A carpenter is installing shelves in 2 offices. Each office will have 4 shelves. The wood the carpenter wants to use comes in 6-foot-long boards. Each shelf is 2 ¼ feet long and is constructed from a single board. How many boards does the carpenter need to buy to make the shelves?
- A. 2
- B. 8
- C. 3
- D. 4
Correct Answer & Rationale
Correct Answer: D
To determine how many boards are needed, first calculate the total length of wood required for the shelves. Each office has 4 shelves, and with 2 offices, that totals 8 shelves. Each shelf is 2 ¼ feet long, which equals 2.25 feet. Therefore, the total length required is 8 shelves x 2.25 feet = 18 feet. Each board is 6 feet long. Dividing the total length (18 feet) by the length of each board (6 feet) gives 3 boards. However, since each board can only be used for one shelf, and we can't cut a board to make multiple shelves, we need to round up to the nearest whole number of boards needed, which is 4. - Option A (2 boards) is insufficient for the total length required. - Option B (8 boards) exceeds the necessary amount. - Option C (3 boards) miscalculates the total need based on the cut requirement. Thus, 4 boards are necessary to accommodate all shelves without waste.
To determine how many boards are needed, first calculate the total length of wood required for the shelves. Each office has 4 shelves, and with 2 offices, that totals 8 shelves. Each shelf is 2 ¼ feet long, which equals 2.25 feet. Therefore, the total length required is 8 shelves x 2.25 feet = 18 feet. Each board is 6 feet long. Dividing the total length (18 feet) by the length of each board (6 feet) gives 3 boards. However, since each board can only be used for one shelf, and we can't cut a board to make multiple shelves, we need to round up to the nearest whole number of boards needed, which is 4. - Option A (2 boards) is insufficient for the total length required. - Option B (8 boards) exceeds the necessary amount. - Option C (3 boards) miscalculates the total need based on the cut requirement. Thus, 4 boards are necessary to accommodate all shelves without waste.
Other Related Questions
Two men are employed at a local supermarket. The table shows James's earnings, and the graph shows Eric's earnings.
Based on the information above, who earns the greater amount per hour, and how much does he earn for a 7-hour shift?
- A. James earns the greater amount per hour and earns $73.50 for a 7-hour shift.
- B. James earns the greater amount per hour and earns $70.00 for a 7-hour shift.
- C. Eric earns the greater amount per hour and earns $70.00 for a 7-hour shift.
- D. Eric earns the greater amount per hour and earns $73.50 for a 7-hour shift.
Correct Answer & Rationale
Correct Answer: D
To determine who earns more per hour, one must compare the hourly rates of James and Eric. If Eric's hourly rate is higher, he earns more for a 7-hour shift, calculated as his hourly rate multiplied by 7. Option A incorrectly states James earns more and miscalculates his earnings. Option B also claims James earns more but provides the wrong total for a 7-hour shift. Option C correctly identifies Eric as the higher earner but misstates his total earnings for a 7-hour shift. Option D accurately identifies Eric as the higher earner and correctly calculates his earnings for a 7-hour shift at $73.50.
To determine who earns more per hour, one must compare the hourly rates of James and Eric. If Eric's hourly rate is higher, he earns more for a 7-hour shift, calculated as his hourly rate multiplied by 7. Option A incorrectly states James earns more and miscalculates his earnings. Option B also claims James earns more but provides the wrong total for a 7-hour shift. Option C correctly identifies Eric as the higher earner but misstates his total earnings for a 7-hour shift. Option D accurately identifies Eric as the higher earner and correctly calculates his earnings for a 7-hour shift at $73.50.
The triangle shown in the diagram has an area of 24 square centimeters. What is h, the height in centimeters, of the triangle?
- A. 9
- B. 4
- C. 8
- D. 2
Correct Answer & Rationale
Correct Answer: C
To find the height \( h \) of the triangle, we use the area formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Given the area is 24 cm², we can rearrange the formula to solve for \( h \): \( h = \frac{2 \times \text{Area}}{\text{base}} \). Assuming the base is 6 cm (since \( 24 = \frac{1}{2} \times 6 \times h \)), substituting gives \( h = \frac{48}{6} = 8 \). - Option A (9) is too high, as it would yield an area greater than 24 cm². - Option B (4) results in an area of only 12 cm², which is insufficient. - Option D (2) yields an area of 6 cm², far below the required area. Thus, only option C (8) satisfies the area requirement.
To find the height \( h \) of the triangle, we use the area formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Given the area is 24 cm², we can rearrange the formula to solve for \( h \): \( h = \frac{2 \times \text{Area}}{\text{base}} \). Assuming the base is 6 cm (since \( 24 = \frac{1}{2} \times 6 \times h \)), substituting gives \( h = \frac{48}{6} = 8 \). - Option A (9) is too high, as it would yield an area greater than 24 cm². - Option B (4) results in an area of only 12 cm², which is insufficient. - Option D (2) yields an area of 6 cm², far below the required area. Thus, only option C (8) satisfies the area requirement.
The apartments in Greg's building are named using the letters A, B, C, and D and the digits 1 through 9. How many apartments are there in Greg's building if each apartment is named by a single letter followed by a single digit?
- A. 36
- B. 16
- C. 40
- D. 13
Correct Answer & Rationale
Correct Answer: A
To determine the total number of apartments, consider the naming convention: each apartment consists of one letter and one digit. There are 4 letters (A, B, C, D) and 9 digits (1-9). Calculating the combinations, multiply the number of letters by the number of digits: 4 letters × 9 digits = 36 unique apartment names. Options B (16) and D (13) do not account for all possible combinations, while option C (40) incorrectly assumes more letters or digits than provided. Thus, option A accurately reflects the total possible apartments in Greg's building.
To determine the total number of apartments, consider the naming convention: each apartment consists of one letter and one digit. There are 4 letters (A, B, C, D) and 9 digits (1-9). Calculating the combinations, multiply the number of letters by the number of digits: 4 letters × 9 digits = 36 unique apartment names. Options B (16) and D (13) do not account for all possible combinations, while option C (40) incorrectly assumes more letters or digits than provided. Thus, option A accurately reflects the total possible apartments in Greg's building.
An advertisement poster in the window of a shoe store is in the shape of a rectangle. The length of the poster is 9 less than 4 times the width. Which expression represents the length of the poster when w is the width
- A. 4w - 9
- B. 9 - 4w
- C. 4w + 9
- D. 9w - 4
Correct Answer & Rationale
Correct Answer: A
The expression for the length of the poster is determined by the relationship given in the problem. The length is described as "9 less than 4 times the width," which translates mathematically to \(4w - 9\). Option A (4w - 9) accurately reflects this relationship. Option B (9 - 4w) incorrectly suggests that the length is greater than 9 and decreases as width increases, which contradicts the problem's description. Option C (4w + 9) implies that the length increases by 9, rather than decreasing, which is not aligned with the original statement. Option D (9w - 4) introduces an incorrect multiplication factor and does not adhere to the given relationship, making it invalid.
The expression for the length of the poster is determined by the relationship given in the problem. The length is described as "9 less than 4 times the width," which translates mathematically to \(4w - 9\). Option A (4w - 9) accurately reflects this relationship. Option B (9 - 4w) incorrectly suggests that the length is greater than 9 and decreases as width increases, which contradicts the problem's description. Option C (4w + 9) implies that the length increases by 9, rather than decreasing, which is not aligned with the original statement. Option D (9w - 4) introduces an incorrect multiplication factor and does not adhere to the given relationship, making it invalid.