To find the number of buttons in each of Jack's five bags, we can use the information given to us and solve for the unknowns. Let's start by finding the total number of buttons:
Total number of buttons = Mean number of buttons × Number of bags
Total number of buttons = 27 × 5
Total number of buttons = 135
Since we know the fewest and greatest number of buttons in a bag, we can find the range:
Range = Greatest number of buttons - Fewest number of buttons
Range = 29 - 21
Range = 8
Now, we can set up two equations to find the two modes:
Mode 1 + Mode 2 + 3 × 27 = 135 (sum of all bags)
Mode 2 - Mode 1 = 8 (difference between the two modes)
Solving these equations simultaneously, we get:
Mode 1 = 23
Mode 2 = 31
Therefore, the number of buttons in each of Jack's five bags is:
Bag 1: 21 buttons
Bag 2: 23 buttons
Bag 3: 25 buttons
Bag 4: 27 buttons
Bag 5: 29 buttons