Factor the expression completely: -3x - 21
- A. -3(x+7)
- B. -3(x-21)
- C. -3(x-7)
- D. -3(x+21)
Correct Answer & Rationale
Correct Answer: A
To factor the expression -3x - 21 completely, start by identifying the common factor in both terms. Here, -3 is the greatest common factor. When factoring out -3 from -3x, you're left with x, and from -21, you have +7. Thus, the expression can be rewritten as -3(x + 7). Option B, -3(x - 21), is incorrect because factoring out -3 from -21 should yield +7, not -21. Option C, -3(x - 7), incorrectly represents the constant term, as it should be +7. Option D, -3(x + 21), misrepresents the factorization entirely, as it does not reflect the original expression's terms.
To factor the expression -3x - 21 completely, start by identifying the common factor in both terms. Here, -3 is the greatest common factor. When factoring out -3 from -3x, you're left with x, and from -21, you have +7. Thus, the expression can be rewritten as -3(x + 7). Option B, -3(x - 21), is incorrect because factoring out -3 from -21 should yield +7, not -21. Option C, -3(x - 7), incorrectly represents the constant term, as it should be +7. Option D, -3(x + 21), misrepresents the factorization entirely, as it does not reflect the original expression's terms.
Other Related Questions
At a local bank, certificates of deposit (CDs) mature every 9 months. At another bank, CDs mature every 12 months. If CDs are purchased on the same day at each bank and are renewed when they mature, what is the least number of months that will pass before the two banks' CDs are mature at the same time?
- A. 72
- B. 36
- C. 108
- D. 3
Correct Answer & Rationale
Correct Answer: B
To find when the CDs from both banks mature simultaneously, we need to determine the least common multiple (LCM) of their maturity periods: 9 months and 12 months. Calculating the LCM, we see that the multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81. The multiples of 12 are 12, 24, 36, 48, 60, 72, and 84. The smallest common multiple is 36 months. Option A (72) is incorrect as it’s not the smallest shared maturity. Option C (108) is also incorrect; it exceeds the LCM. Option D (3) is far too short, as it does not accommodate either maturity period. Thus, 36 months is the earliest point both CDs will mature together.
To find when the CDs from both banks mature simultaneously, we need to determine the least common multiple (LCM) of their maturity periods: 9 months and 12 months. Calculating the LCM, we see that the multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81. The multiples of 12 are 12, 24, 36, 48, 60, 72, and 84. The smallest common multiple is 36 months. Option A (72) is incorrect as it’s not the smallest shared maturity. Option C (108) is also incorrect; it exceeds the LCM. Option D (3) is far too short, as it does not accommodate either maturity period. Thus, 36 months is the earliest point both CDs will mature together.
For an emergency service call, a plumbing company charges a flat fee of $60 plus $40 an hour. A customer remembers paying at least $200 for an emergency service. Which phrase describes the number of hours the plumbing company was at the service call?
- A. at most 2 hours
- B. at most 3.5 hours
- C. at least 3.5 hours
- D. at least 2 hours
Correct Answer & Rationale
Correct Answer: C
To determine the number of hours the plumbing company was on the service call, we start with the total charge of at least $200. The charge consists of a flat fee of $60 plus $40 per hour. First, subtract the flat fee from the total: $200 - $60 = $140. Next, divide this by the hourly rate: $140 ÷ $40 = 3.5 hours. This indicates that the service lasted at least 3.5 hours. Option A (at most 2 hours) is incorrect, as 2 hours would only cost $140. Option B (at most 3.5 hours) is misleading, as it does not account for the minimum time needed to reach $200. Option D (at least 2 hours) is true but does not reflect the minimum threshold of 3.5 hours. Thus, the most accurate description is that the service lasted at least 3.5 hours.
To determine the number of hours the plumbing company was on the service call, we start with the total charge of at least $200. The charge consists of a flat fee of $60 plus $40 per hour. First, subtract the flat fee from the total: $200 - $60 = $140. Next, divide this by the hourly rate: $140 ÷ $40 = 3.5 hours. This indicates that the service lasted at least 3.5 hours. Option A (at most 2 hours) is incorrect, as 2 hours would only cost $140. Option B (at most 3.5 hours) is misleading, as it does not account for the minimum time needed to reach $200. Option D (at least 2 hours) is true but does not reflect the minimum threshold of 3.5 hours. Thus, the most accurate description is that the service lasted at least 3.5 hours.
A cyclist can travel 17.6 feet per second. The cyclist would have a better understanding of her speed if it were measured in miles per hour. Which of these completes the expression used to convert the speed of the cyclist to miles per hour?
- A. 1 hour/60 seconds = 1 mile/5,280 feet
- B. 60 minutes/1 hour = 1 mile/5280 feet
- C. 60 minutes/1 hour = 5280 feet/1 mile
- D. 12 inches/1 foot = 60 minutes/1 hour
Correct Answer & Rationale
Correct Answer: C
To convert speed from feet per second to miles per hour, the conversion factors must relate time and distance appropriately. Option C correctly expresses the relationship between miles and feet, stating that 1 mile equals 5280 feet. Additionally, it includes the conversion of minutes to hours, with 60 minutes equating to 1 hour, which is essential for converting seconds to hours. Option A incorrectly suggests a different time conversion that mixes hours and seconds without properly aligning the units. Option B, while correctly stating the time conversion, mistakenly places the units in an incorrect order. Option D is irrelevant, as it focuses on inches and does not contribute to the necessary conversions for speed.
To convert speed from feet per second to miles per hour, the conversion factors must relate time and distance appropriately. Option C correctly expresses the relationship between miles and feet, stating that 1 mile equals 5280 feet. Additionally, it includes the conversion of minutes to hours, with 60 minutes equating to 1 hour, which is essential for converting seconds to hours. Option A incorrectly suggests a different time conversion that mixes hours and seconds without properly aligning the units. Option B, while correctly stating the time conversion, mistakenly places the units in an incorrect order. Option D is irrelevant, as it focuses on inches and does not contribute to the necessary conversions for speed.
Last weekend, 625 runners entered a 10,000-meter race. A 10,000- meter race is 6.2 miles long. Ruben won the race with a finishing time of 29 minutes 51 seconds.
The graphs show information about the top 10 runners.
Based on the histogram, which statement describes the finishing time of the runner in position 3?
- A. The finishing time was between 30 and 31 minutes
- B. The finishing time was between 33 and 34 minutes
- C. The finishing time was between 31 and 32 minutes
- D. The finishing time was between 32 and 33 minutes
Correct Answer & Rationale
Correct Answer: C
The finishing time for the runner in position 3 falls within the 31 to 32 minutes range, as indicated by the histogram. This range is supported by the data distribution, showing that the majority of runners in the top 10 finished within this timeframe. Option A is incorrect because it suggests a time between 30 and 31 minutes, which does not align with the position 3 runner's time. Option B is inaccurate, as it indicates a finishing time between 33 and 34 minutes, which is too high for this position. Option D, while close, incorrectly suggests a time between 32 and 33 minutes, which does not match the histogram data for the third place runner.
The finishing time for the runner in position 3 falls within the 31 to 32 minutes range, as indicated by the histogram. This range is supported by the data distribution, showing that the majority of runners in the top 10 finished within this timeframe. Option A is incorrect because it suggests a time between 30 and 31 minutes, which does not align with the position 3 runner's time. Option B is inaccurate, as it indicates a finishing time between 33 and 34 minutes, which is too high for this position. Option D, while close, incorrectly suggests a time between 32 and 33 minutes, which does not match the histogram data for the third place runner.