Question: Which of the following expressions has the greatest values?

1. 0.456
2. \(\frac{1}{2}-(\frac{1}{2})^4\)
3. \(\frac{300}{650}\)
4. \(3(\frac{3}{19})\)
5. \(\sqrt{0.17}\)

“Which of the following expressions has the greatest values?”- 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.

Answer:

Approach Solution 1

The basic trick to solve this question is to consider how far each answer choice is away from an easy number. In this way, all the answer choices are very close to \(\frac{1}{2}\)

1. 0.456 is the first option.
   0.044 below 0.5 \(\rightarrow\) \(\frac{11}{250}\) below \(\frac{1}{2}\)\(\rightarrow\) \(\frac{1}{25}\)below \(\frac{1}{2}\)
2. \(\frac{1}{2}-(\frac{1}{2})^4\)is the second option
   \(\frac{1}{2}-\frac{1}{16}\rightarrow \frac{1}{16}\) below \(\frac{1}{2}\)
3. \(\frac{300}{650}\)is the third option
   \(\frac{25}{650}\)below \(\frac{325}{650}\)\(\rightarrow \)\(\frac{1}{26}\) below\(\frac{1}{2}\)
4. \(3(\frac{3}{19})\)is the fourth option
   \(3(\frac{3}{19})\)= \(\frac{9}{19}\rightarrow \frac{0.5}{19}\) below \(\frac{9.5}{19}\rightarrow\frac{1}{38}\) below \(\frac{1}{2}\)
5. \(\sqrt{0.17}\)is the fifth option
   \(\sqrt{0.17}\rightarrow \) closer to \(\sqrt{0.16}\) than to \(\sqrt{0.25}\rightarrow \) closer to 0.4 than 0.5 \(​\rightarrow \)> 0.05 below 0.5 \(​\rightarrow \) > \(\frac{1}{20}\) below \(\frac{1}{2}\)

Since all of these fractions have 1 in the numerator, we don’t need a common denominator to compare them – the one with the largest denominator will be the smallest.

\(\frac{1}{38}\)is the smallest fraction. So, D is the closet to \(\frac{1}{2}\) of all the answer choices.

Correct option: D

Approach Solution 2

Find the best common base for these expressions.

B is \(\frac{1}{2}\)-\(\frac{1}{16}\) , which gives us \(\frac{7}{16}\)

C is \(\frac{300}{650}\) or \(\frac{30}{65}\) or \(\frac{6}{13}\)

D is \(\frac{9}{19}\)

E is slightly greater than 0.4, not much more to surpass A, so we can eliminate it.

Now, let’s see which of these B, C and D is the greatest. We can drop B as it does not add up to C or D, it is further away from \(\frac{1}{2}\) than C and D.

These two fractions are very close to \(\frac{1}{2}\) . If we decrease the values of the denominators of each of the fraction by 1, we will see that they are equal to \(\frac{1}{2}\). For these two fractions \(\frac{6}{13}\) and \(\frac{9}{19}\), the greater value of denominator, the closer will the fraction be to \(\frac{1}{2}\)

Therefore, \(\frac{9}{19}\) is greater.

Correct option: D

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