Bunuel wrote:

The town of Zenith has a population of 50,000. The average income of a person who lives in Zenith is $3,700 per year. Which of the following statements can be inferred from the table and the information about Zenith given above?

I. Every person in Zenith paid $37 in tax.

II. Tax is a linear function of income.

III. The total amount of taxes paid by the people of Zenith was at least $1,850,000.

(A) I only

(B) II only

(C) III only

(D) I and III only

(E) I, II, and III

woodywheel , this might help.

Evaluating:

I. Every person in Zenith paid $37 in tax.

NO. An average income of $3700 (in which an AVERAGE person indeed pays $37) tells you nothing about the income values that were summed to get the income average, and hence taxes that would be paid.

One person, for example, could pay $74, another $37, and a third, $0.

\(\frac{(74 + 37 + 0)}{3}\) = $37

II. Tax is a linear function of income.

NO. Linear functions have a constant rate of change, are a straight line on a graph.

Here, the rate of change is NOT constant. It is marginally progressive: the more you make, the higher percentage you pay. (Now, if only it actually operated that way.)

The slope of each bracket, if x = income and y = taxes, is not identical. Look at the beginning bracket numbers.

$4000 = $40 tax = 1 percent

$6000 = $80 tax = 1.33 percent

$8000 = $149 tax = 1.75 percent

$10,000 = $220 tax = 2.2 percent

Change rate: +0.33, +0.43, +0.45. Rate of change not constant. Steeper slope at $10,000 than at $6,000.

If everyone paid a flat tax of 80%, that would be a linear function. Where y = (.80)(income). Plug in x = income, you will get a linear result for y as a function of x.

At this point you do not have to evaluate III. A, B, D, and E contain option I, or II, or both. Both are false.

Answer C

III. The total amount of taxes paid by the people of Zenith was at least $1,850,000.

YES.

Average income = $3700

\((A*n) = S\)

$3700 per person * 50,000 persons = $185,000,000

Everyone pays at least 1 percent.

$185,000,000 *.01 = $1,850,000

Some people could pay zero (if $0 in income), some people could pay a lot. The point: the SUM of incomes has been determined, then divided by 50,000 to get AVERAGE income of $3,700.

It doesn't matter if some pay zero tax and one honest person pays $500,000. Total and average income have been determined. No matter what, at least 1% of (average income * # of people) will be paid.

IF average income had not been determined, then having a few pay zero, and even fewer paying a high amount, would be an issue. You wouldn't know how to weight the average. This average has already been weighted by income, at least on the absolute minimum side.

Answer C