<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] I do not think it would take a long time to simplify these expression by hand. Rewrite it as
$$\frac {5}{2}\cdot\frac {10}{3}\cdot\frac {17}{4}\cdot\frac {26}{5}\text{ vs }\frac {11}{5}\cdot \frac {13}{4}\cdot\frac {13}{3}\cdot \frac {11}{2}$$
Denominators go away, and the factor $13$ in the numerator as well:
$$1700\text{ vs }13\cdot 121,$$
and the latter seems to be easy to estimate.
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