Bounding exponential ratio

CAT 2005 Slot 1 · QA · Medium · Algebra

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If R=3065−29653064+2964R = \frac{30^{65} - 29^{65}}{30^{64} + 29^{64}}, then

  1. A.

    0<R≤0.10 < R \le 0.1

  2. B.

    0.1<R≤0.50.1 < R \le 0.5

  3. C.

    0.5<R≤1.00.5 < R \le 1.0

  4. D.

    R>1.0R > 1.0

Answer

D

Explanation

Rewrite RR as: R=3065(1−(2930)65)3064(1+(2930)64)=30×1−(0.9667)651+(0.9667)64R = \frac{30^{65}\left(1 - \left(\frac{29}{30}\right)^{65}\right)}{30^{64}\left(1 + \left(\frac{29}{30}\right)^{64}\right)} = 30 \times \frac{1 - (0.9667)^{65}}{1 + (0.9667)^{64}} Since (0.9667)65≈0.116(0.9667)^{65} \approx 0.116, the numerator is approximately 0.8840.884. The denominator is 1+(0.9667)64≈1.121 + (0.9667)^{64} \approx 1.12. Thus R≈30×0.8841.12=30×0.789=23.67>1.0R \approx 30 \times \frac{0.884}{1.12} = 30 \times 0.789 = 23.67 > 1.0. Hence, R>1.0R > 1.0.

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