Word problem? Any help appreciated!

[justwriting]

First problem:

The budget of Central City Chapter of the League of Women Voters was increased by $350. If the dues are raised from $5 to $6, how many new members must be acquired to meet the budget if there were 90 members in the chapter to begin with?

Second problem:

In a run-off between two candidates, a total of 25, 200 votes were cast. If 300 voters had switched from the winner to the loser, the loser would have won by 200 votes. How many votes did each candidate actually receive?

I'm not sure how to set up either of these equations, word problems aren't exactly my preference.
For the second equation, I began with 25,200 = x + y. "y" can be represented by "25,200 - x" right? but the equation "25,200 = x + (25,200 - x)" wouldn't get me anywhere. As for the first problem, I do not understand what the problem is asking for.

These problems do not have to be solved. I just need a little guidance as to how to set these problems up.

Thank you for any help, I appreciate it all.

 

[mmm4444bot]

… for the first problem, I do not understand what the problem is asking for …

Hi Just Writing:

The first exercise asks how many new members are needed, such that the total revenue (from all members paying $6 each) will equal the new budget.

Pick a variable to represent this unknown number of new members, and write it down.

x = the number of new members needed

We have the following information given:

Current membership is 90.

Old dues were $5 per member.

From this, we can calculate the old budget because the 90 members paid $5 each.

90*5 = 450

Therefore, the new budget is $800, right ? (450 + 350)

Now we have enough information to write the equation that models the following goal.

(old members plus new members) times (new dues amount) equals the new budget

Can you write this equation and solve for x ?

If so, then double-check that your answer makes sense, in terms of a number of people, and that it leads to revenue that's at least $800.

'

… For the second [exercise], I began with 25,200 = x + y …

That's an okay start, but nobody will know what x represents or what y represents, unless you say so. Let's go with the following, and write it down.

x = the number of votes for the winner

y = the number of votes for the loser

x + y = 25200

Now, we have a hypothetical senario: the original winner would lose by 200 votes, IF 300 voters were to have voted for this person's opponent, instead.

We can model the phrase "300 voters had switched from the winner to the loser" by the following.

x - 300 = the number of votes for the hypothetical loser

y + 300 = the number of votes for the hypothetical winner

Once this hypothetical switch takes place, the "new" winner has 200 votes more than the "new" loser, right ?

The number of votes (y + 300) is more than the number of votes (x - 300), AND their difference is exactly 200.

This is enough to write the second equation of the system. Can you solve for x and y ?

Please show whatever work that you can, on either of these exercises, if you need more help.

Cheers ~ Mark

MY EDITS: Reworded some stuff because I'm bored

 

[soroban]

Hello, justwriting!

Second problem: In a run-off between two candidates, a total of 25,200 votes were cast.
If 300 voters had switched from the winner to the loser, the loser would have won by 200 votes.
How many votes did each candidate actually receive?

\(\displaystyle \text{Let }\;\begin{Bmatrix}W &=& \text{number of votes the winner got} \\ L &=& \text{number of votes the loser got} \end{array}\)

\(\displaystyle \text{One equation is: }\;W + L \:=\:25,\!200\)


\(\displaystyle \text{If 300 voters switched over: }\:\begin{Bmatrix}\text{The loser got }L + 300\text{ votes.} \\ \text{The winner got }W - 300\text{ votes.} \end{Bmatrix}\)

\(\displaystyle \text{The difference is 200: }\;(L + 300) - (W - 300) \:=\:200 \quad\Longrightarrow\quad W - L \:=\:400\)


\(\displaystyle \text{Solve the system: }\;\begin{array}{ccc}W + L &=& 25,\!200 \\ W - L &=& 400 \end{array} \quad\Rightarrow\quad \boxed{ \begin{array}{ccc}W &=&12,\!800 \\ L &=& 12,\!400 \end{array} }\)