Video Explanation

Text Explanation

We have to begin with some clever noticing.  Notice, first of all, that 409 = 400 + 9.  This suggests a difference of two squares factoring pattern.  Notice that the numerator can be expressed as 159,919 = 160,000 – 81.  That's a difference of two squares that can be factored! 

159,919 = 160,000 – 81 = 400² – 9² = (400 + 9)(400 – 9)

The first parenthesis produces 409.  The second parenthesis contains another difference of two squares:

159,919 = (409)(400 – 9) = (409)(20² – 3²) = (409)(20 + 3)(20 – 3)

159,919 = (409)(23)(17)

Therefore, if we divide by sides by (409)(17), we get that the fraction equals 23. 

Answer = (D).