M02-16

Question

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}

Bunuel · Accepted Answer

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

CPGguyMBA2018 · Answer

Since all the fractions have 5^x in common, we can also easily solve the problem by multiplying the number by 5^6 on all the options.
That gives us a very good visibility that E is indeed the smallest of the given options.

edlc313 · Answer

Bunuel's method is probably the quickest way to solve, but to avoid making mistakes, I broke the denominators down into prime factors - it just requires a bit of focus to make sure to count all the 0's properly, as these represent factors of 2 and 5.

The highest denominator tells us the smallest number. So let's figure out the denominators.

A.  \frac{1}{900 * 5^4} --> 2^2*3^2*5^6 
B.  \frac{1}{80 * 5^5} --> 2^4*5^6 
C.  \frac{1}{5000 * 5^3} --> 2^3*5^7 
D.  \frac{1}{40 * 5^6} --> 2^3*5^7 
E.  \frac{1}{30000 * 5^2} --> 2^4*3*5^6

Why did I choose E?

Well, 5^7 is obviously much greater than 5^6. But, instead of having the greatest exponent of 5, E has an extra factor of 2*3 (6). 
Thus,  5^6 * 6 > 5^7

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Shubham7330 · Answer

I converted the denominators as follows:
a. 9 * 5^6
b. 8 * 5^6 
c. 5 * 5^6 
d. 4 * 5^7
e. 3 * 5^6 
Here, d turned out to be the largest, and hence the corresponding fraction was the smallest, is this not a valid way to look at it? Or am I missing something?

Bunuel · Answer

Your conversions are not correct.

For example, 900 * 5^4 = 9 * 100 * 5^4 = 9 * (2^2 * 5^2) * 5^4 = 9 * 2^2 * 5^6, not 9 * 5^6.

You're missing the powers of 2 in the factorization. The same issue applies to the other terms as well.

cantguardbikes · Answer

I like the solution - it’s helpful.

M02-16

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