A tech consulting firm billed a client for a project completed by senior developers and ju… — GMAT

A tech consulting firm billed a client for a project completed by senior developers and junior developers. Senior developers were billed at a constant hourly rate of $x$ , and junior developers were billed at a constant hourly rate of $y$ . What was the average (arithmetic mean) hourly rate billed across all developer hours on the project?

(1) The total amount billed for senior developer hours was $50\%$ greater than the total amount billed for junior developer hours.
(2) Senior developers worked $40\%$ fewer total hours on the project than junior developers did.

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.Answer

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.

The correct choice is the option stating that both statements together are not sufficient. Statement (1) provides the ratio of total revenues between the two developer types, which is insufficient to find the overall average rate. Statement (2) provides the ratio of hours worked between senior and junior developers, which is also insufficient on its own. Combining both statements allows us to find that senior developers earn

2.5

times the hourly rate of junior developers and to express the average hourly rate as

1.5625 y

. However, because neither statement provides an absolute dollar amount for

x

y

, or total revenue, the numerical average hourly rate cannot be calculated.

Step-by-Step Solution

Rephrase the target question mathematically.

Let

h_s

be the senior hours and

h_j

be the junior hours. Total billing is

x h_s + y h_j

, and total hours is

h_s + h_j

. The average hourly rate is

A = \frac{x h_s + y h_j}{h_s + h_j}

. A specific numerical value for

A

is required.

Clarifying the target prevents confusing relative ratios with absolute dollar values.

Evaluate Statement (1) independently.

Statement (1) states that

x h_s = 1.5 y h_j

. Total billing equals

2.5 y h_j

, so

A = \frac{2.5 y h_j}{h_s + h_j}

Without knowing the ratio of hours

\frac{h_s}{h_j}

or specific dollar values for

x

y

A

cannot be determined. Statement (1) alone is insufficient.

Evaluate Statement (2) independently.

Statement (2) states that

h_s = 0.6 h_j

. Thus

A = \frac{0.6 x h_j + y h_j}{1.6 h_j} = \frac{0.6 x + y}{1.6}

Without the specific values of

x

and

y

A

cannot be determined. Statement (2) alone is insufficient.

Evaluate Statements (1) and (2) together.

From (2), substitute

h_s = 0.6 h_j

into (1):

x (0.6 h_j) = 1.5 y h_j \implies 0.6 x = 1.5 y \implies x = 2.5 y

. Substituting

x = 2.5 y

and

h_s = 0.6 h_j

into

A

yields

A = \frac{(2.5 y)(0.6 h_j) + y h_j}{1.6 h_j} = \frac{1.5 y h_j + y h_j}{1.6 h_j} = \frac{2.5 y}{1.6} = 1.5625 y

Combining both statements allows us to express the average rate as a multiple of

y

(

1.5625 y

), but since no concrete dollar value for

y

x

is provided in either statement, the exact average dollar rate cannot be calculated. Thus, both statements together are insufficient.

Key Concept

Distinguishing relative ratios from absolute numerical values in Data Sufficiency

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

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