If 0.02<x<0.04 and 100<y<250, which of the following could be the valu

\(?\,\,\,:\,\,\,\frac{{y - x}}{{xy}} = \frac{1}{x} - \frac{1}{y}\,\,\,\,\,\underline {{\text{could}}\,\,{\text{be}}}\)

\({2 \over {100}} < x < {4 \over {100}}\,\,\,\,\, \Rightarrow \,\,\,\,\,25 = {{100} \over 4} < {1 \over x} < {{100} \over 2} = 50\,\,\,\,\left( {\rm{I}} \right)\)

\(100 < y < 250\,\,\,\,\, \Rightarrow \,\,\,\,{1 \over {250}} < {1 \over y} < {1 \over {100}}\,\,\,\,\, \Rightarrow \,\,\,\,\, - {1 \over {100}} < - {1 \over y} < - {1 \over {250}}\,\,\,\,\left( {{\rm{II}}} \right)\)

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


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

Regards,
Fabio.

=>

Note that \(\frac{(y-x)}{(xy)} = \frac{1}{x} – \frac{1}{y}.\)
\(0.02 < x < 0.04\)
\(=> \frac{1}{0.02} > \frac{1}{x} > \frac{1}{0.04}\)
\(=> 50 > \frac{1}{x} > 25\)
\(=> 25 < \frac{1}{x} < 50\)

\(100 < y < 250\)
\(=> \frac{1}{100} > \frac{1}{y} > \frac{1}{250}\)
\(=> 0.01 > \frac{1}{y} > 0.004\)
\(=> 0.004 < \frac{1}{y} < 0.01\)

So,
\(25 – 0.01 < \frac{1}{x} – \frac{1}{y} < 50 – 0.004\)
\(24.99 < \frac{(y-x)}{(xy)} < 49.996\)

The only possible value of \(\frac{(y-x)}{(xy)}\) listed is \(30\).

Therefore, the answer is C.
Answer: C

Simplifying the expression, we have:

(y - x)/(xy) = y/(xy) - x/(xy) = 1/x - 1/y

We see that 1/x ranges from 1/0.04 = 25 to 1/0.02 = 50, and 1/y ranges from 1/250 = 0.004 to 1/100 = 0.01. Therefore, 1/x - 1/y ranges from 25 - 0.01 = 24.99 to 50 - 0.004 = 49.996. We see that only 30 falls into this range; thus, the correct choice is C.

Answer: C

guys , what am I donig wrong ?
2/100 < x < 4/100 .....................(1)
-4/100 < -x < -2/100..................(2)
100< y < 250...................(3)
(1) +(2)............ : 100-4/100 < y-x < 250-2/100 (4)

now , by multiblying (1) and (3) , since all values are positive :

100*2/100 < xy < 250* 4/100
2 < xy < 10
that means : 1/10< 1/xy < 1/2 ....................(5)

by multiblying (4) and (5) : we got : ( 100-4/100 ) *1/10 < ( y-x) /xy < 1/2 * ( 250-2/100)
that means : 10< ( y-x) /xy < 125
so where is my mistake ?
MathRevolution
fskilnik

Hi foryearss ,

There were no mistakes. The inferior and superior bounds you found are correct, but unfortunately, they are not good enough for the alternative choices offered.

Regards,
Fabio.

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