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
Which of the following inequalities is true?
- A. 0.7 < 0.1 < 0.11 < 0.101
- B. 0.1 < 0.7 < 0.101 < 0.11
- C. 0.1 < 0.7 < 0.11 < 0.101
- D. 0.1 < 0.101 < 0.11 < 0.7
Correct Answer & Rationale
Correct Answer: D
Option D accurately represents the correct order of the numbers. When comparing the values, 0.1 is the smallest, followed by 0.101, then 0.11, and finally 0.7, which is the largest. Option A is incorrect as it mistakenly places 0.7 as less than both 0.1 and 0.11, which is not true. Option B incorrectly suggests that 0.101 is less than 0.11, which is also inaccurate. Option C places 0.11 before 0.101, misrepresenting their actual values. Thus, D is the only option that correctly orders the numbers from smallest to largest.
Option D accurately represents the correct order of the numbers. When comparing the values, 0.1 is the smallest, followed by 0.101, then 0.11, and finally 0.7, which is the largest. Option A is incorrect as it mistakenly places 0.7 as less than both 0.1 and 0.11, which is not true. Option B incorrectly suggests that 0.101 is less than 0.11, which is also inaccurate. Option C places 0.11 before 0.101, misrepresenting their actual values. Thus, D is the only option that correctly orders the numbers from smallest to largest.
3 × (1/2 + 1/3) =
- A. 2,1/2
- B. 2,5/6
- C. 3,1/6
- D. 3,5/6
Correct Answer & Rationale
Correct Answer: A
To solve 3 × (1/2 + 1/3), first find a common denominator for the fractions 1/2 and 1/3, which is 6. This gives us (3/6 + 2/6) = 5/6. Multiplying by 3 results in 3 × (5/6) = 15/6, which simplifies to 2 1/2 (Option A). Option B (2 5/6) incorrectly adds an extra fraction. Option C (3 1/6) miscalculates the multiplication. Option D (3 5/6) also misinterprets the original problem, leading to an incorrect total. Thus, only Option A accurately represents the solution.
To solve 3 × (1/2 + 1/3), first find a common denominator for the fractions 1/2 and 1/3, which is 6. This gives us (3/6 + 2/6) = 5/6. Multiplying by 3 results in 3 × (5/6) = 15/6, which simplifies to 2 1/2 (Option A). Option B (2 5/6) incorrectly adds an extra fraction. Option C (3 1/6) miscalculates the multiplication. Option D (3 5/6) also misinterprets the original problem, leading to an incorrect total. Thus, only Option A accurately represents the solution.
60 ÷ 3/3 =
- A. 20
- B. 21
- C. 23
- D. 24
Correct Answer & Rationale
Correct Answer: A
To solve 60 ÷ 3/3, first simplify the expression. Dividing by a fraction involves multiplying by its reciprocal. Therefore, 3/3 equals 1, and dividing by 1 does not change the value. Thus, the equation simplifies to 60 ÷ 1, which equals 60. Now, let's analyze the options: A: 20 is incorrect as it does not represent the result of the division. B: 21 is also incorrect, being too low compared to the actual value. C: 23 is incorrect for the same reason, as it underestimates the result. D: 24 is incorrect and does not reflect the correct division outcome. The only accurate interpretation leads to the conclusion that 60 divided by 1 remains 60.
To solve 60 ÷ 3/3, first simplify the expression. Dividing by a fraction involves multiplying by its reciprocal. Therefore, 3/3 equals 1, and dividing by 1 does not change the value. Thus, the equation simplifies to 60 ÷ 1, which equals 60. Now, let's analyze the options: A: 20 is incorrect as it does not represent the result of the division. B: 21 is also incorrect, being too low compared to the actual value. C: 23 is incorrect for the same reason, as it underestimates the result. D: 24 is incorrect and does not reflect the correct division outcome. The only accurate interpretation leads to the conclusion that 60 divided by 1 remains 60.
165 is what percent of 150?
- A. 95%
- B. 110%
- C. 111%
- D. 115%
Correct Answer & Rationale
Correct Answer: B
To find what percent 165 is of 150, divide 165 by 150 and then multiply by 100. This calculation yields 110%, indicating that 165 is 110% of 150. Option A (95%) underestimates the value, as it suggests 165 is less than 150. Option C (111%) slightly overestimates the percentage, as it does not accurately reflect the calculation. Option D (115%) also exaggerates the value, implying that 165 exceeds 150 by a larger margin than it actually does. Therefore, 110% is the precise representation of 165 in relation to 150.
To find what percent 165 is of 150, divide 165 by 150 and then multiply by 100. This calculation yields 110%, indicating that 165 is 110% of 150. Option A (95%) underestimates the value, as it suggests 165 is less than 150. Option C (111%) slightly overestimates the percentage, as it does not accurately reflect the calculation. Option D (115%) also exaggerates the value, implying that 165 exceeds 150 by a larger margin than it actually does. Therefore, 110% is the precise representation of 165 in relation to 150.