How many people voted for each candidate?

Let's follow the steps of George Polya's problem-solving process to analyze and solve the problem:

1) Understand the problem:

The problem states that during an election, a certain number of people voted for three candidates, Mr. M, Mr. K, and Mr. J. We are given the following information:

• 3/7 of the voters voted for Mr. M.
• 2/9 of the remaining voters (after Mr. M's votes) voted for Mr. K.
• 2/3 of what still remained (after Mr. M and Mr. K's votes) voted for Mr. J.
• 1560 people registered but did not vote.

We need to determine how many people voted for each candidate.

2) Devise a plan:

To solve the problem, we can use algebraic equations to represent the given information and find the values we're looking for. Let's assume the total number of voters is "x" and then use the given fractions to express the number of voters who voted for each candidate.

3) Carry out the plan:

Let's calculate the number of voters who voted for each candidate step by step.

Step 1: Number of voters who voted for Mr. M

3/7 of the voters voted for Mr. M, so the equation can be written as:

(3/7) * x

Step 2: Remaining voters after Mr. M's votes

The remaining voters after Mr. M's votes can be calculated by subtracting the votes for Mr. M from the total number of voters:

x - (3/7) * x

Step 3: Number of voters who voted for Mr. K

2/9 of the remaining voters voted for Mr. K, so the equation can be written as:

(2/9) * (x - (3/7) * x)

Step 4: Remaining voters after Mr. K's votes

The remaining voters after Mr. K's votes can be calculated by subtracting the votes for Mr. K from the remaining voters after Mr. M's votes:

(x - (3/7) * x) - (2/9) * (x - (3/7) * x)

Step 5: Number of voters who voted for Mr. J

2/3 of what still remained after Mr. M and Mr. K's votes voted for Mr. J, so the equation can be written as:

(2/3) * ((x - (3/7) * x) - (2/9) * (x - (3/7) * x))

Step 6: Set up the final equation

The sum of the votes for all candidates should equal the total number of voters minus the 1560 people who registered but did not vote:

(3/7) * x + (2/9) * (x - (3/7) * x) + (2/3) * ((x - (3/7) * x) - (2/9) * (x - (3/7) * x)) = x - 1560

Step 7: Solve the equation

Solve the equation to find the value of x, which represents the total number of voters.

4) Look back:

Once we have the value of x, we can substitute it back into the equations for each candidate to calculate the number of voters who voted for Mr. M, Mr. K, and Mr. J.