A football coach divides 42 players and 12 coaches into groups. Each group will have the same number of players and coaches. What is the greatest number of groups that can be formed?
a. 12 groups
b. 6 groups
c. 2 groups
d. 4 groups
My answer is 6 groups because I found out what the greatest common factor is of 42 and 12 which is 6
Can you please double check my answer and let me know if I got it right.
Then the question asks to Use the greatest common factor from the problem above to rewrite the Distributive Property. What numbers am I supposed to use to do the distributive property?
Can you please explain?
Thank you
Yes, your answer of 6 groups is correct. The greatest common factor (GCF) of 42 and 12 is indeed 6. Therefore, you can divide both the players and coaches into groups of 6.
To rewrite the Distributive Property using the GCF, you would use the numbers 6 and the respective quantities of players and coaches. The Distributive Property states that for any numbers a, b, and c, a * (b + c) = a * b + a * c. In this case, you can use the GCF of 6 as the "a" value.
So, if you want to distribute the players and coaches evenly into groups of 6, the Distributive Property can be rewritten as:
6 * (Number of groups) = 6 * (Number of players in each group) + 6 * (Number of coaches in each group)
In this case, the number of groups would be the answer you obtained, which is 6. Let's say the number of players and coaches in each group is "x". The equation would then be:
6 * 6 = 6 * x + 6 * x
36 = 12x
Therefore, the number of players and coaches in each group would be 3, and there would be 6 groups in total.
Your answer of 6 groups is correct! Well done on finding the greatest common factor of 42 and 12, which is 6. Since you want to divide both the players and the coaches into groups of equal size, the maximum number of groups that can be formed is determined by the greatest common factor.
Now, let's explain the Distributive Property and how to apply it using the given greatest common factor. The Distributive Property states that when you have a common factor outside of a set of parentheses, you can distribute or multiply it to each term inside the parentheses.
In this case, since the groups are formed by dividing the players and coaches equally, the number of groups we can form is equal to the greatest common factor, which is 6. So, the Distributive Property can be applied using the number 6 as the common factor.
If we wanted to rewrite the Distributive Property using this common factor, it would look like:
6 × (a + b) = 6a + 6b
Here, the number 6 is being multiplied to each term inside the parentheses, which includes the variables 'a' and 'b'. By applying the Distributive Property, we distribute 6 to each term, resulting in 6a and 6b.
So, in summary, the greatest common factor of 6 can be used to rewrite the Distributive Property as 6 × (a + b) = 6a + 6b.