Question

A dentist is making identical dental care bags for patients using 56 tubes of toothpaste, 112 packets of floss, and 85 toothbrushes. What is the greatest number of identical dental care bags the dentist can make with the least amount of items leftover? Describe the contents of each bag.

Answer 1

Answer : Greatest number of identical dental care bags the dentist can make with the least items leftover is 1.

Number of bags will be 2,4,3 respectively.

Explanation:

Number of tubes of toothpaste = 56

Number of packets of floss = 112

Number of toothbrushes = 85

So, Greatest number of identical dental care bags the dentist can make with the least amount of items leftover from toothbrush  = H.C.F of 56,112, 85=   1   ( ∵ as they are coprime to each other )

Since there is no common factor among 56,112,85 but there is common factor between 56 and 112 i.e. 56 remaining bags from toothbrushes will be 85-56=29 .

So we consider 28 ,

Now, 56 can be written as 28×2

112 can be written as  28×4 and

84 can be written as 28× 3

So, there can be 28 bags in which all the three materials can be present  and there is no leftover from toothpaste and floss but there is leftover from toothbrushes only as we consider 84 instead of 85 to get divisible by 28 so we get the common factor easily.

so, there will be 2 two bags for toothpastes and 4 bags for floss and 3 bags for toothbrushes.