DS: Percentage – Level 2
1.
Did the price of the stock increase by more than 10%?
(1) The price increased by $5.
(2) The original price was less than $40.
(1) The price increased by $5.
(2) The original price was less than $40.
Solution (C):
Analyze Statement (1): Increase = $5. Without knowing the original price, we calculate percent: \(\frac{5}{\text{Original}} \times 100\). If Original is 100, increase is 5%. If Original is 10, increase is 50%. Not Sufficient.
Analyze Statement (2): Original < 40. Without knowing the increase amount, we can't find percentage. Not Sufficient.
Combine (1) and (2): We know Increase = 5 and Original < 40. The percent increase is \(\frac{5}{\text{Original}} \times 100\). Since the denominator is less than 40, the fraction \(\frac{5}{<40}\) must be greater than \(\frac{5}{40}\). \(\frac{5}{40} = \frac{1}{8} = 12.5\%\). So the percent increase is definitely greater than 12.5%, which is greater than 10%. The answer is definitely "Yes". Sufficient.
Analyze Statement (1): Increase = $5. Without knowing the original price, we calculate percent: \(\frac{5}{\text{Original}} \times 100\). If Original is 100, increase is 5%. If Original is 10, increase is 50%. Not Sufficient.
Analyze Statement (2): Original < 40. Without knowing the increase amount, we can't find percentage. Not Sufficient.
Combine (1) and (2): We know Increase = 5 and Original < 40. The percent increase is \(\frac{5}{\text{Original}} \times 100\). Since the denominator is less than 40, the fraction \(\frac{5}{<40}\) must be greater than \(\frac{5}{40}\). \(\frac{5}{40} = \frac{1}{8} = 12.5\%\). So the percent increase is definitely greater than 12.5%, which is greater than 10%. The answer is definitely "Yes". Sufficient.
2.
Is \(x > y\)?
(1) \(x\) increased by 20% equals \(z\).
(2) \(y\) decreased by 20% equals \(z\).
(1) \(x\) increased by 20% equals \(z\).
(2) \(y\) decreased by 20% equals \(z\).
Solution (E):
Analyze Statement (1): \(1.2x = z\). Tells us nothing about \(y\). Not Sufficient.
Analyze Statement (2): \(0.8y = z\). Tells us nothing about \(x\). Not Sufficient.
Combine (1) and (2): \(1.2x = 0.8y\). We can find the ratio: \(\frac{x}{y} = \frac{0.8}{1.2} = \frac{2}{3}\). This means \(x\) is two-thirds of \(y\). If \(x\) and \(y\) are positive (e.g., \(x=2, y=3\)), then \(x < y\) (Answer: No). However, if \(x\) and \(y\) are negative (e.g., \(x=-2, y=-3\)), then \(-2 > -3\), so \(x > y\) (Answer: Yes). Since we don’t know the signs of the numbers, we cannot be certain. Not Sufficient.
Analyze Statement (1): \(1.2x = z\). Tells us nothing about \(y\). Not Sufficient.
Analyze Statement (2): \(0.8y = z\). Tells us nothing about \(x\). Not Sufficient.
Combine (1) and (2): \(1.2x = 0.8y\). We can find the ratio: \(\frac{x}{y} = \frac{0.8}{1.2} = \frac{2}{3}\). This means \(x\) is two-thirds of \(y\). If \(x\) and \(y\) are positive (e.g., \(x=2, y=3\)), then \(x < y\) (Answer: No). However, if \(x\) and \(y\) are negative (e.g., \(x=-2, y=-3\)), then \(-2 > -3\), so \(x > y\) (Answer: Yes). Since we don’t know the signs of the numbers, we cannot be certain. Not Sufficient.
3.
What is the value of \(n\)?
(1) 20% of \(n\) equals 10% of 50.
(2) \(n\) is a positive number less than 30.
(1) 20% of \(n\) equals 10% of 50.
(2) \(n\) is a positive number less than 30.
Solution (A):
Analyze Statement (1): \(0.20n = 0.10(50)\). \(0.20n = 5 \implies n = 5 / 0.20 = 25\). We found a unique value. Sufficient.
Analyze Statement (2): \(0 < n < 30\). There are infinitely many numbers in this range. Not Sufficient.
Analyze Statement (1): \(0.20n = 0.10(50)\). \(0.20n = 5 \implies n = 5 / 0.20 = 25\). We found a unique value. Sufficient.
Analyze Statement (2): \(0 < n < 30\). There are infinitely many numbers in this range. Not Sufficient.
4.
Is \(x\) greater than \(y\)?
(1) \(x\) is 40% of \(k\).
(2) \(y\) is 30% of \(k\).
(1) \(x\) is 40% of \(k\).
(2) \(y\) is 30% of \(k\).
Solution (C):
Analyze Statement (1): \(x = 0.4k\). No info on \(y\). Not Sufficient.
Analyze Statement (2): \(y = 0.3k\). No info on \(x\). Not Sufficient.
Combine (1) and (2): \(x = 0.4k\) and \(y = 0.3k\). If \(k\) is positive (e.g., 100), \(x=40, y=30 \implies x > y\). If \(k\) is negative (e.g., -100), \(x=-40, y=-30 \implies x < y\). If \(k\) is zero, \(x = y\). Since we don't know the sign of \(k\), we cannot answer definitely. **Correction:** Usually in these "Level 2" questions, variables represent physical quantities or prices (positive). However, mathematically, without knowing \(k\)'s sign, it's E. Wait, let me double check the "GMAT standard". GMAT numbers are real numbers. So sign matters. The answer is **E**. *Self-correction*: If I assume \(k\) is positive (like a price), it would be C. But strictly mathematically, it is E. I will mark it as **E** to be rigorous.
Analyze Statement (1): \(x = 0.4k\). No info on \(y\). Not Sufficient.
Analyze Statement (2): \(y = 0.3k\). No info on \(x\). Not Sufficient.
Combine (1) and (2): \(x = 0.4k\) and \(y = 0.3k\). If \(k\) is positive (e.g., 100), \(x=40, y=30 \implies x > y\). If \(k\) is negative (e.g., -100), \(x=-40, y=-30 \implies x < y\). If \(k\) is zero, \(x = y\). Since we don't know the sign of \(k\), we cannot answer definitely. **Correction:** Usually in these "Level 2" questions, variables represent physical quantities or prices (positive). However, mathematically, without knowing \(k\)'s sign, it's E. Wait, let me double check the "GMAT standard". GMAT numbers are real numbers. So sign matters. The answer is **E**. *Self-correction*: If I assume \(k\) is positive (like a price), it would be C. But strictly mathematically, it is E. I will mark it as **E** to be rigorous.
5.
Are there more boys than girls in the class?
(1) 40% of the students are girls.
(2) There are 50 students in the class.
(1) 40% of the students are girls.
(2) There are 50 students in the class.
Solution (A):
Analyze Statement (1): If 40% are girls, then 60% must be boys (assuming only two genders). Since \(60\% > 40\%\), there are always more boys than girls regardless of the total number. We can answer “Yes” definitively. Sufficient.
Analyze Statement (2): Total = 50. We don’t know the breakdown. Not Sufficient.
The answer is **A**. (My label above said C, but logically it is A. I will correct the data-answer attribute to A).
Analyze Statement (1): If 40% are girls, then 60% must be boys (assuming only two genders). Since \(60\% > 40\%\), there are always more boys than girls regardless of the total number. We can answer “Yes” definitively. Sufficient.
Analyze Statement (2): Total = 50. We don’t know the breakdown. Not Sufficient.
The answer is **A**. (My label above said C, but logically it is A. I will correct the data-answer attribute to A).
Score: 0 / 0