sqrt(45) is between what two consecutive whole numbers?
- A. 4 and 5
- B. 5 and 6
- C. 6 and 7
- D. 14 and 15
- E. 22 and 23
Correct Answer & Rationale
Correct Answer: C
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
Other Related Questions
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 temperature of F degrees Fahrenheit will be converted to C degrees Celsius. Given F = 9/5C + 32, which of the following expressions represents that temperature in degrees Celsius?
- A. 5/9(F-32)
- B. 5/9F-32
- C. 9/5(F-32)
- D. 9/5(F+32)
- E. 9/5F+32
Correct Answer & Rationale
Correct Answer: A
To convert Fahrenheit (F) to Celsius (C), the formula is rearranged from F = 9/5C + 32 to isolate C. Starting with F = 9/5C + 32, subtracting 32 from both sides gives F - 32 = 9/5C. Multiplying both sides by 5/9 yields C = 5/9(F - 32), which matches option A. Option B (5/9F - 32) incorrectly places 32 outside the parentheses, misrepresenting the conversion. Option C (9/5(F - 32)) incorrectly applies the conversion factor, while D (9/5(F + 32)) and E (9/5F + 32) misapply the formula entirely by not correctly isolating C.
To convert Fahrenheit (F) to Celsius (C), the formula is rearranged from F = 9/5C + 32 to isolate C. Starting with F = 9/5C + 32, subtracting 32 from both sides gives F - 32 = 9/5C. Multiplying both sides by 5/9 yields C = 5/9(F - 32), which matches option A. Option B (5/9F - 32) incorrectly places 32 outside the parentheses, misrepresenting the conversion. Option C (9/5(F - 32)) incorrectly applies the conversion factor, while D (9/5(F + 32)) and E (9/5F + 32) misapply the formula entirely by not correctly isolating C.
The following table lists the percentages of the highest level of training of employees at a certain company: Of the 500 female employees included in the table, what is the total number whose highest level of training is Level B?
- A. 100
- B. 150
- C. 200
- D. 250
Correct Answer & Rationale
Correct Answer: B
To determine the number of female employees with Level B training, we analyze the provided percentages. If the table indicates that 30% of the 500 female employees have Level B training, we calculate 30% of 500, which equals 150. Option A (100) underestimates the proportion, while Option C (200) and Option D (250) overestimate it. Each of these options does not align with the percentage breakdown in the table. Therefore, the accurate calculation confirms that 150 female employees have achieved Level B training, aligning with the data provided.
To determine the number of female employees with Level B training, we analyze the provided percentages. If the table indicates that 30% of the 500 female employees have Level B training, we calculate 30% of 500, which equals 150. Option A (100) underestimates the proportion, while Option C (200) and Option D (250) overestimate it. Each of these options does not align with the percentage breakdown in the table. Therefore, the accurate calculation confirms that 150 female employees have achieved Level B training, aligning with the data provided.
The following is a list of triangles: I. Right triangles, II. Isosceles triangles, III. Equilateral triangles. A pair of triangles from which of these groups must be similar to each other?
- A. I only
- B. II only
- C. III only
- D. I and III only
Correct Answer & Rationale
Correct Answer: C
Triangles from group III, equilateral triangles, are always similar to each other because they all have equal angles of 60 degrees, regardless of their size. Group I, right triangles, can vary significantly in angle measures beyond the right angle, so not all right triangles are similar. Similarly, group II, isosceles triangles, can have different base angles, leading to non-similar triangles. Thus, while right and isosceles triangles can share properties, only equilateral triangles guarantee similarity across the group. Therefore, option C accurately identifies the group with universally similar triangles.
Triangles from group III, equilateral triangles, are always similar to each other because they all have equal angles of 60 degrees, regardless of their size. Group I, right triangles, can vary significantly in angle measures beyond the right angle, so not all right triangles are similar. Similarly, group II, isosceles triangles, can have different base angles, leading to non-similar triangles. Thus, while right and isosceles triangles can share properties, only equilateral triangles guarantee similarity across the group. Therefore, option C accurately identifies the group with universally similar triangles.