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26 May 2017, 04:43
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If \(0.002 < a < 0.004\) and \(5,000 < b < 10,000\), which of the following can be the value of \(\frac{a^{2}b + b^{2}a}{a^{2}b^{2}}\)?
A. 120
B. 150
C. 225
D. 425
E. 725
A. 120
B. 150
C. 225
D. 425
E. 725
26 May 2017, 04:45
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Official Solution:
If \(0.002 < a < 0.004\) and \(5,000 < b < 10,000\), which of the following can be the value of \(\frac{a^{2}b + b^{2}a}{a^{2}b^{2}}\)?
A. 120
B. 150
C. 225
D. 425
E. 725
You get from \(0.002 < a < 0.004\) to \(\frac{2}{1,000} < a < \frac{4}{1,000}\), then to \(\frac{1}{500} < a < \frac{1}{250}\), and then to \(250 < \frac{1}{a} < 500\). Then, you will get from
\(5,000 < b < 10,000\) to \(\frac{1}{10,000} < \frac{1}{b} < \frac{1}{5,000}\), and then to \(0.0001 < \frac{1}{b} < 0.0002\).
You should get \(\frac{a^{2}b + b^{2}a}{a^{2}b^{2}} = \frac{a^{2}b}{a^{2}b^{2}} + \frac{b^{2}a}{a^{2}b^{2}} = \frac{1}{b} + \frac{1}{a}\), but since \(\frac{1}{b}\) is negligible because it is almost close to 0, so you are left with \(\frac{a^{2}b + b^{2}a}{a^{2}b^{2}} = \frac{a^{2}b}{a^{2}b^{2}} + \frac{b^{2}a}{a^{2}b^{2}} = \frac{1}{b} + \frac{1}{a} = \frac{1}{a}\), and you get \(250 < \frac{1}{a} < 500\). Hence, you get \(425\). Therefore, D is the answer.
Answer: D
If \(0.002 < a < 0.004\) and \(5,000 < b < 10,000\), which of the following can be the value of \(\frac{a^{2}b + b^{2}a}{a^{2}b^{2}}\)?
A. 120
B. 150
C. 225
D. 425
E. 725
You get from \(0.002 < a < 0.004\) to \(\frac{2}{1,000} < a < \frac{4}{1,000}\), then to \(\frac{1}{500} < a < \frac{1}{250}\), and then to \(250 < \frac{1}{a} < 500\). Then, you will get from
\(5,000 < b < 10,000\) to \(\frac{1}{10,000} < \frac{1}{b} < \frac{1}{5,000}\), and then to \(0.0001 < \frac{1}{b} < 0.0002\).
You should get \(\frac{a^{2}b + b^{2}a}{a^{2}b^{2}} = \frac{a^{2}b}{a^{2}b^{2}} + \frac{b^{2}a}{a^{2}b^{2}} = \frac{1}{b} + \frac{1}{a}\), but since \(\frac{1}{b}\) is negligible because it is almost close to 0, so you are left with \(\frac{a^{2}b + b^{2}a}{a^{2}b^{2}} = \frac{a^{2}b}{a^{2}b^{2}} + \frac{b^{2}a}{a^{2}b^{2}} = \frac{1}{b} + \frac{1}{a} = \frac{1}{a}\), and you get \(250 < \frac{1}{a} < 500\). Hence, you get \(425\). Therefore, D is the answer.
Answer: D
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