4/9 (3/16 - 1/12) =
- A. 5/108
- B. 5/48
- C. 2/9
- D. 20/48
Correct Answer & Rationale
Correct Answer: A
To solve \( \frac{4}{9} \left( \frac{3}{16} - \frac{1}{12} \right) \), first calculate \( \frac{3}{16} - \frac{1}{12} \). Finding a common denominator (48), we convert the fractions: \( \frac{3}{16} = \frac{9}{48} \) and \( \frac{1}{12} = \frac{4}{48} \). Thus, \( \frac{9}{48} - \frac{4}{48} = \frac{5}{48} \). Next, multiply \( \frac{4}{9} \) by \( \frac{5}{48} \): \[ \frac{4 \times 5}{9 \times 48} = \frac{20}{432} = \frac{5}{108} \] Option B (5/48) is incorrect as it misrepresents the multiplication step. Option C (2/9) ignores the subtraction and multiplication entirely. Option D (20/48) fails to simplify the fraction correctly.
To solve \( \frac{4}{9} \left( \frac{3}{16} - \frac{1}{12} \right) \), first calculate \( \frac{3}{16} - \frac{1}{12} \). Finding a common denominator (48), we convert the fractions: \( \frac{3}{16} = \frac{9}{48} \) and \( \frac{1}{12} = \frac{4}{48} \). Thus, \( \frac{9}{48} - \frac{4}{48} = \frac{5}{48} \). Next, multiply \( \frac{4}{9} \) by \( \frac{5}{48} \): \[ \frac{4 \times 5}{9 \times 48} = \frac{20}{432} = \frac{5}{108} \] Option B (5/48) is incorrect as it misrepresents the multiplication step. Option C (2/9) ignores the subtraction and multiplication entirely. Option D (20/48) fails to simplify the fraction correctly.
Other Related Questions
Which of the following inequalities is correct?
- A. 2/3 < 3/5 < 5/7
- B. 2/3 < 5/7 < 3/5
- C. 3/5 < 2/3 < 5/7
- D. 3/5 < 5/7 < 2/3
Correct Answer & Rationale
Correct Answer: C
To determine the order of the fractions, we can convert them to decimals or find a common denominator. - **Option A (2/3 < 3/5 < 5/7)** is incorrect because 2/3 (approximately 0.67) is greater than 3/5 (0.6), violating the first inequality. - **Option B (2/3 < 5/7 < 3/5)** is also incorrect, as 5/7 (approximately 0.71) is greater than 2/3, making the first inequality false. - **Option D (3/5 < 5/7 < 2/3)** is incorrect because, while 3/5 is less than 5/7, 5/7 is greater than 2/3, contradicting the second inequality. - **Option C (3/5 < 2/3 < 5/7)** is accurate; 3/5 is indeed less than 2/3, and 2/3 is less than 5/7, maintaining the correct order.
To determine the order of the fractions, we can convert them to decimals or find a common denominator. - **Option A (2/3 < 3/5 < 5/7)** is incorrect because 2/3 (approximately 0.67) is greater than 3/5 (0.6), violating the first inequality. - **Option B (2/3 < 5/7 < 3/5)** is also incorrect, as 5/7 (approximately 0.71) is greater than 2/3, making the first inequality false. - **Option D (3/5 < 5/7 < 2/3)** is incorrect because, while 3/5 is less than 5/7, 5/7 is greater than 2/3, contradicting the second inequality. - **Option C (3/5 < 2/3 < 5/7)** is accurate; 3/5 is indeed less than 2/3, and 2/3 is less than 5/7, maintaining the correct order.
Linda has borrowed 8 more books than Susan from the school library. Richard has borrowed half as many books as Linda has. If Richard has borrowed 17 books from the library, how many books has Susan borrowed?
- A. 25
- B. 26
- C. 34
- D. 42
Correct Answer & Rationale
Correct Answer: B
To determine how many books Susan has borrowed, start with Richard's 17 books. Since Richard has borrowed half as many books as Linda, Linda must have borrowed 34 books (17 x 2). Given that Linda has borrowed 8 more books than Susan, we can set up the equation: Linda's books = Susan's books + 8. Therefore, if Linda has 34 books, we find Susan's total by subtracting 8: 34 - 8 = 26. Option A (25) is incorrect as it underestimates Susan's total. Option C (34) mistakenly suggests Susan borrowed the same amount as Linda. Option D (42) overestimates Susan's total by not accounting for the difference of 8 books. Thus, the only valid option is 26.
To determine how many books Susan has borrowed, start with Richard's 17 books. Since Richard has borrowed half as many books as Linda, Linda must have borrowed 34 books (17 x 2). Given that Linda has borrowed 8 more books than Susan, we can set up the equation: Linda's books = Susan's books + 8. Therefore, if Linda has 34 books, we find Susan's total by subtracting 8: 34 - 8 = 26. Option A (25) is incorrect as it underestimates Susan's total. Option C (34) mistakenly suggests Susan borrowed the same amount as Linda. Option D (42) overestimates Susan's total by not accounting for the difference of 8 books. Thus, the only valid option is 26.
Which of the following numbers is closest to 1?
- A. 4/5
- B. 5/4
- C. 5/6
- D. 6/5
Correct Answer & Rationale
Correct Answer: C
To determine which number is closest to 1, we can convert each option to decimal form: A: 4/5 = 0.8, which is 0.2 away from 1. B: 5/4 = 1.25, which is 0.25 away from 1. C: 5/6 ≈ 0.833, which is approximately 0.167 away from 1. D: 6/5 = 1.2, which is 0.2 away from 1. Among these, 5/6 is the closest to 1, as it has the smallest difference from 1 compared to the other options. The other fractions either exceed or fall short of 1 by a larger margin.
To determine which number is closest to 1, we can convert each option to decimal form: A: 4/5 = 0.8, which is 0.2 away from 1. B: 5/4 = 1.25, which is 0.25 away from 1. C: 5/6 ≈ 0.833, which is approximately 0.167 away from 1. D: 6/5 = 1.2, which is 0.2 away from 1. Among these, 5/6 is the closest to 1, as it has the smallest difference from 1 compared to the other options. The other fractions either exceed or fall short of 1 by a larger margin.
Harriet took 48 minutes to ride her bike the distance from her house to the town library. If she rode at a constant rate, what fraction of the total distance did she ride in the first 12 minutes?
- A. 1/4
- B. 1/3
- C. 1/2
- D. 3/4
Correct Answer & Rationale
Correct Answer: A
To determine the fraction of the total distance Harriet rode in the first 12 minutes, we start by recognizing that she took 48 minutes for the entire trip. Riding at a constant rate means that her distance covered is proportional to the time spent riding. In 12 minutes, which is one-fourth of the total 48 minutes, she would have covered one-fourth of the total distance. Thus, the fraction of the total distance she rode in the first 12 minutes is 1/4. Options B (1/3), C (1/2), and D (3/4) misrepresent the proportion of time to total time. Each suggests a greater fraction than what corresponds to 12 minutes relative to 48 minutes, leading to incorrect conclusions about the distance covered.
To determine the fraction of the total distance Harriet rode in the first 12 minutes, we start by recognizing that she took 48 minutes for the entire trip. Riding at a constant rate means that her distance covered is proportional to the time spent riding. In 12 minutes, which is one-fourth of the total 48 minutes, she would have covered one-fourth of the total distance. Thus, the fraction of the total distance she rode in the first 12 minutes is 1/4. Options B (1/3), C (1/2), and D (3/4) misrepresent the proportion of time to total time. Each suggests a greater fraction than what corresponds to 12 minutes relative to 48 minutes, leading to incorrect conclusions about the distance covered.