Question

The percentage of households with an annual income of more than \$40,000 is higher in Merton County than in any other county. However, the percentage of households with an annual income of \$60,000 or more is higher in Sommer County.
If the statements above are true, which of the following must also be true?

• A. The percentage of households with an annual income of \$80,000 is higher in Sommer County than in Merton County.
• B. Merton County has the second highest percentage of households with an annual income of \$60,000 or more.
• C. Some households in Merton County have an annual income between \$40,000 and \$60,000.
• D. The number of households with an annual income of more than \$40,000 is greater in Merton County than in Sommer County.
• E. Average annual household income is higher in Sommer County than in Merton County.

Hint:

For "Must Be True" questions involving percentages and overlapping categories, try to use proof by contradiction. Assume the opposite of an answer choice is true and see if it logically conflicts with the given premises. If it does, the original answer choice must be correct.

Answer 1

The Correct Option is C

Step 1: Understanding the Premises
Let's represent the given information using symbols to make it clearer.
Let P(County, Income Bracket) be the percentage of households in a given county within a certain income bracket.
Premise 1: P(Merton, >\$40k) > P(Any other county, >\$40k). This implies P(Merton, >\$40k) > P(Sommer, >\$40k).
Premise 2: P(Sommer, \(\geq\)\$60k) > P(Merton, \(\geq\)\$60k).

Step 2: Analyzing the Income Brackets
The question involves three income groups:
1. Income greater than \$40,000 (> \$40k).
2. Income greater than or equal to \$60,000 (\(\geq\) \$60k).
3. The bracket between these two, which we can define as incomes greater than \$40,000 but less than \$60,000 (> \$40k and < \$60k).
The percentage of households in this middle bracket can be calculated as:
P(County, >\$40k and <\$60k) = P(County, >\$40k) - P(County, \(\geq\)\$60k).
The question asks what must be true. This requires a logically certain conclusion.

Step 3: Detailed Explanation
Let's focus on Merton County. We need to determine if there must be any households in the middle bracket (>\$40k and <\$60k). This is equivalent to asking if P(Merton, >\$40k) must be strictly greater than P(Merton, \(\geq\)\$60k).
Let's assume, for the sake of contradiction, that there are no households in Merton County with incomes between \$40,000 and \$60,000.
If this were true, then any household with an income over \$40,000 must also have an income of \$60,000 or more.
This would mean: P(Merton, >\$40k) = P(Merton, \(\geq\)\$60k).
Now let's bring in the premises:
From Premise 1, we know: P(Merton, >\$40k) > P(Sommer, >\$40k).
If our assumption is true, we can substitute to get: P(Merton, \(\geq\)\$60k) > P(Sommer, >\$40k).
We also know that for any county, the percentage of households with income >\$40k must be greater than or equal to the percentage with income \(\geq\)\$60k. So, P(Sommer, >\$40k) \(\geq\) P(Sommer, \(\geq\)\$60k).
Combining these inequalities, we get: P(Merton, \(\geq\)\$60k) > P(Sommer, >\$40k) \(\geq\) P(Sommer, \(\geq\)\$60k).
This leads to the conclusion that P(Merton, \(\geq\)\$60k) > P(Sommer, \(\geq\)\$60k).
However, this directly contradicts Premise 2, which states that P(Sommer, \(\geq\)\$60k) > P(Merton, \(\geq\)\$60k).
Since our initial assumption led to a contradiction, the assumption must be false. Therefore, there must be some households in Merton County with an income between \$40,000 and \$60,000.

Step 4: Evaluating the Options
(A) We are given no information about incomes above \$60,000, so we cannot make any conclusions about the \$80,000 bracket.
(B) We know Sommer has a higher percentage than Merton in the \(\geq\)\$60k bracket, but we don't know how either county ranks against all other counties. Merton could be last.
(C) As proven above, this statement must be true. The premises would be logically inconsistent otherwise.
(D) The premises are about percentages, not absolute numbers. We cannot conclude anything about the number of households without knowing the total population of each county.
(E) We cannot determine the average income. A high percentage of people in one bracket does not guarantee a higher overall average income.