6. Toys were packed into x boxes so that each box contained the same number of toys, with no toys left unpacked. If 3 fewer boxes had been used instead, then 12 toys would have been packed in each box, with 5 toys left unpacked. What is the value of x?
Answer: C
The value of x is 28.
The problem states that if 3 fewer boxes were used, then each box would contain 12 toys with 5 toys left unpacked. This information allows us to set up an equation that leads us to determine that the original number of boxes, x, is 28.
A) 11
Choosing 11 as the value of x would imply that when 3 fewer boxes are used (which would be 8 boxes), each box would have to contain a total of 12 toys. This would only account for 96 toys total, but with 5 toys left unpacked, it contradicts the initial condition of no toys left unpacked, making this option incorrect.
B) 14
If x is 14, then using 3 fewer boxes would mean 11 boxes. With 12 toys per box, this would total 132 toys packed, but there would still be 5 toys left unpacked. This situation is impossible since it implies the existence of toys that are not accounted for, thereby ruling out this option.
C) 28
With 28 boxes, if we reduce the number of boxes by 3, there would be 25 boxes. If each of these 25 boxes contains 12 toys, this accounts for 300 toys, leaving 5 toys unpacked. Hence, the total number of toys is 305, confirming that 28 is indeed correct, as it satisfies all conditions of the problem.
D) 31
If x is 31, then using 3 fewer boxes would mean 28 boxes. Each of these boxes would need to contain 12 toys, which totals 336 toys. However, if there were 5 toys left unpacked, the total would need to be 341, which contradicts the original scenario since it does not align with the required conditions given in the problem.
E) 34
Choosing 34 as the value of x would mean that 31 boxes are used when 3 fewer boxes are considered. Each box would have to contain 12 toys, leading to a total of 372 toys packed. With 5 left unpacked, it would mean there are 377 toys in total, which again contradicts the problem's conditions, thus rendering this option invalid.
Conclusion
The analysis shows that the only value of x that meets all the criteria outlined in the problem is 28. All other options either result in discrepancies in the toy count or do not satisfy the conditions of having no toys left unpacked originally.