You have 72 books and 90 DVD's to pack. Your client insists that the books and DVD's have to have the same number of items in them.
a)What are the possible numbers of items in each box?
b)If you could make the most profit by packing the fewest number of boxes, how many boxes would you need for books? How many for DVD's?
To determine the possible numbers of items in each box, we need to find the common factors of 72 and 90. The factors of a number are the numbers that divide evenly into it without leaving any remainder.
a) Finding the common factors of 72 and 90:
- The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
- The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
The common factors of 72 and 90 are 1, 2, 3, 6, 9, 18. These are the possible numbers of items that can be packed in each box to meet the client's requirement.
b) To find the fewest number of boxes needed for books and DVDs, we can consider the highest common factor of 72 and 90, which is 18.
For books:
- Divide the total number of books (72) by the common factor (18).
- The result is 4, meaning you can pack 4 boxes of books with 18 items in each box.
For DVDs:
- Divide the total number of DVDs (90) by the common factor (18).
- The result is 5, meaning you can pack 5 boxes of DVDs with 18 items in each box.
Therefore, to make the most profit by using the fewest number of boxes, you would need 4 boxes for books and 5 boxes for DVDs.