What is the value of 2/5 multiplied by ¾ divide by 8/5
- A. 12\25
- B. 1\3
- C. 3\16
- D. 64/75
Correct Answer & Rationale
Correct Answer: C
To solve \( \frac{2}{5} \times \frac{3}{4} \div \frac{8}{5} \), first, convert the division into multiplication by flipping the second fraction: \[ \frac{2}{5} \times \frac{3}{4} \times \frac{5}{8} \] Next, multiply the fractions: \[ \frac{2 \times 3 \times 5}{5 \times 4 \times 8} = \frac{30}{160} \] Simplifying \( \frac{30}{160} \) gives \( \frac{3}{16} \), confirming option C. Option A (12/25) is incorrect as it does not simplify correctly from the original operation. Option B (1/3) results from an incorrect multiplication or division process. Option D (64/75) does not match the calculated result and suggests an error in fraction handling.
To solve \( \frac{2}{5} \times \frac{3}{4} \div \frac{8}{5} \), first, convert the division into multiplication by flipping the second fraction: \[ \frac{2}{5} \times \frac{3}{4} \times \frac{5}{8} \] Next, multiply the fractions: \[ \frac{2 \times 3 \times 5}{5 \times 4 \times 8} = \frac{30}{160} \] Simplifying \( \frac{30}{160} \) gives \( \frac{3}{16} \), confirming option C. Option A (12/25) is incorrect as it does not simplify correctly from the original operation. Option B (1/3) results from an incorrect multiplication or division process. Option D (64/75) does not match the calculated result and suggests an error in fraction handling.
Other Related Questions
A manufacturing plant makes dog toys in the shape of a sphere. The diameter of each dog toy is 3 inches. What is the surface area, in square inches of each dog toy?
- A. 113.04
- B. 75.36
- C. 28.26
- D. 37.68
Correct Answer & Rationale
Correct Answer: C
To find the surface area of a sphere, the formula used is \(4\pi r^2\). Given the diameter of the dog toy is 3 inches, the radius \(r\) is half of that, which is 1.5 inches. Plugging this into the formula: \[ Surface Area = 4\pi (1.5)^2 = 4\pi (2.25) \approx 28.26 \text{ square inches.} \] Option A (113.04) results from incorrectly using the diameter instead of the radius. Option B (75.36) arises from miscalculating the radius or misapplying the formula. Option D (37.68) likely results from a miscalculation of the surface area formula, possibly using an incorrect value for \(r\).
To find the surface area of a sphere, the formula used is \(4\pi r^2\). Given the diameter of the dog toy is 3 inches, the radius \(r\) is half of that, which is 1.5 inches. Plugging this into the formula: \[ Surface Area = 4\pi (1.5)^2 = 4\pi (2.25) \approx 28.26 \text{ square inches.} \] Option A (113.04) results from incorrectly using the diameter instead of the radius. Option B (75.36) arises from miscalculating the radius or misapplying the formula. Option D (37.68) likely results from a miscalculation of the surface area formula, possibly using an incorrect value for \(r\).
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A truck driver sees a road sign warning of an 8% road incline. To the nearest tenth of a foot, what will be the change in the truck's vertical position, in feet, during the time it takes the truck's horizontal position to change by 1 mile? (1 mile = 5,280 ft)
Correct Answer & Rationale
Correct Answer: 422.4
To determine the vertical change during a 1-mile horizontal distance on an 8% incline, we calculate the vertical rise using the formula: vertical rise = incline percentage × horizontal distance. Here, 8% as a decimal is 0.08, and the horizontal distance is 5,280 feet. Therefore, the vertical change is 0.08 × 5,280 = 422.4 feet. Other options are incorrect as they either miscalculate the incline percentage or the conversion of miles to feet. For instance, values significantly lower than 422.4 feet suggest a misunderstanding of the incline's impact, while options above this value imply an overestimation of the incline's effect on vertical change.
To determine the vertical change during a 1-mile horizontal distance on an 8% incline, we calculate the vertical rise using the formula: vertical rise = incline percentage × horizontal distance. Here, 8% as a decimal is 0.08, and the horizontal distance is 5,280 feet. Therefore, the vertical change is 0.08 × 5,280 = 422.4 feet. Other options are incorrect as they either miscalculate the incline percentage or the conversion of miles to feet. For instance, values significantly lower than 422.4 feet suggest a misunderstanding of the incline's impact, while options above this value imply an overestimation of the incline's effect on vertical change.
Solve the inequality for x: (1/8)x ? (1/2)x + 15
- A. x ? -24
- B. x ? -40
- C. x ? -40
- D. x ? -24
Correct Answer & Rationale
Correct Answer: C
To solve the inequality \((1/8)x < (1/2)x + 15\), first, subtract \((1/2)x\) from both sides, yielding \(-\frac{3}{8}x < 15\). Next, multiply both sides by \(-\frac{8}{3}\) (remembering to reverse the inequality), resulting in \(x > -40\). Option A (\(x < -24\)) and Option D (\(x < -24\)) suggest \(x\) values that are too high, contradicting the derived solution. Option B (\(x < -40\)) incorrectly indicates that \(x\) must be less than \(-40\), rather than greater. Thus, Option C accurately represents the solution \(x > -40\).
To solve the inequality \((1/8)x < (1/2)x + 15\), first, subtract \((1/2)x\) from both sides, yielding \(-\frac{3}{8}x < 15\). Next, multiply both sides by \(-\frac{8}{3}\) (remembering to reverse the inequality), resulting in \(x > -40\). Option A (\(x < -24\)) and Option D (\(x < -24\)) suggest \(x\) values that are too high, contradicting the derived solution. Option B (\(x < -40\)) incorrectly indicates that \(x\) must be less than \(-40\), rather than greater. Thus, Option C accurately represents the solution \(x > -40\).
Simplify: (3x - 5) + (-7x + 2)
- A. -4x^2 - 3
- B. -4x - 3
- C. 28
- D. -4x^2 - 10
Correct Answer & Rationale
Correct Answer: B
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.