If a:b=2:3, then the value of `(5a^(3)-2a^(2)b):(3ab^(2)-b^(3))` is :

Options

A. `16:27`
B. `32:29`
C. `34:19`
D. `27:16`

Answer

To solve the problem, we start with the given ratio \( a:b = 2:3 \). We need to find the value of the expression \( (5a^3 - 2a^2b):(3ab^2 - b^3) \).

### Step 1: Express \( a \) and \( b \) in terms of a common variable
From the ratio \( a:b = 2:3 \), we can express \( a \) and \( b \) as:
- Let \( a = 2k \)
- Let \( b = 3k \)

### Step 2: Substitute \( a \) and \( b \) into the expression
Now we substitute \( a \) and \( b \) into the expression \( (5a^3 - 2a^2b):(3ab^2 - b^3) \).

1. **Calculate \( 5a^3 \)**:
\[ 5a^3 = 5(2k)^3 = 5(8k^3) = 40k^3 \]

2. **Calculate \( 2a^2b \)**:
\[ 2a^2b = 2(2k)^2(3k) = 2(4k^2)(3k) = 24k^3 \]

3. **Calculate \( 3ab^2 \)**:
\[ 3ab^2 = 3(2k)(3k)^2 = 3(2k)(9k^2) = 54k^3 \]

4. **Calculate \( b^3 \)**:
\[ b^3 = (3k)^3 = 27k^3 \]

### Step 3: Substitute these values into the expression
Now we substitute these calculated values back into the expression:
\[ (5a^3 - 2a^2b) = 40k^3 - 24k^3 = 16k^3 \]
\[ (3ab^2 - b^3) = 54k^3 - 27k^3 = 27k^3 \]

### Step 4: Form the ratio
Now we have:
\[ (5a^3 - 2a^2b):(3ab^2 - b^3) = 16k^3 : 27k^3 \]

### Step 5: Simplify the ratio
We can simplify this ratio by canceling \( k^3 \):
\[ 16 : 27 \]

### Final Answer
Thus, the value of \( (5a^3 - 2a^2b):(3ab^2 - b^3) \) is \( 16 : 27 \).

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