Final answer:
Alice originally had 32 marbles, and after Karen gave her 12 more marbles, she ended up with a total of 76 marbles. Therefore, Karen gave Alice 44 marbles.
Step-by-step explanation:
The question involves solving a simple algebraic equation to find out how many marbles Karen gave to Alice.
Let's denote the number of marbles Alice originally had as x. According to the question, Karen gave Alice 12 more marbles than she already had, which means Karen gave her x + 12 marbles. After receiving the marbles from Karen, Alice now has a total of 76 marbles.
So, we can set up the following equation to represent the situation:
x + (x + 12) = 76
Combining like terms, we get:
2x + 12 = 76
Subtracting 12 from both sides of the equation gives us:
2x = 64
Dividing both sides by 2 yields:
x = 32
This means Alice originally had 32 marbles. To find the number of marbles Karen gave to Alice, we add 12 to Alice's original number of marbles:
32 + 12 = 44
Karen gave Alice 44 marbles.