Given that `ab = 12 and c/a + 10 = 15`, what is bc?

Answer

To solve the problem step by step, we start with the given equations:

1. **Given equations:**
- \( ab = 12 \) (Equation 1)
- \( \frac{c}{a} + 10 = 15 \) (Equation 2)

2. **Rearranging Equation 2:**
- We can isolate \( \frac{c}{a} \) by subtracting 10 from both sides:
\[ \frac{c}{a} = 15 - 10 \]
\[ \frac{c}{a} = 5 \]

3. **Expressing \( c \) in terms of \( a \):**
- We can multiply both sides of the equation by \( a \) to express \( c \):
\[ c = 5a \]

4. **Substituting \( c \) into the product \( bc \):**
- We need to find \( bc \). From Equation 1, we know \( ab = 12 \), so we can express \( b \) in terms of \( a \):
\[ b = \frac{12}{a} \]

5. **Now substituting \( b \) and \( c \) into \( bc \):**
- We can now substitute \( b \) and \( c \) into the expression \( bc \):
\[ bc = b \cdot c = \left(\frac{12}{a}\right) \cdot (5a) \]

6. **Simplifying the expression:**
- When we multiply, the \( a \) in the numerator and denominator cancels out:
\[ bc = \frac{12 \cdot 5a}{a} = 12 \cdot 5 = 60 \]

7. **Final answer:**
- Therefore, the value of \( bc \) is:
\[ \boxed{60} \]