This question can be solved either algebraically or by picking numbers.

ALGEBRA:

let N = net income, G = gross income, D = deductions
N = G - D

let x = change in net income, y = change in gross income, and z = change in deductions
N(1 + x/100) = G(1 + y/100) - D(1 + z/100)

If we want to know the value of x, we need information about G, D, y, and z. (You could solve for x algebraically here, but that would make things needlessly complicated).

1. Guy's gross income increased by 4% on Jan 1, 1989

This tells us that y = 4, but tells us nothing about z, G, or D. Insufficient.

2. Guy's deductions increased by 15% on Jan 1, 1989

This tells us that z = 15, but tells us nothing about y, G, or D. Insufficient.

1 & 2 together:

We can now express the new net income as:

N(1 + x/100) = 1.04G - 1.15D

substitute G - D for N:
(G - D)(1 + x/100) = 1.04G - 1.15D
1 + x/100 = (1.04G - 1.15D)/(G - D)

Since we cannot simplify the righthand side of the equation and cancel G and D, we cannot solve for x. Insufficient.

The answer is E.

If you found the algebra confusing or weren't sure how to interpret it, picking numbers is a good option.

PICKING NUMBERS:

It's easy to see that the statements are insufficient on their own, as they each give only partial information. Therefore, let's start with the 2 statements together:

1 & 2 together:

If Net = Gross - Deductions, let:
original Gross = 100
original Deductions = 10
thus, original Net = 90

After a 4% increase to Gross and a 15% increase to deductions,
new Gross = 104
new Deductions = 11.5
thus, new Net = 92.5
the percent change in the Net is a little more than a 2.5% increase

Now, let's see if we get the same result with different values.

To keep things simple, keep the same Gross, but pick a much larger number for deductions:
original Gross = 100
original Deductions = 80
thus, original Net = 20

After a 4% increase to Gross and a 15% increase to deductions,
new Gross = 104
new Deductions = 92
thus, new Net = 12
the percent change in the Net is now a 40% decrease!

We got vastly different results in different cases, so the answer must be E.

The third possibility - circumventing algebra and picking numbers - is to THINK CONCEPTUALLY about the relationships given.

Because net income is defined as a difference between gross and deductions, we don't know anything about the proportional relationship of gross to deductions. If gross is HUGE but deductions are tiny, the 4% increase in gross will be bigger than a 15% increase in deductions. If those numbers are much closer to each other, though, the 15% increase in deductions will weigh more heavily than the 4% increase in gross, and the net income could actually decrease.

To have sufficient information here, we'd also need a PROPORTION of gross income to net income.

Consider the difference between this situation and the following:

Simple income is defined as wage times number of hours works. By how much did Guy's simple income increase when his wage went up 20% and his hours worked went up 10%?

Here, the relationship is multiplicative: S = (w)(h)
Multiplicative relationships are proportional.

We can express the change as:
(1.2w)(1.1h)
= 1.32wh
Since we know that wh = S, there must have been a 32% increase in S.

So, if you know the percent change in each term in a multiplicative relationship (a product or ratio), you'll know the percent change in the total. If it's an additive relationship (a difference or sum), just knowing the percent change in each term is not enough to know the percent change in the total.

amina.shaikh309 wrote:Guy's net income equals his gross income minus his deductions. By what % did Guy's net income change on Jan 1,1989, when both his gross income and his deductions increased?

1. Guy's gross income increased by 4% on Jan 1, 1989
2. Guy's deductions increased by 15% on Jan 1, 1989

We are given that Guy's net income is equal to his gross income minus his deductions, and we want to determine by what percent his net income changed on January 1, 1989, when both his gross income and his deductions increased.

Statement One Alone:

Guy's gross income increased by 4 percent on January 1, 1989.

Since we don't know anything about his deductions, statement one does not provide sufficient information for us to determine the percent change in his net income.

Statement Two Alone:

Guy's deductions increased by 15 percent on January 1, 1989.

Since we don't know anything about his gross income, statement two does not provide sufficient information for us to determine the percent change in his net income.

Statements One and Two Together:

Let g = his old gross income and d = his old deductions. Using the information from statements one and two we can express his new gross income as 1.04g and his new deductions as 1.15d.

The percent change in his net income is

(new net income)/(old net income) x 100.

In other words, the percent change is

(new gross income - new deductions)/(old gross income - old deductions) x 100.

Thus, the percent change is

(1.04g - 1.15d)/(g - d) x 100.

However, we see that we still do not have enough information to determine the percent change in his net income since we still don't know the values of g and d.

Answer: E