D01-07

Question

Which of the following numbers is the greatest?

A.  \frac{1,883,453}{1,883,456}

B.  \frac{1,876,452}{1,876,455}

C.  \frac{1,883,456}{1,883,459}

D.  \frac{1,883,491}{1,883,494}

E.  \frac{1,883,446}{1,883,449}

Bunuel · Accepted Answer

Official Solution:

Which of the following numbers is the greatest?

A.  \frac{1,883,453}{1,883,456}

B.  \frac{1,876,452}{1,876,455}

C.  \frac{1,883,456}{1,883,459}

D.  \frac{1,883,491}{1,883,494}

E.  \frac{1,883,446}{1,883,449}

To solve this problem, notice that the difference between the numerator and the denominator in each fraction is equal to 3. Since all of the numbers are positive and less than 1, we know that the larger the denominator, the larger the fraction:
 
 \frac{1}{4} < \frac{2}{5} < \frac{3}{6} < \frac{4}{7} < \frac{5}{8} < \frac{6}{9} < \frac{7}{10} < ... 
 
Using this logic, we can see that the largest fraction is  \frac{1,883,491}{1,883,494} , which has the largest denominator among all the options.
 
We can also take an algebraic approach by writing the fractions as  \frac{n-3}{n} . Simplifying this expression, we get:
 
  \frac{n-3}{n}= 
 
 =\frac{n}{n}-\frac{3}{n}= 
 
 =1-\frac{3}{n} 
 
As  n  increases, the value of  \frac{3}{n}  decreases, making the value of  1-\frac{3}{n}  larger. Therefore, we reach the same conclusion as before: the larger the denominator, the larger the fraction.

Answer: D

Bunuel · Answer

Yes, your understanding is on point. When the difference between the numerator and the denominator is the same, and the fractions are between 0 and 1 (positive but less than 1), the fraction with the larger denominator is bigger. For instance, 1/2 (0.5) is smaller than 99/100 (0.99) because 2 is less than 100. However, for fractions greater than 1 with the same difference between the numerator and the denominator, the fraction with the smaller denominator is larger. As an example, 3/2 (1.5) is greater than 101/100 (1.01) because 2 is less than 100.

rickyric395 · Answer

If you see, all numbers have 3 as difference between numerator and denominator(with numerator on lesser side). All the numbers can be written in format \(\frac{Denominator - 3}{Denominator}\) (As already mentioned). So it comes down to \(1-\frac{3}{Denominator}\). Hence the greater the denominator , the bigger is the number. We will get E as answer.

Eg- \(\frac{1876452}{1876455}\) = \(\frac{1876455-3}{1876455}\) = \(1-\frac{3}{1876455}\)

Eg-  \frac{1876452}{1876455}  =  \frac{1876455-3}{1876455}  =  1-\frac{3}{1876455}

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.

oman573 · Answer

This took me a second but I see more clearly. 97/100 is clearly smaller than 100/103. And I understand more about the 3/n portion as well.

Tanisha2819 · Answer

Hi  Bunuel  thank you so much for the solution.

So we can say as a rule that the larger the denominator, the larger the fraction when the difference between the numerator and the denominator in each fraction is equal.

But here an important catch also comes that the numerator is also lesser than the denominator in each fraction.

If we have the numerator greater than the denominator then would the same rule apply? I guess not, if the numerator is greater than the denominator and the difference between the numerator and denominator is equal in each fraction then the fraction with the least denominator should be the greatest.

Am i correct in my understanding here.

Thanks again  Bunuel

vn049636 · Answer

I think this is a  high-quality  question and I agree with explanation.

odd_major · Answer

I think this is a  high-quality  question and I agree with explanation.

AnkurSinghania · Answer