Minimum Value Given Average

Question

I am struggling with this (very embarrassingly basic) minimization problem:

If $a \leqslant b \leqslant c \leqslant d \leqslant e \leqslant 110$, and the average (arithmetic mean) of $a,b,c,d,e$ is $100$, what is the least possible value of $a$?

Multiple Choices:

• $0$
• $20$
• $40$
• $60 \quad \bigstar$
• $80$

I have the answer, which is great, but I'm oblivious as to how to go about solving this short of using calculus. The creator of the question has strongly hinted that derivatives are not needed.

Thanks in advance for any and all assistance.

Answer 1

The basic idea is if you want the least value for $a$ you have to get the greatest value for $b$,...,$e$, so $b=c=d=e=110$. Then the average is $$\frac{a+4\cdot 110}{5}= 100$$.

Now you can solve this for $a$.

Answer 2

Because the arithmetic mean is $100$, the sum of the five numbers must be $500$. The largest that $b,c,d$, and $e$ can be is $110$, so they can contribute at most $440$ to the total; that leaves a minimum of $500-440=60$ that must come from $a$.