Brian hosts a podcast about computers, and Nate hosts a podcast about parenting. Both podcasts have a set duration for each episode. Last year, Brian released 32 episodes and Nate released 16 episodes, for a total of 4,720 minutes of content. This year, Brian released 40 episodes and Nate released 16 episodes, which lasted a total of 5,392 minutes. How long is each episode?
A
Each episode of Brian's podcast is 104 minutes long, and each episode of Nate's podcast is 157 minutes long.
B
Each episode of Brian's podcast is 84 minutes long, and each episode of Nate's podcast is 127 minutes long.
C
Each episode of Brian's podcast is 127 minutes long, and each episode of Nate's podcast is 84 minutes long.
D
Each episode of Brian's podcast is 104 minutes long, and each episode of Nate's podcast is 127 minutes long.
Let's assume that each episode of Brian's podcast is 'x' minutes long, and each episode of Nate's podcast is 'y' minutes long.
According to the information given in the question:
Last year, Brian released 32 episodes and Nate released 16 episodes, for a total of 4,72x + 16y = 4,720 minutes of content.
This year, Brian released 40 episodes and Nate released 16 episodes, for a total of 40x + 16y = 5,392 minutes of content.
To find the duration of each episode, we need to solve these two equations simultaneously.
Multiplying the first equation by 2 and subtracting from the second equation:
(80x + 32y) - (4,72x + 16y) = 5,392 - 4,720
80x + 32y - 4,72x - 16y = 672
(80 - 4,72)x + (32 - 16)y = 672
75,28x + 16y = 672 ...........(i)
Re-arranging the first equation:
4,72x + 16y = 4,720
(16 - 4,72)x + 16y = 16(157)
11,28x + 16y = 2,512 ...........(ii)
Multiplying equation (ii) by 4 and subtracting equation (i) from it:
(45,12x + 64y) - (75,28x + 16y) = 4.025 - 672
45,12x + 64y - 75,28x - 16y = 3.353
(45,12 - 75,28)x + (64 - 16)y = 3.353
-30,16x + 48y = 3.353 ...........(iii)
Dividing equation (iii) by (-2):
(30,16x - 48y)/(-2) = (3.353)/(-2)
-15,08x + 24y = -1,676 ...........(iv)
Adding equation (iv) to equation (ii):
11,28x + 16y + (-15,08x + 24y) = 2.512 + (-1,676)
-3.8x + 40y = 836
Multiplying equation (ii) by 4:
4(11.28x + 16y) = 4(2,512)
45,12x + 64y = 10,048 ...........(v)
Adding equation (iv) to equation (v):
-3,8x + 45,12x + 40y + 64y = 836 + 10,048
41,32x + 104y = 10,884 ...........(vi)
Dividing equation (vi) by 4:
(41,32x + 104y)/4 = (10,884)/4
10,33x + 26y = 2,721 ...........(vii)
Multiplying equation (v) by (-2.5) and adding equation (vii) to it:
-2,5(45,12x + 64y) + 10,33x + 26y = -2,5(10,048) + 2,721
-112,8x - 160y + 10,33x + 26y = -25,12 - 2,721
-102,47x - 134y = -27,841
Dividing both sides of the equation by (-2.47):
(-102,47x - 134y)/(-2.47) = (-27,841)/(-2.47)
41.5x + 54.27y = 11.288
The only option that satisfies the algebraic equations and represents positive, whole numbers is option C: Each episode of Brian's podcast is 127 minutes long, and each episode of Nate's podcast is 84 minutes long.
A truck driver drove 1380 km from Edmonton to a drop-off point in Vancouver in 16 hours , spending 2 of the 16 hours at rest stops. The truck driver’s average speed was 110 km/h on divided highways and 90 km/h on undivided highways.
If x represents the number of hours spent on divided highways and y represents the number of hours spent on undivided highways, then which of the following systems of equations could be used to determine the number of hours the truck driver spent on each type of highway?
Let's assume that the truck driver spent 'x' hours on divided highways and 'y' hours on undivided highways.
The total distance traveled on divided highways can be calculated as x multiplied by the average speed on divided highways, which is 110 km/h.
Similarly, the total distance traveled on undivided highways can be calculated as y multiplied by the average speed on undivided highways, which is 90 km/h.
According to the given information, the total distance traveled is 1380 km.
So, the first equation representing the total distance traveled can be written as:
110x + 90y = 1380
The truck driver traveled for a total of 16 hours, out of which 2 hours were spent at rest stops.
So, the second equation representing the total time spent can be written as:
x + y = 16 - 2
x + y = 14
Therefore, the system of equations that could be used to determine the number of hours the truck driver spent on each type of highway is:
110x + 90y = 1380
x + y = 14
Hence, option B is the correct answer.
Three times one number minus another number equals 34. If the sum of the numbers is 22, what are the two numbers?
Let's assume the two numbers are 'x' and 'y'.
According to the given information,
Three times one number minus another number equals 34:
3x - y = 34 ...........(1)
And the sum of the numbers is 22:
x + y = 22 ...........(2)
To solve this system of equations, we can use the method of substitution.
From equation (2), we can express x in terms of y:
x = 22 - y
Substituting this value of x in equation (1), we get:
3(22 - y) - y = 34
66 - 3y - y = 34
66 - 4y = 34
-4y = 34 - 66
-4y = -32
y = -32/-4
y = 8
Substituting the value of y back into equation (2), we get:
x + 8 = 22
x = 22 - 8
x = 14
Therefore, the two numbers are x = 14 and y = 8.
Hence, the two numbers are 14 and 8.
The equations 5x + 2y = 48 and 3x + 2y = 32 represent the money collected from school concert tickets sales during two class periods. If x represents the cost for each adult ticket and y represents the cost for each student ticket, what is the cost for each adult ticket?
To find the cost for each adult ticket, we can solve the given system of equations.
The equations are:
5x + 2y = 48 ...........(1)
3x + 2y = 32 ...........(2)
We will use the method of elimination to solve the system.
Multiplying equation (2) by -2:
-2(3x + 2y) = -2(32)
-6x - 4y = -64 ...........(3)
Adding equation (1) and equation (3):
(5x + 2y) + (-6x - 4y) = 48 + (-64)
5x - 6x + 2y - 4y = -16
Simplifying the equation:
-x - 2y = -16 ...........(4)
We can multiply equation (4) by -5:
-5(-x - 2y) = -5(-16)
5x + 10y = 80 ...........(5)
Adding equation (1) and equation (5):
(5x + 2y) + (5x + 10y) = 48 + 80
10x + 12y = 128
Dividing equation (5) by 2, we get:
10x/2 + 12y/2 = 128/2
5x + 6y = 64
Now we have two equations:
10x + 12y = 128 ...........(6)
5x + 6y = 64 ...........(7)
Multiplying equation (7) by -2:
-2(5x + 6y) = -2(64)
-10x - 12y = -128 ...........(8)
Adding equation (6) and equation (8):
(10x + 12y) + (-10x - 12y) = 128 + (-128)
0 = 0
This equation tells us that the two equations are dependent and represent the same line. This means that there are infinitely many solutions to the system, including a range of values for x and y.
Therefore, we cannot determine a unique value for the cost of each adult ticket based on the given information.
The equations 5x + 2y = 48 and 3x + 2y = 32 represent the money collected from school concert tickets sales during two class periods. If x represents the cost for each adult ticket and y represents the cost for each student ticket, what is the cost for each adult ticket?
A
The cost of an adult ticket is $10.
B
The cost of an adult ticket is $8.
D
The cost of an adult ticket is $12.
To find the cost for each adult ticket, we can solve the given system of equations.
The equations are:
5x + 2y = 48 ...........(1)
3x + 2y = 32 ...........(2)
We can subtract equation (2) from equation (1) to eliminate y:
(5x + 2y) - (3x + 2y) = 48 - 32
5x - 3x + 2y - 2y = 16
2x = 16
x = 16/2
x = 8
Therefore, the cost for each adult ticket is $8.
Hence, the correct answer is option B: The cost of an adult ticket is $8.