If 0.02 < x < 0.04 and 100 < y < 250, which of t

[Max@Math Revolution]

[GMAT math practice question]

If 0.02 < x < 0.04 and 100 < y < 250, which of the following could be the value of (y-x)/(xy)?

A. 10
B. 20
C. 30
D. 50
E. 100

[Brent@GMATPrepNow]

[GMAT math practice question]

If 0.02 < x < 0.04 and 100 < y < 250, which of the following could be the value of (y-x)/(xy)?

A. 10
B. 20
C. 30
D. 50
E. 100

We can use a nice (and often tested) fraction property that says: (a - b)/c = a/c - bc
So, (y-x)/(xy) = y/xy - x/xy
= 1/x - 1/y
That's better!
So, we're now looking for a possible value of 1/x - 1/y

Let's look at some EXTREME values.
We can MAXIMIZE the value of 1/x - 1/y by MAXIMIZING the value of 1/x , and MINIMIZING the value of 1/y

1/x is maximized when the value of x is a small as possible
We're told that 0.02 < x < 0.04, so the smallest value of x is just a tiny bit bigger than 0.02
To make things easier on ourselves, let's see what happens when x = 0.02
ASIDE: 0.02 = 1/50
When x = 0.02, we see that 1/x =1/0.02 = 1/(1/50) = 50

Conversely, 1/y is minimized when the value of y is a big as possible
We're told that 100 < y < 250, so the biggest value of y is just a tiny bit less than 250
To make things easier on ourselves, let's see what happens when y = 250
We get: 1/y = 1/250 = 0.004

So, the MAXIMUM value of 1/x - 1/y = 50 - 0.004 = 49.996
So, b]1/x - 1/y[/b] must be LESS THAN 49.996

At this point, we can see that the correct answer must be C.
How do we know this?
We already know that the MAXIMUM value 1/x - 1/y is approximately 49.996
IF we were to also find the MINIMUM value of 1/x - 1/y, then we'd have a range of possible values for 1/x - 1/y
The range would look like this: some smaller value < 1/x - 1/y < 49.996

Since there can be only 1 correct answer, the correct answer must be biggest value that is less than 49.996

Answer: C

Cheers,
Brent

[fskilnik@GMATH]

[GMAT math practice question]

If 0.02 < x < 0.04 and 100 < y < 250, which of the following could be the value of (y-x)/(xy)?

A. 10
B. 20
C. 30
D. 50
E. 100

$$?\,\,\,:\,\,\,\frac{{y - x}}{{xy}} = \frac{1}{x} - \frac{1}{y}\,\,\,\,\,\underline {{\text{could}}\,\,{\text{be}}} $$
$${2 \over {100}} < x < {4 \over {100}}\,\,\,\,\, \Rightarrow \,\,\,\,\,25 = {{100} \over 4} < {1 \over x} < {{100} \over 2} = 50\,\,\,\,\left( {\rm{I}} \right)$$
$$100 < y < 250\,\,\,\,\, \Rightarrow \,\,\,\,{1 \over {250}} < {1 \over y} < {1 \over {100}}\,\,\,\,\, \Rightarrow \,\,\,\,\, - {1 \over {100}} < - {1 \over y} < - {1 \over {250}}\,\,\,\,\left( {{\rm{II}}} \right)$$

$$\left( {\rm{I}} \right) + \left( {{\rm{II}}} \right)\,\,\,25 - {1 \over {100}} < {1 \over x} - {1 \over y} < 50 - {1 \over {250}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{C}} \right)$$


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[Max@Math Revolution]

=>

Note that (y-x)/(xy) = 1/x - 1/y.

0.02 < x < 0.04
=> 1/0.02 > 1/x > 1/0.04
=> 50 > 1/x > 25
=> 25 < 1/x < 50

100 < y < 250
=> 1/100 > 1/y > 1/250
=> 0.01 > 1/y > 0.004
=> 0.004 < 1/y < 0.01

So,
25 - 0.01 < 1/x - 1/y < 50 - 0.004
24.99 < (y-x)/(xy) < 49.996

The only possible value of (y-x)/(xy) listed is 30.

Therefore, the answer is C.
Answer: C