If 2A = 3B and 3B = 2C, then what is `A:B: C`?

Options

A. `3 : 2 : 3 `
B. `2 : 3 : 2`
C. `1 : 3 : 1`
D. `2 : 3 : 2`

Answer

To solve the problem where \( 2A = 3B \) and \( 3B = 2C \), we need to express the ratios \( A:B:C \). Here’s how we can do it step by step:

### Step 1: Express \( A \) in terms of \( B \)
From the equation \( 2A = 3B \), we can isolate \( A \):
\[ A = \frac{3B}{2} \]

**Hint:** To isolate a variable, divide both sides of the equation by the coefficient of that variable.

### Step 2: Express \( B \) in terms of \( C \)
From the equation \( 3B = 2C \), we can isolate \( B \):
\[ B = \frac{2C}{3} \]

**Hint:** Similar to the first step, isolate \( B \) by dividing both sides by the coefficient of \( B \).

### Step 3: Substitute \( B \) in the expression for \( A \)
Now, we substitute \( B \) from Step 2 into the expression for \( A \):
\[ A = \frac{3 \left(\frac{2C}{3}\right)}{2} \]
This simplifies to:
\[ A = \frac{2C}{2} = C \]

**Hint:** When substituting, make sure to simplify the expression carefully.

### Step 4: Write the ratios \( A:B:C \)
Now we have:
- \( A = C \)
- \( B = \frac{2C}{3} \)

To express \( A:B:C \), we can substitute \( C \) for \( A \):
\[ A:B:C = C : \frac{2C}{3} : C \]

### Step 5: Remove the common factor \( C \)
Since \( C \) is common in all terms, we can simplify the ratio:
\[ A:B:C = 1 : \frac{2}{3} : 1 \]

To express this in whole numbers, we can multiply all parts by 3 to eliminate the fraction:
\[ A:B:C = 3 : 2 : 3 \]

### Final Answer
Thus, the ratio \( A:B:C \) is:
\[ \boxed{3:2:3} \]