Darren McFadden of Arkansas placed second overall in the Heisman Trophy voting. Players are given 3 points for every first-place vote, 2 point for every second-place vote, and 1 point for every third place vote. McFadden received 490 total votes for first, second, and third place, for a total of 878 points. If he had 4 more than twice as many second-place votes as third-place votes, how many votes did he receive each place?

Question

Darren McFadden of Arkansas placed second overall in the Heisman Trophy voting. Players are given 3 points for every first-place vote, 2 point for every second-place vote, and 1 point for every third place vote. McFadden received 490 total votes for first, second, and third place, for a total of 878 points. If he had 4 more than twice as many second-place votes as third-place votes, how many votes did he receive each place?

Answer 1

Well, if we go by the information provided, let's do some math-robatics, shall we?

Let's say Darren received x number of first-place votes, y number of second-place votes, and z number of third-place votes.

We know that Darren received a total of 490 votes overall, which includes first, second, and third place votes. So, the equation would look like this:

3x + 2y + 1z = 490

We are also told that Darren received 4 more than twice as many second-place votes as third-place votes. So, we can set up another equation:

y = 2z + 4

Now, let's use the power of algebra to solve these equations. We could also use the power of comedy, but that might solve the equations too quickly. Let's save some laughs for later!

Using substitution method, we can substitute the value of y from the second equation into the first equation:

3x + 2(2z + 4) + 1z = 490
3x + 4z + 8 + 1z = 490
3x + 5z + 8 = 490

Now, let's simplify this equation:

3x + 5z = 482

Hmm, this is starting to look quite tricky. But fear not! We're here to solve it!

Since there are no other equations or values given, we'll need some more information to find the exact values of x, y, and z. It seems like we're missing a piece of the puzzle here. Without more information, we can't determine the exact number of votes Darren received in each place.

But hey, don't be too disappointed! Math can be a bit of a juggling act sometimes. If you have any more questions or jokes you'd like me to perform, feel free to ask!

Answer 2

Let's assume the number of first-place votes for Darren McFadden is F, the number of second-place votes is S, and the number of third-place votes is T.

According to the given information, we know that Darren McFadden received a total of 878 points. Since 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, we can set up an equation:

3F + 2S + 1T = 878 ...(Equation 1)

Now, we are also given that McFadden had 4 more than twice as many second-place votes as third-place votes. Mathematically, we can express this as:

S = 2T + 4 ...(Equation 2)

To solve these equations, we will use substitution.

Substituting Equation 2 into Equation 1, we have:

3F + 2(2T + 4) + 1T = 878
3F + 4T + 8 + 1T = 878
3F + 5T + 8 = 878
3F + 5T = 870 ...(Equation 3)

Now, we can express the number of first-place votes in terms of T by rearranging Equation 2:

S = 2T + 4
S - 4 = 2T
2T = S - 4
T = (S - 4)/2 ...(Equation 4)

Substituting Equation 4 into Equation 3, we have:

3F + 5((S - 4)/2) = 870
4F + 5S - 20 = 870
4F + 5S = 890 ...(Equation 5)

To simplify Equation 5, we can multiply it by 2 to eliminate the fraction:

8F + 10S = 1780 ...(Equation 6)

Now, we can solve Equations 5 and 6 simultaneously.
Multiplying Equation 5 by 2 gives us:

8F + 10S = 1780

Subtracting Equation 6 from it, we get:

12S - 10S = 1250 - 1780
2S = -530
S = -265

Since the number of votes cannot be negative, there must be an error in the given information or calculations as the solution is not valid. Please double-check the information or calculations provided.

Answer 3

To solve this problem, let's break it down step by step.

Let's assume Darren McFadden received x first-place votes, y second-place votes, and z third-place votes.

Based on the information given, we can create two equations:

Equation 1: x + y + z = 490 (Total number of first, second, and third-place votes adds up to 490)

Equation 2: 3x + 2y + z = 878 (Total points obtained by multiplying the number of votes in each place)

Now let's solve these equations simultaneously to find the values of x, y, and z.

First, let's simplify equation 2 by distributing the coefficients:

3x + 2y + z = 878
3x + 2(y + z) = 878

Next, let's use the information given that "he had 4 more than twice as many second-place votes as third-place votes." We can rewrite this relationship as:

y = 2z + 4

Now we can substitute this value of y into equation 1:

x + (2z + 4) + z = 490
x + 3z + 4 = 490
x + 3z = 486

Now we have two equations:

x + 3z = 486
3x + 2(y + z) = 878

Using the information provided, we can solve these equations simultaneously to find the values of x, y, and z.