M02-16

Official Solution:

Which of the following fractions has the smallest value?

A. \(\frac{1}{900 * 5^4}\)
B. \(\frac{1}{80 * 5^5}\)
C. \(\frac{1}{5,000 * 5^3}\)
D. \(\frac{1}{40 * 5^6}\)
E. \(\frac{1}{30,000 * 5^2}\)


Observe that each denominator can be expressed in the form of \(something*5^4\). This uniform format aids in comparing the fractions:

A. \(\frac{1}{900*5^4}= \frac{1}{900 * 5^4}\)

B. \(\frac{1}{80*5^5}= \frac{1}{80*5*5^4} = \frac{1}{400*5^4}\)

C. \(\frac{1}{5,000*5^3}= \frac{1}{1,000*5*5^3} = \frac{1}{1,000*5^4}\)

D. \(\frac{1}{40*5^6}= \frac{1}{40*5^2*5^4} = \frac{1}{1,000*5^4}\)

E. \(\frac{1}{30,000*5^2}= \frac{1}{1,200*5^2*5^2} = \frac{1}{1,200 * 5^4}\)

Option E has the largest denominator, \(5^4 * 1,200\), making its fraction the smallest in value.


Answer: E