How to know how many of each number are present

[IBstudent]

Hello.

I am having a hard time solving questions that go like:

A class of 60 pupils will be divided into 8 groups each consisting of 7 or 8 students.

How many groups contain 8 students?


Note: I am not sure of I wrote this question correctly, I just want to know the way in which to solve such problems.

Thanks in advance

 

[pka]

A class of 60 pupils will be divided into 8 groups each consisting of 7 or 8 students.
How many groups contain 8 students?

Solve \(\displaystyle 7m+8n=60\) where \(\displaystyle m~\&~n\) are positive integers.
Then \(\displaystyle n\) is your answer.

 

[IBstudent]

Solve \(\displaystyle 7m+8n=60\) where \(\displaystyle m~\&~n\) are positive integers.
Then \(\displaystyle n\) is your answer.

Are you insinuating the I should use trial and error? I simply cant do that if the number is bigger... Isnt there a more systemmatic approach to such questions? Even with the use of GDCs?

 

[IBstudent]

\(\displaystyle \dfrac{60}{7} = 8 + \dfrac{4}{7}\)

If each group had 7, you would need more than 8 groups, and the ninth would have 4. Not a solution. Obviously at least 4 groups must have
7 + 1 = 8. So at most 4 groups must have 7.

4 groups of 7 and 4 groups of 8 = 4(7) + 4(8) = 28 + 32 = 60.

By the way, that is what pka hinted you should do.

Thanks... Very helpful

I guess he should have elucidated his method by being more explicit! But thanks alot to u all