A group of people was surveyed about whether the council should collect recycling once a week instead of once a fortnight. Two-thirds of the people said ‘Yes’, and one-third said ‘No’.

A year later the same group of people was surveyed again and this time one-quarter of them had changed their minds, resulting in a tie between the ‘Yes’ and ‘No’ votes.
Of the people who originally voted ‘Yes’, what fraction changed their minds?

Let x be the number who changed from No to Yes

Let y be the number who changed from Yes to No
If there were n people altogether, then
1/3 n - x + y = 2/3 n - y + x
x + y = 1/4 n
or,
y-x = 1/6 n
y+x = 1/4 n
So, y = 5/24 n
But we want the fraction of the original Yes voters, so
(5/24 n) / (2/3 n) = 5/16

Well, it seems like the recycling debate really took a toll on these people's minds! Let's do some math, but don't worry, I won't recycle any old jokes for you.

Let's assume there were 100 people in the group initially (easier math!). So, two-thirds, or 2/3, of them voted 'Yes', which means 2/3 * 100 = 66.67 people voted 'Yes'.

Now, a year later, one-quarter, or 1/4, of the total group changed their minds. Since the votes were tied, 50 people voted 'Yes' this time. This means that 50 - 66.67 = -16.67 people who originally voted 'Yes' changed their minds.

As you can see, we have a negative fraction of people who changed their minds! Seems like there was some messy arithmetic going on here. Perhaps some people started voting from the recycling bin? Yikes!

In all seriousness, it's not possible to have a negative fraction in this context. So, let's conclude that there was an error in the calculations, and we'll need more accurate information to determine the exact fraction of people who changed their minds.

Let's assume there were initially a total of "N" people in the group surveyed.

According to the information provided, two-thirds of the people initially said 'Yes', which means (2/3) * N said 'Yes'.
This implies that (1/3) * N said 'No'.

A year later, one-quarter of the people changed their minds, resulting in an equal number of 'Yes' and 'No' votes.

Let's calculate the number of people who voted 'Yes' a year later:
Initially, (2/3) * N said 'Yes'.
Out of these, one-quarter changed their minds, so the number of 'Yes' voters who changed their minds is (1/4) * (2/3) * N.

To find the fraction of people who changed their minds out of those who originally voted 'Yes', we need to divide the number of 'Yes' voters who changed their minds by the total number of people who initially voted 'Yes'.

Fraction of people who changed their minds = (number of 'Yes' voters who changed their minds) / (number of people who initially voted 'Yes')
Fraction of people who changed their minds = (1/4) * (2/3) * N / (2/3) * N

The fraction simplifies to:
Fraction of people who changed their minds = 1/4

Therefore, one-quarter of the people who originally voted 'Yes' changed their minds.

To find the fraction of people who originally voted 'Yes' and changed their minds, we need to follow a step-by-step approach:

Step 1: Determine the total number of people surveyed. Let's assume the total number of people surveyed is denoted by 'x'.

Step 2: Calculate the number of people who initially voted 'Yes'. Since two-thirds of the people voted 'Yes', we can calculate it as (2/3) * x.

Step 3: Calculate the number of people who initially voted 'No'. Since one-third of the people voted 'No', we can calculate it as (1/3) * x.

Step 4: Determine the number of people who changed their minds. If we assume that one-quarter of the people changed their minds, we can calculate it as (1/4) * x.

Step 5: Calculate the fraction of people who originally voted 'Yes' and changed their minds. To do this, divide the number of people who changed their minds by the number of people who originally voted 'Yes'. Therefore, the fraction can be calculated as [(1/4) * x] / [(2/3) * x].

Simplifying the fraction, we get:
[(1/4) * x] / [(2/3) * x] = (1/4) * (3/2) = 3/8.

Therefore, the fraction of people who originally voted 'Yes' and changed their minds is 3/8.