Question: When one number is increased by $6$ and the other decreased by $2$, the change in product is
a) $6b - 2a$
b) $6b - 2a - 12$
c) $ab - 12$
d) $6a - 2b$
1. **State the problem:**
We have two numbers, say $a$ and $b$. One number is increased by $6$, so it becomes $a + 6$. The other is decreased by $2$, so it becomes $b - 2$.
2. **Find the original product:**
The original product of the two numbers is $$ab$$.
3. **Find the new product after changes:**
The new product after increasing one number by $6$ and decreasing the other by $2$ is:
$$ (a + 6)(b - 2) $$
4. **Expand the new product:**
$$ (a + 6)(b - 2) = ab - 2a + 6b - 12 $$
5. **Calculate the change in product:**
Change in product = New product $-$ Original product
$$ = (ab - 2a + 6b - 12) - ab $$
$$ = ab - 2a + 6b - 12 - ab $$
$$ = -2a + 6b - 12 $$
6. **Rewrite the change in product:**
$$ 6b - 2a - 12 $$
7. **Conclusion:**
The change in product is $$6b - 2a - 12$$, which corresponds to option (b).