A retail store sets the selling price, P, of a custom item based on its wholesale cost, C,… — GMAT

A retail store sets the selling price, $P$ , of a custom item based on its wholesale cost, $C$ , according to the linear equation $P = 1.4C + 20$ . The store's profit on each item is defined as $P - C$ . If the profit on a certain item is also equal to $0.2P + 40$ , what is the wholesale cost, $C$ , of the item in dollars?

Answer: 200 dollars

Answer

The wholesale cost

C

of the item is $200.

By writing profit both as

P - C = 0.4C + 20

and as

0.2P + 40 = 0.28C + 44

, we obtain a single linear equation in terms of

C

0.4C + 20 = 0.28C + 44

. Subtracting

0.28C

and

20

from both sides gives

0.12C = 24

, which simplifies to

C = 200

Step-by-Step Solution

Substitute

P = 1.4C + 20

into the standard profit expression

P - C

Profit =

0.4C + 20

This expresses the profit solely as a linear function of the wholesale cost

C

Substitute

P = 1.4C + 20

into the alternative profit expression

0.2P + 40

Profit =

0.28C + 44

This converts the given percentage-based profit condition into an expression dependent only on

C

Set the two profit expressions equal to each other and solve the resulting single-variable linear equation.

0.12C = 24 \implies C = 200

Equating two valid expressions for the same quantity allows solving for the unknown variable

C

Key Concept

Solving Systems of Linear Equations by Algebraic Substitution

Estimated Time:2m 0s

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