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
Alexia, Bob, and Comelia recorded the number of pages of books they read last month. Alexia read 135 pages, Bob read 26 pages less than Alexia, and Comelia read 3 and one-half times more pages than Alexia and Bob combined. Which of the following represents the total number of pages that Alexia, Bob, and Comelia read last month?
- A. 3.5(135 + 26)
- B. 3.5[2(135) - 26]
- C. 4.5[2(135) - 26]
- D. 4.5[2(135) + 26]
Correct Answer & Rationale
Correct Answer: C
To determine the total number of pages read, first calculate Bob's pages: he read 135 - 26 = 109 pages. The combined pages of Alexia and Bob is 135 + 109 = 244 pages. Comelia read 3.5 times this total, resulting in 3.5 × 244. Option A incorrectly uses 135 + 26, which does not account for Bob's actual pages read. Option B mistakenly uses a subtraction instead of addition for the combined total. Option D incorrectly adds Bob's pages instead of using the correct combined total for Comelia's calculation. Thus, C accurately represents the total with 3.5(244), leading to the correct final total.
To determine the total number of pages read, first calculate Bob's pages: he read 135 - 26 = 109 pages. The combined pages of Alexia and Bob is 135 + 109 = 244 pages. Comelia read 3.5 times this total, resulting in 3.5 × 244. Option A incorrectly uses 135 + 26, which does not account for Bob's actual pages read. Option B mistakenly uses a subtraction instead of addition for the combined total. Option D incorrectly adds Bob's pages instead of using the correct combined total for Comelia's calculation. Thus, C accurately represents the total with 3.5(244), leading to the correct final total.
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.
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.
2,3/8 + 5,5/6 =
- A. 7,5/24
- B. 7,4/7
- C. 8,5/24
- D. 8,4/7
Correct Answer & Rationale
Correct Answer: C
To solve 2,3/8 + 5,5/6, first convert the mixed numbers into improper fractions. For 2,3/8, this becomes (2 * 8 + 3)/8 = 19/8. For 5,5/6, it is (5 * 6 + 5)/6 = 35/6. Next, find a common denominator, which is 24. Convert the fractions: 19/8 becomes 57/24, and 35/6 becomes 140/24. Adding these gives 197/24, which converts back to a mixed number as 8,5/24. Options A and B do not match this result. Option D, while close, inaccurately represents the fraction.
To solve 2,3/8 + 5,5/6, first convert the mixed numbers into improper fractions. For 2,3/8, this becomes (2 * 8 + 3)/8 = 19/8. For 5,5/6, it is (5 * 6 + 5)/6 = 35/6. Next, find a common denominator, which is 24. Convert the fractions: 19/8 becomes 57/24, and 35/6 becomes 140/24. Adding these gives 197/24, which converts back to a mixed number as 8,5/24. Options A and B do not match this result. Option D, while close, inaccurately represents the fraction.