An art gallery displays works in three mediums: oil paintings, sculptures, and watercolors… — GMAT

An art gallery displays works in three mediums: oil paintings, sculptures, and watercolors. Initially, the ratio of oil paintings to sculptures to watercolors is $3 : 4 : 5$ , respectively. After the gallery acquires 12 additional oil paintings and sells 6 watercolors, the ratio of oil paintings to watercolors becomes $1 : 1$ . If no sculptures were acquired or sold, how many sculptures are currently on display at the gallery?

A
27
36Answer
C
45
D
48
E
108

Answer

By defining the initial quantities of oil paintings, sculptures, and watercolors as

3x

4x

, and

5x

respectively, we update the modified amounts to

3x + 12

for oil paintings and

5x - 6

for watercolors. Equating these gives

3x + 12 = 5x - 6

, leading to

2x = 18

and

x = 9

. The number of sculptures, which remained unchanged, is

4x = 4 \times 9 = 36

Step-by-Step Solution

Define variables based on the initial ratio

Let the multiplier be

x

. Initial quantities are Oil Paintings

= 3x

, Sculptures

= 4x

, and Watercolors

= 5x

Ratios expressed as

3 : 4 : 5

can be represented using a common multiplier

x

to find actual quantities.

Set up an equation incorporating the changes in inventory

New Oil Paintings

= 3x + 12

; New Watercolors

= 5x - 6

. The ratio between them is

1 : 1

, so

3x + 12 = 5x - 6

1 : 1

ratio means the updated quantity of oil paintings equals the updated quantity of watercolors.

Solve for the common multiplier

x

5x - 3x = 12 + 6 \implies 2x = 18 \implies x = 9

Linear algebraic manipulation isolates

x

Calculate the quantity of sculptures

Number of Sculptures

= 4x = 4 \times 9 = 36

Substitute the value of

x

back into the expression for sculptures.

Key Concept

Ratio modification and single-variable setting for multi-part ratios

Estimated Time:2m 0s

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