Question

If 0 < a < b < c < d < e in the equation above, then the greatest increase in S would result from adding 1 to the value of which variable?

Answer 1

Equation?

Answer 2

s=a/b + c/d + 1/e

Answer 3

Increasing \(b\), \(d\), or \(e\) will decrease the value of \(S\), so you can ignore those.

Now using the fact that \(a<c\), what can you say about \(S\) when one of these increases? (Which results in a greater increase?)

Answer 4

a because c is a greater numerator?

Answer 5

@SithsAndGiggles

Answer 6

Right! If you're dealing with natural numbers, \(a\) is less than \(c\) by at least 2, since \(a<b<c\). For example, consider a=1, b=2, c=3, d=4:
\[\frac{1+1}{2}+\frac{3}{4}~\cdots~\frac{1}{2}+\frac{3+1}{4}\\
\frac{7}{4}{\bf>}\frac{6}{4}\]
The difference caused by a change in \(a\) is much larger than the difference caused by a change in \(c\).