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.
Other Related Questions
A campground rents canoes for either $20 per day or $4 per hour. For what number or numbers of hours, h, is it more expensive to rent a canoe at the daily rate than at the hourly rate?
- A. h = 5
- B. h >= 25
- C. h > 5
- D. h < 5
- E. h ≤ 5
Correct Answer & Rationale
Correct Answer: C
To determine when renting a canoe at the daily rate exceeds the hourly rate, we compare the costs. The daily rate is $20, while the hourly rate is $4 per hour. Setting up the inequality, we have: \[ 20 > 4h \] Dividing both sides by 4 gives: \[ 5 > h \] This means that renting for more than 5 hours makes the daily rate more economical. Option A (h = 5) is incorrect since at 5 hours, both rates are equal. Option B (h ≥ 25) is incorrect because it's not relevant to the threshold we calculated. Option D (h < 5) suggests a scenario where the daily rate is not more expensive, which contradicts our findings. Option E (h ≤ 5) includes values where the rates are equal or less, which doesn't satisfy the condition.
To determine when renting a canoe at the daily rate exceeds the hourly rate, we compare the costs. The daily rate is $20, while the hourly rate is $4 per hour. Setting up the inequality, we have: \[ 20 > 4h \] Dividing both sides by 4 gives: \[ 5 > h \] This means that renting for more than 5 hours makes the daily rate more economical. Option A (h = 5) is incorrect since at 5 hours, both rates are equal. Option B (h ≥ 25) is incorrect because it's not relevant to the threshold we calculated. Option D (h < 5) suggests a scenario where the daily rate is not more expensive, which contradicts our findings. Option E (h ≤ 5) includes values where the rates are equal or less, which doesn't satisfy the condition.
Let f(x) = 3x². What is f(-2x)?
- A. -36x²
- B. -12x²
- C. -6x²
- D. 12x²
- E. 36x²
Correct Answer & Rationale
Correct Answer: D
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.
Each month, the charge for a lawn care service consists of a flat fee of $25, plus $5 each time the lawn is mowed. Which of the following equations represents the total monthly charge, A(m), in dollars, as a function of the number of times the lawn is mowed, m?
- A. A(m) = 5(25)m
- B. A(m) = 5 + 25m
- C. A(m) = 5m + 25
- D. A(m) = 25m + 5
- E. A(m) = m + 5 + 25
Correct Answer & Rationale
Correct Answer: C
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.
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.