Mallory loaded 200 digital pictures into a digital picture frame. 78 are pictures of family members, 26 are pictures of pets, the rest are pictures of friends. The frame displays one picture every 10 seconds. Which value is closest to the probability that the next picture the frame displays will be a picture of a friend?
- A. 0.33
- B. 0.43
- C. 0.48
- D. 0.52
- E. 0.96
Correct Answer & Rationale
Correct Answer: C
To find the probability that the next picture displayed is of a friend, first calculate the total number of friend pictures. There are 200 total pictures, with 78 family and 26 pet pictures, leaving 200 - 78 - 26 = 96 pictures of friends. The probability is then the number of friend pictures divided by the total: 96/200 = 0.48. Option A (0.33) underestimates the proportion of friend pictures. Option B (0.43) is also lower than the calculated probability. Option D (0.52) slightly overestimates it, and option E (0.96) is far too high, misrepresenting the actual count. Thus, 0.48 accurately reflects the likelihood of displaying a friend picture next.
To find the probability that the next picture displayed is of a friend, first calculate the total number of friend pictures. There are 200 total pictures, with 78 family and 26 pet pictures, leaving 200 - 78 - 26 = 96 pictures of friends. The probability is then the number of friend pictures divided by the total: 96/200 = 0.48. Option A (0.33) underestimates the proportion of friend pictures. Option B (0.43) is also lower than the calculated probability. Option D (0.52) slightly overestimates it, and option E (0.96) is far too high, misrepresenting the actual count. Thus, 0.48 accurately reflects the likelihood of displaying a friend picture next.
Other Related Questions
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.
What are the solutions to (x-2)(x+4) = 0?
- A. -4 and 2
- B. -3 and 1
- C. -2 and 4
- D. -1 and 1
- E. -1 and 3
Correct Answer & Rationale
Correct Answer: A
To solve the equation (x-2)(x+4) = 0, we apply the zero product property, which states that if a product of factors equals zero, at least one of the factors must equal zero. Setting each factor to zero gives us the equations x - 2 = 0 and x + 4 = 0. Solving these yields x = 2 and x = -4, confirming that the solutions are -4 and 2. Options B, C, D, and E provide incorrect pairs of solutions that do not satisfy the original equation when substituted back in. Each of these pairs results in non-zero products for the factors, thus failing to meet the requirement of the equation.
To solve the equation (x-2)(x+4) = 0, we apply the zero product property, which states that if a product of factors equals zero, at least one of the factors must equal zero. Setting each factor to zero gives us the equations x - 2 = 0 and x + 4 = 0. Solving these yields x = 2 and x = -4, confirming that the solutions are -4 and 2. Options B, C, D, and E provide incorrect pairs of solutions that do not satisfy the original equation when substituted back in. Each of these pairs results in non-zero products for the factors, thus failing to meet the requirement of the equation.
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.
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.