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.

We have that
$$a=\left(2+\frac{1}{2}\right)\left(3+\frac{1}{3}\right)\left(4+\frac{1}{4}\right)\left(5+\frac{1}{5}\right)=$$
$$=\frac12\left(4+1\right)\frac13\left(9+1\right)\frac14\left(16+1\right)\frac15\left(25+1\right)=$$
$$=\frac{5\cdot 10\cdot17\cdot26}{120}$$
and similarly
$$b=\left(2+\frac{1}{5}\right)\left(3+\frac{1}{4}\right)\left(4+\frac{1}{3}\right)\left(5+\frac{1}{2}\right)=\frac{11\cdot 13\cdot13\cdot11}{120}$$
with
$$5\cdot 10\cdot17\cdot26 > 11\cdot 13\cdot13\cdot11$$
$$5\cdot 10\cdot 17\cdot 2 > 11\cdot 13\cdot11$$
$$100 \cdot 17 > 121 \cdot 13$$
