Darren McFadden of Arkansas placed second overall in the Heisman Tophy voting. Players are given 3 points for every first place vote, 2 points for every second-place vote, and 1point for every third place vote. McFadden received 490 total votes for a total of 878 points. If he had 4 more than twice as many second-place votes as third place votes , how many vote did he receive for each place.
f+s+t = 490
3f+2s+t = 878
s = 4+2t
Now you can solve for f,s,t.
f= 45
s= 298
t= 147
Oh, the Heisman Trophy, the ultimate popularity contest for college football players. Let me put on my clown nose and calculate this for you.
Let's assume that Darren McFadden received x first-place votes, y second-place votes, and z third-place votes. According to the problem, we can set up the following system of equations:
x + y + z = 490 (since he received a total of 490 votes)
3x + 2y + z = 878 (since first-place votes are worth 3 points, second-place votes 2 points, and third-place votes 1 point)
From the second piece of information, we're told that y (second-place votes) is 4 more than twice z (third-place votes). So we can write it as:
y = 2z + 4
Now let's solve this crazy circus math. We'll start by eliminating y from the equation.
Plugging in y = 2z + 4 into the first equation, we get:
x + (2z + 4) + z = 490
x + 3z + 4 = 490
x + 3z = 486 <-- Equation 1
Now, let's substitute y = 2z + 4 into the second equation:
3x + 2(2z + 4) + z = 878
3x + 4z + 8 + z = 878
3x + 5z = 870 <-- Equation 2
We now have two equations with two variables. To make it easier, let's multiply Equation 1 by 3:
3(x + 3z) = 3(486)
3x + 9z = 1458 <-- Equation 3
Now we have two equations with the same x term:
3x + 5z = 870
3x + 9z = 1458
Subtracting Equation 3 from Equation 2 will allow us to eliminate the x term:
(3x + 9z) - (3x + 5z) = 1458 - 870
4z = 588
z = 147
Now, plugging this value of z back into Equation 1:
x + 3(147) = 486
x + 441 = 486
x = 45
Since x represents the number of first-place votes, y the number of second-place votes, and z the number of third-place votes, we have our results:
Darren McFadden received 45 first-place votes, 147 second-place votes, and 147 third-place votes.
Whew! Math can be quite the circus act.
Let's assume Darren McFadden received x first-place votes, y second-place votes, and z third-place votes.
According to the given information, we can form three equations based on the given information:
Equation 1: x + y + z = 490 (Total number of votes)
Equation 2: 3x + 2y + z = 878 (Total points received)
Now, we are given that he had 4 more than twice as many second-place votes as third-place votes. Mathematically, this can be represented as:
y = 2z + 4
Now, we can solve these three equations to find the values of x, y, and z.
Substituting the value of y in the first equation, we get:
x + (2z + 4) + z = 490
x + 3z + 4 = 490
x + 3z = 490 - 4
x + 3z = 486 ----(Equation 3)
Substituting the values of x, y, and z in the second equation, we get:
3x + 2(2z + 4) + z = 878
3x + 4z + 8 + z = 878
3x + 5z = 878 - 8
3x + 5z = 870 ----(Equation 4)
We now have two equations (Equation 3 and Equation 4) with two variables (x and z). We can solve these equations simultaneously.
Now, multiplying Equation 3 by 3, we get:
3(x + 3z) = 3(486)
3x + 9z = 1458 ----(Equation 5)
Subtracting Equation 4 from Equation 5, we get:
(3x + 9z) - (3x + 5z) = 1458 - 870
3x - 3x + 9z - 5z = 588
4z = 588
z = 147
Substituting the value of z in Equation 3, we get:
x + 3(147) = 486
x + 441 = 486
x = 486 - 441
x = 45
Substituting the values of x and z in y = 2z + 4, we get:
y = 2(147) + 4
y = 294 + 4
y = 298
Therefore, Darren McFadden received 45 first-place votes, 298 second-place votes, and 147 third-place votes.
I hope this helps! Let me know if you have any other questions.
To find out how many votes Darren McFadden received for each place, we can set up a system of equations based on the given information.
Let x be the number of first-place votes, y be the number of second-place votes, and z be the number of third-place votes.
From the statement "Darren McFadden received 490 total votes," we can write the equation:
x + y + z = 490 (Equation 1)
From the statement "he had 4 more than twice as many second-place votes as third place votes," we can write the equation:
y = 2z + 4 (Equation 2)
Now let's use the statement "Players are given 3 points for every first-place vote, 2 points for every second-place vote, and 1 point for every third-place vote" to form another equation.
The total points Darren McFadden received is 878. We can express this using the number of votes for each place:
3x + 2y + 1z = 878 (Equation 3)
To solve this system of equations, we'll substitute equation 2 into equations 1 and 3.
Substituting equation 2 into equation 1:
x + (2z + 4) + z = 490
x + 3z + 4 = 490
Simplifying equation 4:
x + 3z = 486 (Equation 4)
Substituting equation 2 into equation 3:
3x + 2(2z + 4) + z = 878
3x + 4z + 8 + z = 878
3x + 5z = 870 (Equation 5)
Now we can solve equations 4 and 5 simultaneously to find the values of x and z.
Multiplying equation 4 by 5 and equation 5 by 3 to create a system of equations:
5x + 15z = 2430 (Equation 6)
9x + 15z = 2610 (Equation 7)
Subtracting equation 6 from equation 7 to eliminate z:
(9x + 15z) - (5x + 15z) = 2610 - 2430
9x - 5x = 180
4x = 180
x = 45
Now we substitute the value of x into equation 4:
45 + 3z = 486
3z = 441
z = 147
So Darren McFadden received 45 first-place votes, 147 third-place votes, and based on equation 2, he received 2z + 4 = 2(147) + 4 = 298 + 4 = 302 second-place votes.
In summary, Darren McFadden received 45 votes for first place, 302 votes for second place, and 147 votes for third place.