When Henry plays the songs on the playlist in a random order, what is the probability a rock song will be played first?
- A. 3/4
- B. 1/3
- C. 1/4
- D. 3/10
- E. 5/16
Correct Answer & Rationale
Correct Answer: D
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.
Other Related Questions
Mallory loaded 200 digital pictures into a digital picture frame. 78 are pictures of family members, 26 are pictures of pets, the rest are pictures of friends. The frame displays one picture every 10 seconds. Which value is closest to the probability that the next picture the frame displays will be a picture of a friend?
- A. 0.33
- B. 0.43
- C. 0.48
- D. 0.52
- E. 0.96
Correct Answer & Rationale
Correct Answer: C
To find the probability that the next picture displayed is of a friend, first calculate the total number of friend pictures. There are 200 total pictures, with 78 family and 26 pet pictures, leaving 200 - 78 - 26 = 96 pictures of friends. The probability is then the number of friend pictures divided by the total: 96/200 = 0.48. Option A (0.33) underestimates the proportion of friend pictures. Option B (0.43) is also lower than the calculated probability. Option D (0.52) slightly overestimates it, and option E (0.96) is far too high, misrepresenting the actual count. Thus, 0.48 accurately reflects the likelihood of displaying a friend picture next.
To find the probability that the next picture displayed is of a friend, first calculate the total number of friend pictures. There are 200 total pictures, with 78 family and 26 pet pictures, leaving 200 - 78 - 26 = 96 pictures of friends. The probability is then the number of friend pictures divided by the total: 96/200 = 0.48. Option A (0.33) underestimates the proportion of friend pictures. Option B (0.43) is also lower than the calculated probability. Option D (0.52) slightly overestimates it, and option E (0.96) is far too high, misrepresenting the actual count. Thus, 0.48 accurately reflects the likelihood of displaying a friend picture next.
Isabel earns $15.80 per hour for the first 8 hours she works each day. She earns 1.5 times her hourly rate for all time after the first 8 hours. How much does Isabel earn on a day when she works 8.5 hours?
- A. 126.4
- B. 138.25
- C. 189.6
- D. 201.45
- E. 237
Correct Answer & Rationale
Correct Answer: B
To determine Isabel's earnings for an 8.5-hour workday, first calculate her earnings for the first 8 hours at $15.80 per hour, which totals $126.40 (8 hours × $15.80/hour). For the additional 0.5 hours, she earns 1.5 times her hourly rate, which is $23.70 (1.5 × $15.80). Therefore, for the extra half hour, she earns $11.85 (0.5 hours × $23.70/hour). Adding these amounts together gives $138.25 ($126.40 + $11.85). Option A ($126.40) only accounts for the first 8 hours. Option C ($189.60) incorrectly assumes full-time pay without considering the overtime rate. Option D ($201.45) miscalculates the overtime pay, while Option E ($237) overestimates by not applying the correct hourly rates.
To determine Isabel's earnings for an 8.5-hour workday, first calculate her earnings for the first 8 hours at $15.80 per hour, which totals $126.40 (8 hours × $15.80/hour). For the additional 0.5 hours, she earns 1.5 times her hourly rate, which is $23.70 (1.5 × $15.80). Therefore, for the extra half hour, she earns $11.85 (0.5 hours × $23.70/hour). Adding these amounts together gives $138.25 ($126.40 + $11.85). Option A ($126.40) only accounts for the first 8 hours. Option C ($189.60) incorrectly assumes full-time pay without considering the overtime rate. Option D ($201.45) miscalculates the overtime pay, while Option E ($237) overestimates by not applying the correct hourly rates.
The number of years the employee has been employed by the city is at least 25 years. The sum of the employee's age and number of years employed by the city is at least 90 years. Larry has been employed by the city since his 38th birthday. Assuming he continues to work for the city, at what age will he first qualify for full retirement benefits?
- A. 52
- B. 55
- C. 62
- D. 63
- E. 64
Correct Answer & Rationale
Correct Answer: E
To qualify for full retirement benefits, Larry must be at least 25 years employed and have a combined age and years of service of at least 90 years. Since he started working at age 38, he will reach 25 years of employment at age 63. At that point, his age (63) plus his years of service (25) totals 88, which does not meet the 90-year requirement. At age 64, he will have 26 years of service, bringing the total to 90 years (64 + 26), thus meeting both criteria. Options A (52), B (55), and C (62) do not allow for 25 years of service, while D (63) fails to meet the age and service sum requirement.
To qualify for full retirement benefits, Larry must be at least 25 years employed and have a combined age and years of service of at least 90 years. Since he started working at age 38, he will reach 25 years of employment at age 63. At that point, his age (63) plus his years of service (25) totals 88, which does not meet the 90-year requirement. At age 64, he will have 26 years of service, bringing the total to 90 years (64 + 26), thus meeting both criteria. Options A (52), B (55), and C (62) do not allow for 25 years of service, while D (63) fails to meet the age and service sum requirement.
A medium-sized grain of sand can be approximated as a cube with an edge length of 5×10â»â´ meters. Which expression best represents the number of medium-sized sand grains that could be lined up side by side to result in a total length of 1 meter?
- A. 2×10³
- B. 2×10â´
- C. 2×10âµ
- D. 5×10³
- E. 5×10â´
Correct Answer & Rationale
Correct Answer: B
To determine how many medium-sized sand grains can be lined up to equal 1 meter, we first calculate the volume of one grain, approximated as a cube with an edge length of 5×10⁻⁴ meters. The length of one grain is 5×10⁻⁴ meters. To find the number of grains in 1 meter, divide 1 meter (1×10⁰) by the length of one grain: 1×10⁰ / 5×10⁻⁴ = 2×10³. Thus, option B (2×10³) accurately represents the number of grains. Options A (2×10³) and D (5×10³) are incorrect due to miscalculating the division. Option C (2×10⁻) and E (5×10⁵) misrepresent the scale entirely, either by underestimating or overestimating the number of grains.
To determine how many medium-sized sand grains can be lined up to equal 1 meter, we first calculate the volume of one grain, approximated as a cube with an edge length of 5×10⁻⁴ meters. The length of one grain is 5×10⁻⁴ meters. To find the number of grains in 1 meter, divide 1 meter (1×10⁰) by the length of one grain: 1×10⁰ / 5×10⁻⁴ = 2×10³. Thus, option B (2×10³) accurately represents the number of grains. Options A (2×10³) and D (5×10³) are incorrect due to miscalculating the division. Option C (2×10⁻) and E (5×10⁵) misrepresent the scale entirely, either by underestimating or overestimating the number of grains.