If 3 < a < 7 < b, which of the following must be greater than 20?
- A. a²
- B. 2b
- C. ab
- D. b + a
Correct Answer & Rationale
Correct Answer: C
To determine which option must be greater than 20, we analyze each one based on the inequalities provided (3 < a < 7 < b). **Option A: a²** Since a is less than 7, the maximum value for a² is 49 (when a=7), and the minimum value is 16 (when a=4). Thus, a² can be less than 20. **Option B: 2b** With b being greater than 7, the minimum value for 2b is 16 (when b=8). Therefore, 2b can also be less than 20. **Option C: ab** Given a is at least 4 and b is at least 8, the minimum value of ab is 32 (4*8). This must be greater than 20. **Option D: b + a** The minimum value for b + a is 11 (when a=4 and b=7), which is less than 20. Thus, only ab must consistently exceed 20.
To determine which option must be greater than 20, we analyze each one based on the inequalities provided (3 < a < 7 < b). **Option A: a²** Since a is less than 7, the maximum value for a² is 49 (when a=7), and the minimum value is 16 (when a=4). Thus, a² can be less than 20. **Option B: 2b** With b being greater than 7, the minimum value for 2b is 16 (when b=8). Therefore, 2b can also be less than 20. **Option C: ab** Given a is at least 4 and b is at least 8, the minimum value of ab is 32 (4*8). This must be greater than 20. **Option D: b + a** The minimum value for b + a is 11 (when a=4 and b=7), which is less than 20. Thus, only ab must consistently exceed 20.
Other Related Questions
A record store sold 100 copies of a CD in January. In February, the store's sales of the CD increased by 10 percent over the January sales. In March, the store sold 20 percent more copies of the CD than it sold in February. How many copies of the CD did the store sell in March?
- A. 120
- B. 122
- C. 130
- D. 132
Correct Answer & Rationale
Correct Answer: D
To find the number of CDs sold in March, start with January's sales of 100 copies. February's sales increased by 10%, resulting in 100 + (10% of 100) = 110 copies sold. In March, the store sold 20% more than February's sales: 110 + (20% of 110) = 110 + 22 = 132 copies. Option A (120) incorrectly assumes a lower percentage increase in February. Option B (122) miscalculates the increase in March. Option C (130) underestimates the sales for March by not applying the correct percentage increase. Thus, the accurate calculation leads to 132 copies sold in March.
To find the number of CDs sold in March, start with January's sales of 100 copies. February's sales increased by 10%, resulting in 100 + (10% of 100) = 110 copies sold. In March, the store sold 20% more than February's sales: 110 + (20% of 110) = 110 + 22 = 132 copies. Option A (120) incorrectly assumes a lower percentage increase in February. Option B (122) miscalculates the increase in March. Option C (130) underestimates the sales for March by not applying the correct percentage increase. Thus, the accurate calculation leads to 132 copies sold in March.
John worked at a bookstore for two weeks. The second week he earned 20 percent more than he did the first week. If he earned $300 the second week, how much did he earn the first week?
- A. 240
- B. 250
- C. 280
- D. 380
Correct Answer & Rationale
Correct Answer: B
To determine John’s earnings for the first week, we know that his second week earnings were 20% more than the first week. If he earned $300 in the second week, we can calculate his first week earnings by setting up the equation: Let x be the first week’s earnings. Then, x + 0.2x = 300. This simplifies to 1.2x = 300. Dividing both sides by 1.2 gives x = 250. Option A ($240) is too low, as it would not result in a $300 second week. Option C ($280) would imply a second week earning of $336, which exceeds $300. Option D ($380) is also incorrect as it suggests a second week earning of $456. Thus, $250 is the only viable answer.
To determine John’s earnings for the first week, we know that his second week earnings were 20% more than the first week. If he earned $300 in the second week, we can calculate his first week earnings by setting up the equation: Let x be the first week’s earnings. Then, x + 0.2x = 300. This simplifies to 1.2x = 300. Dividing both sides by 1.2 gives x = 250. Option A ($240) is too low, as it would not result in a $300 second week. Option C ($280) would imply a second week earning of $336, which exceeds $300. Option D ($380) is also incorrect as it suggests a second week earning of $456. Thus, $250 is the only viable answer.
1 is 3 percent of what number?
- A. 1/3
- B. 3
- C. 30
- D. 33,1/3
Correct Answer & Rationale
Correct Answer: D
To find the number of which 1 is 3 percent, we set up the equation: 1 = 0.03 * x. Solving for x gives x = 1 / 0.03, which equals 33.33 (or 33 1/3). Option A (1/3) is incorrect as it represents a much smaller value, specifically 0.33. Option B (3) misinterprets the percentage, suggesting that 1 is 33.33% of 3, which is not accurate. Option C (30) also fails, as 3% of 30 is 0.9, not 1. Thus, only option D correctly identifies the number as 33 1/3.
To find the number of which 1 is 3 percent, we set up the equation: 1 = 0.03 * x. Solving for x gives x = 1 / 0.03, which equals 33.33 (or 33 1/3). Option A (1/3) is incorrect as it represents a much smaller value, specifically 0.33. Option B (3) misinterprets the percentage, suggesting that 1 is 33.33% of 3, which is not accurate. Option C (30) also fails, as 3% of 30 is 0.9, not 1. Thus, only option D correctly identifies the number as 33 1/3.
The fraction x/24 is equal to 0.75. What is the value of x?
- A. 3
- B. 6
- C. 9
- D. 18
Correct Answer & Rationale
Correct Answer: D
To find the value of x in the equation x/24 = 0.75, we start by converting 0.75 to a fraction, which is 75/100 or 3/4. Setting the two fractions equal gives us x/24 = 3/4. Cross-multiplying leads to 4x = 72. Dividing both sides by 4 results in x = 18. Option A (3) is too low; substituting it back yields 3/24 = 0.125. Option B (6) also falls short, as 6/24 = 0.25. Option C (9) gives 9/24 = 0.375, still incorrect. Only option D (18) satisfies the original equation, confirming its validity.
To find the value of x in the equation x/24 = 0.75, we start by converting 0.75 to a fraction, which is 75/100 or 3/4. Setting the two fractions equal gives us x/24 = 3/4. Cross-multiplying leads to 4x = 72. Dividing both sides by 4 results in x = 18. Option A (3) is too low; substituting it back yields 3/24 = 0.125. Option B (6) also falls short, as 6/24 = 0.25. Option C (9) gives 9/24 = 0.375, still incorrect. Only option D (18) satisfies the original equation, confirming its validity.