Bunuel wrote:
Which of the following fractions has a value less than \(\frac{6}{3^2*7^3}\) ?
A. \(\frac{14}{3*7^4}\)
B. \(\frac{2}{3*7^3}\)
C. \(\frac{45}{3^5*7^3}\)
D. \(\frac{19}{3^3*7^3}\)
E. \(\frac{132}{3^3*7^4}\)
We see that the common denominator of the given fraction and all of the answer choices is 3^5 x 7^4. So we can get common denominators of each value and compare the numerators. Let’s start with the given value.
Numerator of given value becomes: 6 x 3^3 x 7 = 2 x 3^4 x 7^1
Numerator of choice A becomes: 14 x 3^4 = 2 x 3^4 x 7^1
We see that answer choice A is equal to the given value.
Numerator of choice B becomes: 2 x 3^4 x 7^1
We see that answer choice B is equal to the given value.
Numerator of choice C becomes: 3^2 x 5^1 x 7^1
We see that answer choice C is less than the given value.
Although we can stop here, let’s continue to verify that the last two answer choices have a numerator greater than or equal to the numerator of the given fraction.
Numerator of choice D becomes 19 x 3^2 x 7^1
We see that answer choice D is greater than the given value.
Numerator of choice E becomes 132 x 3^2
We see that answer choice E is greater than the given value.
Alternate Solution:
As we are looking for a value that is less than 6/(3^2 * 7^3), we know that all but one of the answer choices will be greater than or equal to 6/(3^2 * 7^3). We can therefore ignore the given fraction and simply find the smallest fraction among the given fractions.
Notice that the fraction in answer choice A simplifies to 2/(3 * 7^3); which is the same as the fraction in answer choice B. As there can be only one correct answer, we can eliminate both of A and B.
Next, notice that the fraction in answer choice C simplifies to 5/(3^3 * 7^3). As this simplified expression has the same denominator as the fraction in answer choice D, we can easily compare the numerators and conclude that C is smaller than D. We only need to compare the fractions in answer choices C and E. To do that, we can multiply the numerator and the denominator of the simplified form of the fraction in answer choice C by 7 to obtain that it is equivalent to 35/(3^3 * 7^4). Comparing the numerators of C and E, we conclude that C is the smallest fraction among the answer choices and thus, it must also be smaller than 6/(3^2 * 7^3).
Answer: C
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