So, Lena sent 24 messages, Alan sent 32 messages, and Bill sent 48 messages.
Let's denote the number of messages Lena, Alan, and Bill sent as L, A, and B, respectively. We are given the following information:
L + A + B = 104 (Total messages)
L = A - 8 (Lena sent 8 fewer messages than Alan)
B = 2L (Bill sent 2 times as many messages as Lena)
Now, we'll use the second equation to express A in terms of L:
A = L + 8
Next, substitute the expressions for A and B from equations 2 and 3 into equation 1:
L + (L + 8) + 2L = 104
Combine like terms:
4L + 8 = 104
Subtract 8 from both sides:
4L = 96
Divide by 4:
L = 24
Now that we have the number of messages Lena sent, we can find the number of messages Alan and Bill sent:
A = L + 8 = 24 + 8 = 32
B = 2L = 2 * 24 = 48