Question

greatest common factor - applications - item 992217
william has 24 cans of fruit and 60 cans of vegetables that he will be putting into bags for a food drive. he wants each bag to have the same number of cans of each type of food. he uses all the cans.
use the drop - down menus to complete the statements below about the number of bags of food william can make.
the greatest number of bags william can make is . each of these bags will have cans of fruit and cans of vegetables.
if he made fewer bags, .
he could use bags, but this is not the greatest number he could use.

Explanation:

Step1: Find GCF of 24 and 60

To find the greatest common factor (GCF) of 24 and 60, we can use prime factorization.
Prime factors of 24: $24 = 2^3 \times 3$
Prime factors of 60: $60 = 2^2 \times 3 \times 5$
The common prime factors are $2^2$ and $3$, so GCF = $2^2 \times 3 = 12$.

Step2: Calculate cans per bag

For fruit: $\frac{24}{12} = 2$ cans per bag.
For vegetables: $\frac{60}{12} = 5$ cans per bag.

Step3: Analyze fewer bags

Fewer bags mean a factor of 12 (since number of bags must divide 24 and 60). Factors of 12 (other than 12) are 1, 2, 3, 4, 6. Choosing a smaller factor (e.g., 6) would mean more cans per bag, but the key is the GCF gives the greatest number of bags. A non - GCF factor (like 6) is a valid number of bags but not the greatest.

Answer:

The greatest number of bags William can make is 12. Each of these bags will have 2 cans of fruit and 5 cans of vegetables.
If he made fewer bags, each bag would have more cans of each type of food.
He could use 6 bags (or other factors of 12 like 1, 2, 3, 4), but this is not the greatest number he could use. (For the last blank, a possible answer is 6, but other valid factors of 12 except 12 are also correct.)