When Henry plays the songs on the playlist in a random order, what is the probability a rock song will be played first?
- A. 3/4
- B. 1/3
- C. 1/4
- D. 3/10
- E. 5/16
Correct Answer & Rationale
Correct Answer: D
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.
Other Related Questions
Which of the following expressions is equivalent to (4x²)(5x³)?
- A. 9xâµ
- B. 9xâ¶
- C. 20xâµ
- D. 20xâ¶
- E. 20xâ¹
Correct Answer & Rationale
Correct Answer: C
To find the equivalent expression for (4x²)(5x³), multiply the coefficients (4 and 5) and add the exponents of x (2 and 3). Thus, 4 × 5 equals 20, and x² × x³ results in x^(2+3) = x⁵. This gives us 20x⁵. Option A (9x⁶) is incorrect because it miscalculates both the coefficient and the exponent. Option B (9x⁷) also miscalculates both the coefficient and exponent. Option D (20x⁶) correctly identifies the coefficient but incorrectly adds the exponents. Option E (20x¹) miscalculates the exponent entirely. Only option C accurately represents the expression as 20x⁵.
To find the equivalent expression for (4x²)(5x³), multiply the coefficients (4 and 5) and add the exponents of x (2 and 3). Thus, 4 × 5 equals 20, and x² × x³ results in x^(2+3) = x⁵. This gives us 20x⁵. Option A (9x⁶) is incorrect because it miscalculates both the coefficient and the exponent. Option B (9x⁷) also miscalculates both the coefficient and exponent. Option D (20x⁶) correctly identifies the coefficient but incorrectly adds the exponents. Option E (20x¹) miscalculates the exponent entirely. Only option C accurately represents the expression as 20x⁵.
Which of the following expressions is equivalent to: 1200 × (5 × 10â·)?
- A. 12×10¹â°
- B. 6.0×10¹â°
- C. 6.0×10¹¹
- D. 7.2×10¹³
- E. 9.4×10¹â´
Correct Answer & Rationale
Correct Answer: B
To find an equivalent expression for \( 1200 \times (5 \times 10^n) \), we first simplify \( 1200 \) as \( 1.2 \times 10^3 \). Thus, the expression becomes \( 1.2 \times 10^3 \times 5 \times 10^n = 6.0 \times 10^{3+n} \). Option A incorrectly simplifies the coefficient and exponent. Option C miscalculates the exponent, not aligning with the original multiplication. Option D has an incorrect coefficient and exponent combination. Option E also miscalculates the coefficient and exponent. Therefore, only option B accurately reflects the simplified expression.
To find an equivalent expression for \( 1200 \times (5 \times 10^n) \), we first simplify \( 1200 \) as \( 1.2 \times 10^3 \). Thus, the expression becomes \( 1.2 \times 10^3 \times 5 \times 10^n = 6.0 \times 10^{3+n} \). Option A incorrectly simplifies the coefficient and exponent. Option C miscalculates the exponent, not aligning with the original multiplication. Option D has an incorrect coefficient and exponent combination. Option E also miscalculates the coefficient and exponent. Therefore, only option B accurately reflects the simplified expression.
Which of the following equations does not represent y as a function of x in the standard (x, y) coordinate plane?
- A. y = x
- B. y = x + 2
- C. y = x² + 2
- D. x = y + 2
- E. x = y² + 2
Correct Answer & Rationale
Correct Answer: E
Option E, \( x = y^2 + 2 \), does not represent \( y \) as a function of \( x \) because it can yield multiple \( y \) values for a single \( x \) value. For example, when \( x = 6 \), \( y \) can be either 2 or -2, violating the definition of a function. In contrast, options A, B, and C express \( y \) explicitly in terms of \( x \), allowing only one output for each input. Option D, while rearranging the equation, can also be transformed into a function of \( y \) in terms of \( x \) (i.e., \( y = x - 2 \)). Thus, options A, B, C, and D all represent \( y \) as a function of \( x \).
Option E, \( x = y^2 + 2 \), does not represent \( y \) as a function of \( x \) because it can yield multiple \( y \) values for a single \( x \) value. For example, when \( x = 6 \), \( y \) can be either 2 or -2, violating the definition of a function. In contrast, options A, B, and C express \( y \) explicitly in terms of \( x \), allowing only one output for each input. Option D, while rearranging the equation, can also be transformed into a function of \( y \) in terms of \( x \) (i.e., \( y = x - 2 \)). Thus, options A, B, C, and D all represent \( y \) as a function of \( x \).
A home improvement store offers to finance the purchase of any single item with zero interest for one year, with a down payment of $50. The remainder of the purchase price will be split into 12 equal monthly payments. Which of the following equations represents the relationship between an item's purchase price, s dollars, and the amount, a dollars, of each monthly payment under this offer?
- A. s = a-50/12
- B. s = a/12 -50
- C. s = 12a + 50
- D. s = 12a - 50
- E. s = 12 (a + 50)
Correct Answer & Rationale
Correct Answer: C
To determine the relationship between the item's purchase price \( s \) and the monthly payment \( a \), consider the financing terms. After a $50 down payment, the remaining amount to finance is \( s - 50 \). This amount is divided into 12 equal monthly payments, leading to the equation \( s - 50 = 12a \). Rearranging this gives \( s = 12a + 50 \), confirming option C. Options A and B misrepresent the relationship by incorrectly adjusting the down payment or monthly payments. Option D incorrectly subtracts the down payment from the total, while option E miscalculates the total by incorrectly adding the down payment to the monthly payment before multiplying.
To determine the relationship between the item's purchase price \( s \) and the monthly payment \( a \), consider the financing terms. After a $50 down payment, the remaining amount to finance is \( s - 50 \). This amount is divided into 12 equal monthly payments, leading to the equation \( s - 50 = 12a \). Rearranging this gives \( s = 12a + 50 \), confirming option C. Options A and B misrepresent the relationship by incorrectly adjusting the down payment or monthly payments. Option D incorrectly subtracts the down payment from the total, while option E miscalculates the total by incorrectly adding the down payment to the monthly payment before multiplying.