a, 3b, c), (x, 2y, z), (p, 5, 5):}| = 3 |{:(a, b, c), (x, 2y, z), (p, 5, 5):}| \]
This is because the first row has been multiplied by 3.

### Step 3: Factor Out the Scalar from the Second Row
Next, we can factor out the 2 from the second row:
\[ = 3 \cdot 2 |{:(a, b, c), (x, y, z), (p, 5, 5):}| = 6 |{:(a, b, c), (x, y, z), (p, 5, 5):}| \]

### Step 4: Rearranging the Original Determinant
Now, we need to relate \( |{:(a, 5x, p), (b, 10y, 5), (c, 15z, 15):}| \) to \( |{:(a, b, c), (x, y, z), (p, 5, 5):}| \).

We can express the original determinant as:
\[ |{:(a, 5x, p), (b, 10y, 5), (c, 15z, 15):}| = 5 \cdot 2 \cdot 3 |{:(a, b, c), (x, y, z), (p, 5, 5):}| = 30 |{:(a, b, c), (x, y, z), (p, 5, 5):}| \]

### Step 5: Setting Up the Equation
Given that this determinant equals 125, we can write:
\[ 30 |{:(a, b, c), (x, y, z), (p, 5, 5):}| = 125 \]
Thus,
\[ |{:(a, b, c), (x, y, z), (p, 5, 5):}| = \frac{125}{30} = \frac{25}{6} \]

### Step 6: Final Calculation
Now substituting back into our expression for the new determinant:
\[ |{:(3a, 3b, c), (x, 2y, z), (p, 5, 5):}| = 6 \cdot |{:(a, b, c), (x, y, z), (p, 5, 5):}| = 6 \cdot \frac{25}{6} = 25 \]

### Final Answer
Thus, the value of \( |{:(3a, 3b, c), (x, 2y, z), (p, 5, 5):}| \) is \( \boxed{25} \).