D01-07

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