An animal shelter has 24 large cages and 42 small cages that need to be cleaned. The director of the shelter wants to divide the work so that each person cleans the same number of large cages and the same number of small cages. What is the largest number of people he can use to do the work and how many large and small cages will each person clean? explain your reasoning
24 = 6*4
42 = 6*7
So, 6 people each clean 4 small and 7 large cages. No other larger number divides both 24 and 42, since 4 and 7 have no common factors.
BIG E
To find the largest number of people the director can use to clean the cages, we need to find the greatest common divisor (GCD) of the number of large cages and the number of small cages. The GCD represents the maximum number of cages that can be evenly divided between a certain number of people.
In this case, the number of large cages is 24, and the number of small cages is 42. To find the GCD, we can list out the factors of each number and find the largest common factor.
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.
The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, and 42.
From the list above, we can see that the largest common factor (GCD) between 24 and 42 is 6.
So, the director can use a maximum of 6 people to clean the cages. Now let's determine how many large and small cages each person will clean.
To do this, we divide the total number of cages by the number of people.
For the large cages:
24 (total number of large cages) / 6 (number of people) = 4 large cages per person.
For the small cages:
42 (total number of small cages) / 6 (number of people) = 7 small cages per person.
Therefore, the largest number of people the director can use to do the work is 6, and each person will clean 4 large cages and 7 small cages.