A varies directly as B and inversely as C. A is 12 when B is 6 and C is 2. What is the value of A when B is 12 and C is 3?

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the relationship
We know that A varies directly as B and inversely as C. This can be expressed mathematically as:
\[ A = k \cdot \frac{B}{C} \]
where \( k \) is a constant of proportionality.

### Step 2: Find the constant \( k \)
We are given that \( A = 12 \) when \( B = 6 \) and \( C = 2 \). We can substitute these values into the equation to find \( k \):
\[ 12 = k \cdot \frac{6}{2} \]

### Step 3: Simplify the equation
Now, simplify the fraction:
\[ \frac{6}{2} = 3 \]
So the equation becomes:
\[ 12 = k \cdot 3 \]

### Step 4: Solve for \( k \)
To find \( k \), divide both sides by 3:
\[ k = \frac{12}{3} = 4 \]

### Step 5: Use \( k \) to find \( A \) for new values of \( B \) and \( C \)
Now we need to find the value of \( A \) when \( B = 12 \) and \( C = 3 \). Substitute \( k = 4 \), \( B = 12 \), and \( C = 3 \) into the equation:
\[ A = 4 \cdot \frac{12}{3} \]

### Step 6: Simplify the expression
Calculate the fraction:
\[ \frac{12}{3} = 4 \]
So the equation becomes:
\[ A = 4 \cdot 4 \]

### Step 7: Calculate the final value of \( A \)
Now multiply:
\[ A = 16 \]

Thus, the value of \( A \) when \( B = 12 \) and \( C = 3 \) is **16**.

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