Jim had (5w + 10) marbles. The number of marbles Joe had was 1/5 of what Jim had and Mike had (3w + 12) marbles. How many marbles, in terms of w, did each of them have if they shared the marbles equally?
The number of marbles Joe had was 1/5 of what Jim had, which is (1/5)(5w + 10) = w + 2.
Together, Jim and Joe had (5w + 10) + (w + 2) = 6w + 12 marbles.
If Mike had (3w + 12) marbles, then the total number of marbles was (6w + 12) + (3w + 12) = 9w + 24.
If they shared the marbles equally, each of them would have (9w + 24)/(3) = 3w + 8 marbles.
Therefore, Jim had 5w + 10 = (3w + 8) - (w + 2) marbles.
Joe had w + 2 = (3w + 8)/(5) marbles.
Mike had 3w + 12 = (3w + 8) + 4 marbles.
To find out how many marbles, in terms of w, each person had when they shared the marbles equally, we need to equalize the number of marbles each person had. We know that Jim had (5w + 10) marbles and Joe had 1/5 of what Jim had.
Let's solve this step-by-step:
Step 1: Find the number of marbles Joe had:
Joe had 1/5 of what Jim had, so the number of marbles Joe had = 1/5 * (5w + 10) = (5w + 10)/5 = w + 2
Step 2: Find the equal number of marbles they all had:
To find the equal number of marbles they all had, we need to calculate the least common multiple (LCM) of (5w + 10) and (3w + 12).
Step 2.1: Factorize the expressions (5w + 10) and (3w + 12):
(5w + 10) = 5(w + 2)
(3w + 12) = 3(w + 4)
Step 2.2: Calculate the LCM:
The LCM of 5(w + 2) and 3(w + 4) will be the product of the highest power of each prime factor, which is w in this case. So, the LCM = 5 * 3 * w * (w + 2) = 15w(w + 2)
Step 3: Divide the LCM by the total number of people (Jim, Joe, and Mike) to find out how many marbles each person had:
Each person had (15w(w + 2) / 3) = 5w(w + 2) marbles.
Therefore, in terms of w, Jim had 5w(w + 2) marbles, Joe had w + 2 marbles, and Mike had 5w(w + 2) marbles.