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.
Other Related Questions
How many solutions does the equation 3x + 9 = 3x - 12 have?
- B. 1
- C. 2
- D. 3
- E. Infinitely many
Correct Answer & Rationale
Correct Answer: A
To determine the number of solutions for the equation 3x + 9 = 3x - 12, we can simplify both sides. Subtracting 3x from each side results in 9 = -12, which is a false statement. Since the equation leads to a contradiction, it indicates that there are no values of x that can satisfy it. Option B (1 solution) suggests a single value exists, which is incorrect. Option C (2 solutions) and D (3 solutions) imply multiple valid values, which is also false. Option E (infinitely many solutions) suggests that any x would satisfy the equation, which is not true given the contradiction. Thus, the equation has no solutions.
To determine the number of solutions for the equation 3x + 9 = 3x - 12, we can simplify both sides. Subtracting 3x from each side results in 9 = -12, which is a false statement. Since the equation leads to a contradiction, it indicates that there are no values of x that can satisfy it. Option B (1 solution) suggests a single value exists, which is incorrect. Option C (2 solutions) and D (3 solutions) imply multiple valid values, which is also false. Option E (infinitely many solutions) suggests that any x would satisfy the equation, which is not true given the contradiction. Thus, the equation has no solutions.
The expression 6a + 4c represents the total price, in dollars, of admission to an air show for a adults and c children. On Saturday, 380 adults and 120 children paid admission to the air show. What was the total price of admission for those people?
- A. 524
- B. 2240
- C. 2760
- D. 5000
- E. 12000
Correct Answer & Rationale
Correct Answer: C
To find the total price of admission, substitute the values of adults (a) and children (c) into the expression 6a + 4c. Here, a = 380 and c = 120. Calculating: 6(380) + 4(120) = 2280 + 480 = 2760. Thus, the total price is 2760 dollars. Option A (524) is too low, as it doesn't account for the number of attendees. Option B (2240) underestimates the total, likely misunderstanding the pricing structure. Option D (5000) and Option E (12000) are excessively high, suggesting a miscalculation or misunderstanding of the pricing per adult and child.
To find the total price of admission, substitute the values of adults (a) and children (c) into the expression 6a + 4c. Here, a = 380 and c = 120. Calculating: 6(380) + 4(120) = 2280 + 480 = 2760. Thus, the total price is 2760 dollars. Option A (524) is too low, as it doesn't account for the number of attendees. Option B (2240) underestimates the total, likely misunderstanding the pricing structure. Option D (5000) and Option E (12000) are excessively high, suggesting a miscalculation or misunderstanding of the pricing per adult and child.
What is the sum of the two polynomials? 4x² + 3x + 5 + x² + 6x - 3?
- A. 4x² + 9x + 2
- B. 5x² + 9x + 2
- C. 5x² + 9x + 8
- D. 4x² + 9x² + 2
- E. 5x² + 9x² + 8
Correct Answer & Rationale
Correct Answer: B
To find the sum of the polynomials \(4x^2 + 3x + 5\) and \(x^2 + 6x - 3\), we combine like terms. 1. For \(x^2\) terms: \(4x^2 + x^2 = 5x^2\). 2. For \(x\) terms: \(3x + 6x = 9x\). 3. For constant terms: \(5 - 3 = 2\). Thus, the resulting polynomial is \(5x^2 + 9x + 2\), which corresponds to option B. Option A incorrectly adds the \(x^2\) terms, leading to an incorrect polynomial. Option C miscalculates the constant term. Option D mistakenly adds the \(x^2\) terms incorrectly and does not follow proper polynomial addition. Option E also miscalculates by incorrectly summing the \(x^2\) terms and the constants.
To find the sum of the polynomials \(4x^2 + 3x + 5\) and \(x^2 + 6x - 3\), we combine like terms. 1. For \(x^2\) terms: \(4x^2 + x^2 = 5x^2\). 2. For \(x\) terms: \(3x + 6x = 9x\). 3. For constant terms: \(5 - 3 = 2\). Thus, the resulting polynomial is \(5x^2 + 9x + 2\), which corresponds to option B. Option A incorrectly adds the \(x^2\) terms, leading to an incorrect polynomial. Option C miscalculates the constant term. Option D mistakenly adds the \(x^2\) terms incorrectly and does not follow proper polynomial addition. Option E also miscalculates by incorrectly summing the \(x^2\) terms and the constants.
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.