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
3/8 expressed as a percent is
- A. 3.75%
- B. 37.50%
- C. 38%
- D. 38,1/3%
Correct Answer & Rationale
Correct Answer: B
To convert a fraction to a percent, multiply by 100. For 3/8, the calculation is (3 ÷ 8) × 100, which equals 37.5%. This aligns with option B: 37.50%. Option A (3.75%) results from miscalculating the fraction, likely confusing the decimal representation. Option C (38%) rounds up incorrectly, as it does not accurately reflect the precise conversion. Option D (38, 1/3%) misrepresents the fraction by suggesting a value that exceeds the actual percentage, further indicating a misunderstanding of the conversion process. Thus, option B is the only accurate representation of 3/8 as a percent.
To convert a fraction to a percent, multiply by 100. For 3/8, the calculation is (3 ÷ 8) × 100, which equals 37.5%. This aligns with option B: 37.50%. Option A (3.75%) results from miscalculating the fraction, likely confusing the decimal representation. Option C (38%) rounds up incorrectly, as it does not accurately reflect the precise conversion. Option D (38, 1/3%) misrepresents the fraction by suggesting a value that exceeds the actual percentage, further indicating a misunderstanding of the conversion process. Thus, option B is the only accurate representation of 3/8 as a percent.
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.
The large square above has area 1 and is divided into 25 squares of equal area. Which of the following represents the area of the shaded region?
- A. 0.8
- B. 0.16
- C. 0.24
- D. 0.32
Correct Answer & Rationale
Correct Answer: D
In a large square with an area of 1, each of the 25 smaller squares has an area of \( \frac{1}{25} = 0.04 \). To find the area of the shaded region, count the number of shaded squares. If there are 8 shaded squares, then the area of the shaded region is \( 8 \times 0.04 = 0.32 \). Option A (0.8) is incorrect as it exceeds the total area of the large square. Option B (0.16) represents 4 shaded squares, which is not consistent with the given information. Option C (0.24) suggests 6 shaded squares, which also does not match. Thus, the area of the shaded region is accurately represented by option D, 0.32.
In a large square with an area of 1, each of the 25 smaller squares has an area of \( \frac{1}{25} = 0.04 \). To find the area of the shaded region, count the number of shaded squares. If there are 8 shaded squares, then the area of the shaded region is \( 8 \times 0.04 = 0.32 \). Option A (0.8) is incorrect as it exceeds the total area of the large square. Option B (0.16) represents 4 shaded squares, which is not consistent with the given information. Option C (0.24) suggests 6 shaded squares, which also does not match. Thus, the area of the shaded region is accurately represented by option D, 0.32.
6 + 5,1/3 ÷ (6 - 5,1/3) =
- A. 1,1/3
- B. 5,1/3
- C. 16
- D. 17
Correct Answer & Rationale
Correct Answer: C
To solve the equation, first evaluate the expression in the parentheses: \(6 - 5\frac{1}{3}\) equals \(6 - \frac{16}{3} = \frac{18}{3} - \frac{16}{3} = \frac{2}{3}\). Next, compute \(5\frac{1}{3}\) as \(\frac{16}{3}\). The equation now reads \(6 + \frac{16}{3} \div \frac{2}{3}\). Dividing \(\frac{16}{3}\) by \(\frac{2}{3}\) gives \(8\). Adding this to \(6\) results in \(14\), leading to the final answer of \(16\). Option A (1\(\frac{1}{3}\)) is incorrect due to miscalculating the operations. Option B (5\(\frac{1}{3}\)) fails to account for the division correctly. Option D (17) mistakenly adds an extra unit instead of properly evaluating the expression.
To solve the equation, first evaluate the expression in the parentheses: \(6 - 5\frac{1}{3}\) equals \(6 - \frac{16}{3} = \frac{18}{3} - \frac{16}{3} = \frac{2}{3}\). Next, compute \(5\frac{1}{3}\) as \(\frac{16}{3}\). The equation now reads \(6 + \frac{16}{3} \div \frac{2}{3}\). Dividing \(\frac{16}{3}\) by \(\frac{2}{3}\) gives \(8\). Adding this to \(6\) results in \(14\), leading to the final answer of \(16\). Option A (1\(\frac{1}{3}\)) is incorrect due to miscalculating the operations. Option B (5\(\frac{1}{3}\)) fails to account for the division correctly. Option D (17) mistakenly adds an extra unit instead of properly evaluating the expression.