Problem Solving

mathewmithun wrote:if 0<1-f/g<1 and g>0, which of the followiing must be true?
1. f^2+g^2>1
2. f>0
3. f/g<1

A. 1 only
B. 3 only
C. 1 and 3 only
D. 2 and 3 only
E. 1,2 and 3 only

Let's deal with this in parts:

Part 1: 0 < 1-(f/g)
Add f/g to both sides to get: f/g < 1
Since we're told that g is positive, we'll multiply both sides by g to get: f < g

Part 2: 1-(f/d) < 1
Add f/g to both sides to get: 1 < 1 + f/g
Subtract 1 from both sides to get: 0 < f/g
If g is positive and f/g is positive, then f must be positive.
In other words, 0 < f

Combining the two parts, we get 0 < f < g

What kinds of conclusions can we make from this information?

1. f^2+g^2 >1 (no, this need not be true. For example, we could have f=0.1 and g=0.2, in which case f^2 + g^2 is not greater than 1.)
2. f>0 (yes, this must be true)
3. f/g < 1 (since f < g, and since f and g are both positive, this must be true)

Answer = D

Cheers,
Brent