In which of the following pairs are the two numbers...

swerve wrote:In which of the following pairs are the two numbers reciprocals of one another?

$$I.\ \ 15\ and \ -1/15$$
$$II.\ \sqrt{2}\ and\ \frac{\sqrt{2}}{2}$$
$$III.\ \ 4\ and\ \frac{1}{4}$$

A. I only
B. III only
C. I and II
D. II and III
E. I and III

Some examples of reciprocals:
2 and 1/2
13 and 1/13
-8 and -1/8
2/3 and 3/2
x and 1/x
a/b and b/a

Okay, now let's examine the 3 statements....

I) 15 and -1/15
Since reciprocals MAINTAIN the same sign (positive or negative), 15 and -1/15 are NOT reciprocals
This means we can ELIMINATE A, C and E

IMPORTANT: Notice that the two remaining answer choices (B and C) both suggest that III is true.
So, it MUST be the case that III is true, which means we need not bother to check III

II) √2 and (√2)/2
Since (√2)(√2) = 2, we can rewrite (√2)/2 as follows:
Notice that (√2)/2 = (√2)/(√2)(√2) = 1/√2
At this point, we can see that = √2 and 1/√2 are, indeed, reciprocals.

Answer: D

Cheers,
Brent

swerve wrote:In which of the following pairs are the two numbers reciprocals of one another?

$$I.\ \ 15\ and \ -1/15$$
$$II.\ \sqrt{2}\ and\ \frac{\sqrt{2}}{2}$$
$$III.\ \ 4\ and\ \frac{1}{4}$$

A. I only
B. III only
C. I and II
D. II and III
E. I and III

The reciprocal of 15 is 1/15; so 15 and -1/15 are not reciprocals of each other.

The reciprocal of √2 is 1/√2 = √2/2 (after rationalizing the denominator); so √2 and √2/2 are reciprocals of each other.

The reciprocal of 4 is 1/4; so 4 and 1/4 are reciprocals of each other.

Answer: D