If \(x=10^{10},\frac{x^2+2x+7}{3x^2-10x+200}\) is closest to GMAT Problem Solving

Question: If \(x=10^{10},\frac{x^2+2x+7}{3x^2-10x+200}\) is closest to:

1. \(\frac{1}{6}\)
2. \(\frac{1}{3}\)
3. \(\frac{2}{5}\)
4. \(\frac{1}{2}\)
5. \(\frac{2}{3}\)

Correct Answer: (B)

Solutions and Explanation

Approach Solution (1):

\(\frac{x^2+2x+7}{3x^2-10x+200}=\frac{1+\frac{2}{x}+\frac{7}{x^2}}{3-\frac{10}{x}+\frac{200}{x^2}}\)

When x is larger, \(\frac{2}{7},\frac{7}{x^2},\frac{10}{x},and,\frac{200}{x^2}\) are small and the fraction is very close to\(\frac{1}{3}\)

Approach Solution (2):

\(\frac{x^2+2x+7}{3x^2-10x+200}=\frac{10^{20}+2*10^{10}+7}{3*10^{20}-10*10^{10}+200}\)

Note that we need the approximate value of the given expression. Now, since \(10^{20}\) is much larger number than \(2*10^{10}+7\) then \(2*10^{10}+7\) is pretty much negligible in this case. Similarly, \(3*10^{20}\) is much larger number \(-10*10^{10}+200\) than \(-10*10^{10}+200\), sois also negligible in this case.

So, \(\frac{10^{20}+2*20^{10}+7}{3*10^{20}-10*10^{10}+20}\approx\frac{10^{20}}{3*10^{20}}=\frac{1}{3}\)

Approach Solution (3):

We can safely disregard the small numbers because 10^10 is a very, very large number (7 & 200).

That leaves us with, (x^2 + 2x) / (3x^2 - 10x) = x(x+2)/(3x-10)

Subtract the single digits and x's to get x/3x or 1/3.

“If \(x=10^{10},\frac{x^2+2x+7}{3x^2-10x+200}\) is closest to:”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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