Problem:
Which of the following is closest to 65−63?
Answer Choices:
A. 0.12
B. 0.13
C. 0.14
D. 0.15
E. 0.16
Solution:
Since
65−63=65+63(65−63)(65+63)=65+632
and since 65 and 63 are each equal to about 8, the answer must be near 1/8. Specifically,
65+63>7.5+7.5=15, so 65−63<2/15≈.1333
Thus the answer is (A) or (B). To determine which, we must decide whether 65+63 is larger or smaller than 8+8, since 2/16=.125 exactly. In fact, it is smaller: this is the case n=64 and a=1 of the inequality
n+a+n−a<2n
valid whenever 0<∣a∣≤n, which one may verify by squaring. (Alternately, the fact that the graph of y=x is concave down shows geometrically that 64 is greater than the average of 63 and 65 - how?) Therefore, 65−63>2/16 and the answer is (B).
Note. 65−63≈.125004, as can be verified using the Binomial series or (after the test) a calculator.