An investor holds x shares of Stock X and y shares of Stock Y in a financial portfolio. Th… — GMAT

An investor holds $x$ shares of Stock X and $y$ shares of Stock Y in a financial portfolio. The total monetary value of these holdings is $\$6,300$ , modeled by the linear equation $40x + 20y = 6,300$ . If the ratio of the number of shares of Stock X to the total number of shares held is $2$ to $5$ , what is the total number of shares of Stock X and Stock Y in the portfolio?

A
90
B
135
C
210
225Answer
E
245

Answer

225

The correct answer is 225. Expressing the part-to-whole ratio

\frac{x}{x+y} = \frac{2}{5}

algebraically gives

3x = 2y

, or

y = 1.5x

. Substituting

1.5x

for

y

in the value equation

40x + 20y = 6,300

yields

70x = 6,300

, so

x = 90

. Then

y = 135

, and the total number of shares is

90 + 135 = 225

Step-by-Step Solution

Set up the linear relationship between

x

and

y

using the given ratio.

\frac{x}{x + y} = \frac{2}{5} \implies 5x = 2x + 2y \implies 3x = 2y \implies y = 1.5x

The problem specifies that the ratio of Stock X shares (

x

) to total shares (

x + y

) is

2:5

Substitute

y = 1.5x

into the portfolio value equation

40x + 20y = 6,300

40x + 20(1.5x) = 6,300 \implies 40x + 30x = 6,300 \implies 70x = 6,300 \implies x = 90

Replacing

y

with an equivalent expression in terms of

x

reduces the system to a single linear equation in one variable.

Calculate

y

and determine the total number of shares

x + y

y = 1.5(90) = 135 \implies x + y = 90 + 135 = 225

The question asks for the total combined number of shares of both stocks held in the portfolio.

Key Concept

Solving systems of linear equations formed by combining a linear value equation with a ratio relationship.

Question

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