Jay’s Income

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Jay's net income equals his gross income minus his deductions. By what percent did Jay's net income change as of December 31, when both his gross income and his deductions increased?

(1) Jay's gross income increased by 8 percent on December 31.

(2) Jay's deductions increased by 16 percent on December 31.

Review: Jay's Income




Explanation

In this question, we can translate what we are told in the question into algebra as the equation on our scratch work. Furthermore, we can notate it informally in this way to indicate that G and D have increased:



This informal trick is used by some scientists and economists to indicate what's happening when there are changes to the entities in an equation. It clarifies something: that N goes up only if G goes up more than D goes up, since G is added and D is subtracted. Let's go to the statements, evaluating them separately first.

Statement (1): this information is definitely insufficient, because anything could be going on with deductions. Consider analysis by cases: they could have barely increased or wildly increased, changing everything. Insufficient.

Statement (2) is insufficient by comparable logic.

Combining them, we need to get more specific. We know that . Now, the new N, call it , is equal to



If we could factor out the multiples, we could substitute in for and then be able to get in terms of . But we can't do that, and we still don't know the values of G and D. We can evaluate by cases. If G and D were both $100, then there would be a gain of $8, a loss of $16, and a net loss of $8. Another case: if G were $100 and D were $200, the gain would be the same, but the loss doubled, for a different net loss. We have insufficient information to determine the change.

The correct answer is (E).


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