<s>[INST] Comparing $(2+\frac{1}{2})(3+\frac{1}{3})(4+\frac{1}{4})(5+\frac{1}{5})$ with $(2+\frac{1}{5})(3+\frac{1}{4})(4+\frac{1}{3})(5+\frac{1}{2})$ This question appeared in one of the national exams (MCQs) in Saudi Arabia.
In this exam;

*

*Using calculators is not allowed,

*The student have $72$ seconds on average to answer one question.


PROBLEM:
Compare $a=(2+\frac{1}{2})(3+\frac{1}{3})(4+\frac{1}{4})(5+\frac{1}{5})$ with $b=(2+\frac{1}{5})(3+\frac{1}{4})(4+\frac{1}{3})(5+\frac{1}{2})$.
CHOICES:
A) $a>b$
B) $a<b$
C) $a=b$
D) Given information is not enough

Using algebra to evaluate each expression is easy, and the correct choice is $A$, but that will take a long time.
Any suggestion to solve this problem in a short time? THANKS.
 [/INST] My intuition is to notice that
$$\frac{2+\frac{1}{2}}{2+\frac{1}{5}}\ ?\ \frac{5+\frac{1}{2}}{5+\frac{1}{5}}$$
(I have used $?$ since I do not know how these expressions relate)
$$\frac{5}{2}\frac{5}{11}\ ?\ \frac{11}{2}\frac{5}{25}$$
$$\frac{25}{22}\ ?\ \frac{55}{50}$$
$$\frac{50}{44}\ ?\ \frac{55}{50}$$
Since $50-44=6>5=55-50$, we know that $?$ is actually $>$. Then
$$\left(\frac{2+\frac{1}{2}}{2+\frac{1}{5}}\right)(3+\frac{1}{3})(4+\frac{1}{4})>(2+\frac{1}{5})(3+\frac{1}{4})\left(\frac{5+\frac{1}{2}}{5+\frac{1}{5}}\right)$$
and the expression simplifies, giving answer $A$.
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