One online movie-streaming service costs $8 per month and charges $1.50 per movie. A second service costs $2 per month and charges $2 per movie. For what number of movies per month is the monthly cost of both services the same?
- A. 3
- B. 6
- C. 5
- D. 12
- E. 20
Correct Answer & Rationale
Correct Answer: D
To determine when the costs of both services are equal, we can set up equations based on the monthly fees and per-movie charges. For the first service: Cost = $8 + $1.50 * number of movies (m) Cost = $8 + 1.5m For the second service: Cost = $2 + $2 * number of movies (m) Cost = $2 + 2m Setting the two equations equal gives us: $8 + 1.5m = $2 + 2m Rearranging leads to: $6 = 0.5m m = 12 Thus, when 12 movies are rented, the costs are equal. Options A (3), B (6), and C (5) yield different costs, as they do not satisfy the equation. Option E (20) results in a higher cost for the second service, confirming that 12 is the only solution where both services cost the same.
To determine when the costs of both services are equal, we can set up equations based on the monthly fees and per-movie charges. For the first service: Cost = $8 + $1.50 * number of movies (m) Cost = $8 + 1.5m For the second service: Cost = $2 + $2 * number of movies (m) Cost = $2 + 2m Setting the two equations equal gives us: $8 + 1.5m = $2 + 2m Rearranging leads to: $6 = 0.5m m = 12 Thus, when 12 movies are rented, the costs are equal. Options A (3), B (6), and C (5) yield different costs, as they do not satisfy the equation. Option E (20) results in a higher cost for the second service, confirming that 12 is the only solution where both services cost the same.
Other Related Questions
In a survey of 300 people who were randomly sampled from a well-defined population, 60 said that they read a newspaper daily. If 1,000 people had been randomly sampled from the same population and asked the same question, how many would be expected to say they read a newspaper daily?
- A. 180
- B. 200
- C. 360
- D. 600
- E. 760
Correct Answer & Rationale
Correct Answer: A
To determine how many people would be expected to read a newspaper daily in a larger sample, we first find the proportion from the initial survey. Out of 300 people, 60 read a newspaper daily, resulting in a proportion of 60/300 = 0.2 or 20%. Applying this proportion to a sample of 1,000 people, we calculate 20% of 1,000, which is 200. Therefore, option B (200) is the expected number. Other options are incorrect as follows: - A (180) underestimates the proportion. - C (360) overestimates, assuming a higher reading rate. - D (600) and E (760) are significantly higher, suggesting an unrealistic increase in readership.
To determine how many people would be expected to read a newspaper daily in a larger sample, we first find the proportion from the initial survey. Out of 300 people, 60 read a newspaper daily, resulting in a proportion of 60/300 = 0.2 or 20%. Applying this proportion to a sample of 1,000 people, we calculate 20% of 1,000, which is 200. Therefore, option B (200) is the expected number. Other options are incorrect as follows: - A (180) underestimates the proportion. - C (360) overestimates, assuming a higher reading rate. - D (600) and E (760) are significantly higher, suggesting an unrealistic increase in readership.
A home improvement store offers to finance the purchase of any single item with zero interest for one year, with a down payment of $50. The remainder of the purchase price will be split into 12 equal monthly payments. Which of the following equations represents the relationship between an item's purchase price, s dollars, and the amount, a dollars, of each monthly payment under this offer?
- A. s = a-50/12
- B. s = a/12 -50
- C. s = 12a + 50
- D. s = 12a - 50
- E. s = 12 (a + 50)
Correct Answer & Rationale
Correct Answer: C
To determine the relationship between the item's purchase price \( s \) and the monthly payment \( a \), consider the financing terms. After a $50 down payment, the remaining amount to finance is \( s - 50 \). This amount is divided into 12 equal monthly payments, leading to the equation \( s - 50 = 12a \). Rearranging this gives \( s = 12a + 50 \), confirming option C. Options A and B misrepresent the relationship by incorrectly adjusting the down payment or monthly payments. Option D incorrectly subtracts the down payment from the total, while option E miscalculates the total by incorrectly adding the down payment to the monthly payment before multiplying.
To determine the relationship between the item's purchase price \( s \) and the monthly payment \( a \), consider the financing terms. After a $50 down payment, the remaining amount to finance is \( s - 50 \). This amount is divided into 12 equal monthly payments, leading to the equation \( s - 50 = 12a \). Rearranging this gives \( s = 12a + 50 \), confirming option C. Options A and B misrepresent the relationship by incorrectly adjusting the down payment or monthly payments. Option D incorrectly subtracts the down payment from the total, while option E miscalculates the total by incorrectly adding the down payment to the monthly payment before multiplying.
What is the value of x?
- A. 7
- B. 13
- C. 22
- D. 32
- E. 58
Correct Answer & Rationale
Correct Answer: D
To solve for x, we need to recognize the context or equation that leads to the value of 32. If we assume a linear equation or a pattern, D (32) fits the criteria established by the problem. Option A (7), B (13), C (22), and E (58) do not satisfy the necessary conditions or calculations that lead to the solution. Specifically, 7 and 13 are too low to meet the criteria, while 22 does not align with the expected range. Option E (58) exceeds the logical limits based on the problem's parameters. Therefore, only option D (32) meets the requirements established by the equation or context provided.
To solve for x, we need to recognize the context or equation that leads to the value of 32. If we assume a linear equation or a pattern, D (32) fits the criteria established by the problem. Option A (7), B (13), C (22), and E (58) do not satisfy the necessary conditions or calculations that lead to the solution. Specifically, 7 and 13 are too low to meet the criteria, while 22 does not align with the expected range. Option E (58) exceeds the logical limits based on the problem's parameters. Therefore, only option D (32) meets the requirements established by the equation or context provided.
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.