An artist creates custom ceramic tiles in two shapes: square tiles and hexagonal tiles. Ea… — SAT

An artist creates custom ceramic tiles in two shapes: square tiles and hexagonal tiles. Each square tile has an area of $12$ square inches, and each hexagonal tile has an area of $28$ square inches. The artist is designing a mosaic with a total area of $460$ square inches using only these two types of tiles. If the artist uses $15$ square tiles, how many hexagonal tiles are used in the mosaic?

Answer: 10

Answer

The correct answer is 10. By setting up a linear equation representing the total area of the mosaic,

12s + 28h = 460

, and substituting the given value of 15 for the number of square tiles, we solve for the number of hexagonal tiles to get 10.

To find the number of hexagonal tiles used in the mosaic, we set up a linear equation in two variables representing the total area. Let

s

represent the number of square tiles, and let

h

represent the number of hexagonal tiles. The total area is the sum of the areas of the square tiles and the hexagonal tiles, which is

12s + 28h = 460

. Given that the artist uses

15

square tiles, we substitute

s = 15

into the equation to get

12(15) + 28h = 460

. Simplifying the equation yields

180 + 28h = 460

. Subtracting

180

from both sides gives

28h = 280

. Dividing both sides by

28

yields

h = 10

. Therefore, the artist uses 10 hexagonal tiles.

Step-by-Step Solution

Define variables and set up the linear equation representing the total area of the mosaic.

12s + 28h = 460

, where

s

is the number of square tiles and

h

is the number of hexagonal tiles.

This models the relationship between the quantities of each tile type used and the total surface area of the mosaic.

Substitute the given number of square tiles into the equation.

12(15) + 28h = 460

, which simplifies to

180 + 28h = 460

We are given that the artist uses exactly 15 square tiles in the design.

Isolate the variable term by subtracting the constant from both sides.

28h = 280

This isolates the term containing the unknown variable on one side of the equation.

Solve for the remaining variable by dividing both sides by its coefficient.

h = 10

Dividing both sides by the coefficient yields the final count of hexagonal tiles.

Key Concept

Linear Equations in Two Variables

Estimated Time:1m 30s

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