GMAT Quantitative Practice Test: Practice Test 9

Question 1

1. Warehouse X, because its weights are distributed over a wider interval. (correct answer)
2. Warehouse Y, because its tallest bar is at the center of the distribution.
3. Both warehouses must have exactly the same standard deviation.
4. Neither; without computing numerical values, no comparison is possible.

Explanation: Warehouse X's histogram covers a spread from roughly 10 kg to 50 kg, while Warehouse Y's data are tightly clustered between about 20 kg and 30 kg. The broader range and more even spread of X imply a larger standard deviation. B – A central peak alone does not imply greater variability. C – The differing spreads contradict identical standard deviations. D – Visual evidence about dispersion is usually sufficient to compare variability qualitatively.

Question 2

1. $\dfrac16$ (correct answer)
2. $\dfrac19$
3. $\dfrac13$
4. $\dfrac14$

Explanation: When you encounter conditional probability questions, you're dealing with situations where additional information changes the sample space. The phrase "given that the first roll is even" means you only consider scenarios where the first die shows 2, 4, or 6. To find this probability, you need to identify favorable outcomes within the restricted sample space. Since the first roll must be even, you have three possible first rolls: 2, 4, or 6. For each, determine what second roll creates a sum of 8:

• First roll = 2: Need second roll = 6 (one way)
• First roll = 4: Need second roll = 4 (one way)
• First roll = 6: Need second roll = 2 (one way)

Question 3

1. It will be between 6 and 9, closer to 7 since Class 2 has more students and lower variability
2. It will be between 6 and 9, but could be closer to either value depending on the distribution of individual scores
3. It will be approximately 7.4, calculated as the weighted average of the two standard deviations based on class sizes
4. It will likely be greater than 9 due to the difference in means between the classes creating additional variability (correct answer)

Explanation: When combining data from multiple groups, you need to consider not just the variability within each group, but also how the groups differ from each other. This question tests whether you understand that pooled standard deviation can exceed the individual group standard deviations when group means are sufficiently different. The key insight is that when you combine classes with different means (78 vs. 82), you're introducing additional variability beyond what exists within each class. The combined dataset will include scores clustered around 78 AND scores clustered around 82, creating a more spread-out distribution than either class alone. This extra variability from the mean differences can push the pooled standard deviation above the higher individual standard deviation of 9. Choice A incorrectly assumes the pooled standard deviation must fall between the individual values, ignoring the impact of different group means. Choice B makes the same fundamental error, suggesting the result depends only on score distributions rather than recognizing that different means automatically create additional spread. Choice C represents a common misconception—you cannot calculate pooled standard deviation by simply taking a weighted average of individual standard deviations. This approach completely ignores the between-group variability. Choice D correctly recognizes that the 4-point difference in means (78 vs. 82) will likely contribute enough additional variability to push the combined standard deviation above 9, despite Class 2's lower individual variability. Study tip: Remember that pooled standard deviation isn't bounded by the individual group standard deviations when group means differ significantly. Always consider both within-group and between-group sources of variability.

Question 4

1. $x-1$
2. $1-x$ (correct answer)
3. $\dfrac{1}{x}-1$
4. $1-\dfrac{1}{x}$

Explanation: When you encounter complex fractions with multiple layers, the key is to simplify from the inside out, working systematically through each nested fraction. Let's start with the innermost fraction and build outward. First, simplify $1-\frac{1}{x}$: $1-\frac{1}{x} = \frac{x}{x} - \frac{1}{x} = \frac{x-1}{x}$ Now we can rewrite the middle fraction: $1-\frac{1}{1-\frac{1}{x}} = 1-\frac{1}{\frac{x-1}{x}}$ When dividing by a fraction, multiply by its reciprocal: $1-\frac{1}{\frac{x-1}{x}} = 1-\frac{x}{x-1}$ Converting to a common denominator: $1-\frac{x}{x-1} = \frac{x-1}{x-1} - \frac{x}{x-1} = \frac{x-1-x}{x-1} = \frac{-1}{x-1}$ Finally, the entire expression becomes: $\frac{1}{1-\frac{1}{1-\frac{1}{x}}} = \frac{1}{\frac{-1}{x-1}} = \frac{1 \cdot (x-1)}{-1} = -(x-1) = 1-x$ The answer is (B) $1-x$. Looking at the wrong choices: (A) $x-1$ is the negative of our answer, likely resulting from a sign error. (C) $\frac{1}{x}-1$ might come from incorrectly simplifying the innermost fraction. (D) $1-\frac{1}{x}$ is just the value of the innermost expression, not the full simplification. Strategy tip: With nested fractions, always work inside-out and double-check your algebra at each step. Sign errors are particularly common in these problems.