In a town survey residents were asked whether they would approve of a tax increase to build a new library. four-sevenths of the residents were in support of the new library and the rest were opposed.?

Answer 1

Not necessarily.

There is also (at least) the possibility that people don't care one way or the other.

It could be:

• 4/7 support, none oppose, 3/7 don't care
• 4/7 support, 1/7 oppose, 2/7 don't care
• 4/7 support, 2/7 oppose, 1/7 don't care
• 4/7 support, 3/7 oppose, none don't care

or any other split of the 3/7 rest of the residents between oppose and don't care.

I suspect that was not was you were intending to ask but that is the question that you did ask as any more of the question you blindly copied without any thought as to what may be the relevant information was truncated at the question length limit. If there is more to the question, please resubmit it by concisely including the relevant information.

For example, if the full question was:

"In a town survey, residents were asked whether they would approve of a tax increase to build a new library. Four-sevenths of the residents were in support of the new library and the rest of the residents were opposed. There were 912 residents who were opposed. Find the fraction of the residents who were opposed to the building of a new library. Then find the total number of residents in the town."

You should ask:

"What is the fraction of residents who oppose a new library and the total number of residents when a town survey found 912 residents representing four sevenths in favour and the rest opposed?"

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In fact I highly suspect that was the full question example above is that which was intended, and will answer it - will I get the full credit by fulfilling the "Need an equation for full credit."?

If you look at the full question, the survey was to see if they would approve a tax increase to build a new library. The survey results shows those in favour of a tax rise to build a new library, NOT those in favour of a new library; there could be those in favour of a new library but oppose a tax rise to fund it.

Now we have a problem as it is IMPOSSIBLE with the given data to find the number of residents in the town. What was done was a survey not a census of the residents. As such only a sample of the residents were asked and it is usual for not all people surveyed to respond. The data we have is that less than 100% of the residents were surveyed, no indication of how many of them responded and 4/7 of those that did respond represents 912 were in favour and the rest were opposed. The best we can achieve is to give the minimum number of residents in the town.

However, I assume that what was intended in the question was that a census of the town was done with a 100% response and 4/7 were in favour of a tax rise to fund a new library and this was 912 residents.

So ignoring the inconsistencies in the hypothetical situation that the mathematical problem has been dressed up in:

There are 4/7 who are in favour and the rest oppose.

4/7 + opposers = 1

→ opposers = 1 - 4/7

= 7/7 - 4/7

= (7-4)/7

= 3/7

So 3/7 oppose.

4/7 of the residents is 912 residents

→ 1/7 of the residents is 912 ÷ 4 residents

→ all the residents is 912 ÷ 4 × 7 residents = 1596 residents.