I'm having a problem with this number and digits problem. What to do?

Question

During the last election , the total number of votes recorded in the municipality of San Juan was 8600. Had one-third of Estrada’s supporters stayed away from the polls and one-half of Arroyo’s behaved likewise , Estrada’s majority would have been reduced by 200.

1. How many votes did Estrada actually received?
2. How many votes did Arroyo actually received?
3. Determine Estrada’s new majority over Arroyo when both their supporters stayed away from the polls.

Answer 1

HINT

Let $E$ denote the number of voters from Estrada and $A$ - from Arrayo. Then, the total is $A+E = 8600$. Translate the second sentence into an equation and solve them together.

Answer 2

Let E = number of Estrada supporters
Let A = number of Arrayo supporters

Then E + A = 8600

Estrada's majority -- we don't know what it is yet -- is E - A

In the second scenario, 1/3 of Estrada's stay home, so he gets 2/3 E votes.
Likewise, 1/2 of Arrayo's fans are at home watching Dr. Who, so he's left with 1/2 A votes.

The new spread, still a majority for Estrada is this: 2/3 E - 1/2 A

We don't know what Estrado's majority is here, either. But we do know it's decreased by 200:

E - A - 200 = 2/3 E - 1/2 A

Group like terms and simplify:

200 = 1/3 E - 1/2 A

Now we have 2 equations in two unknowns:

200 = 1/3 E - 1/2 A
8600 = E + A

Can you solve a system of two equations and two unknowns? Doing so will provide you with values for E and A which you should then go back and plug into the other equations to see if they give the predicted values.