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. (A) Find the fraction of residents who were opposed to the building of a new library. (B) Then find the total number of residents in the town.

Answer: (A) The fraction of residents opposed to the project was three-sevenths (3/7)

(B) The total number of residents is 2,128.

Step-by-step explanation: If the number of residents in support of the building project is expressed as a fraction of the total number, then the total number can be expressed as 1. That means we should have two fractions, those in support and those not in support and their addition should be equal to 1.

Hence if the total is 1 and 4/7 are in support, the fraction of those opposed (let that be y) can be calculated as

y = 1 - 4/7

y = 7/7 - 4/7

y = (7 - 4) / 7

y = 3/7

Also, the number of residents not in support is 912 (which is 3/7 of the total). That means 912 is three out of seven of f the total population. In other words,

3/7 = 912/x

(Where x is the total population)

By cross multiplication we now have

3x = 912 x 7

x = 6384/3

x = 2128

Therefore the total population of residents is 2,128.