Problem:
Which of the following is the correct order of the fractions 1115, 1519, 1317, from least to greatest?
Answer Choices:
A. 1115<1317<1519
B. 1115<1519<1317
C. 1317<1519<1115
D. 1519<1115<1317
E. 1519<1317<1115
Solution:
To determine the order of 1519 and 1317, rewrite the fractions using a common
denominator: 15⋅1319⋅13 and 13⋅1517⋅15. Because
19⋅13=(16+3)(16−3)=162−32
17⋅15=(16+1)(16−1)=162−12
and 162−32<162−12 , it follows that 1519 < 1317.
Similarly, to determine the order of 1317 and 1115, rewrite the fractions using a common denomi-nator:13⋅1117⋅11 and 11⋅1315⋅13. Because
17⋅11=(14+3)(14−3)=142−32,
15⋅13=(14+1)(14−1)=142−12,
and 142−32<142−12 , it follows that 1317 < 1115.
OR
Subtracting 1 from each fraction results in the fractions 114,154, and 134. Because 154 < 134 < 114 it follows that 1519 < 1317 < 1115.
OR
Given a fraction ba where 0 < b < a, if n is a positive integer, then bn < an and so b(a+n) = ab + bn < ab + an = a(b + n). Thus b+na+n < ba. Therefore 1519< 1317< 1115.
Answer: E.
The problems on this page are the property of the MAA's American Mathematics Competitions