Question

19. A box contains pieces of wood cut into triangular, pentagonal, and hexagonal shapes. There are 53

pieces of wood in the box and the pieces total 253 sides. The box contains 5 more pentagons than
triangles. How many of each shape are in the box?

Answer 1

There are 5 triangular, 10 pentagonal, and 38 hexagonal pieces in the box.

Step-by-step explanation:

Let's denote the number of triangular pieces as x, the number of pentagonal pieces as y, and the number of hexagonal pieces as z. From the given information, we can form the following equations:

x + y + z = 53 --> (Equation 1)

3x + 5y + 6z = 253 --> (Equation 2)

We also know that there are 5 more pentagons than triangles, so we can write the equation:

y = x + 5 --> (Equation 3)

To solve this system of equations, we can substitute Equation 3 into Equations 1 and 2.

After substituting and simplifying, we get:

4x + z = 48 --> (Equation 4)

x = 5

Substituting this value back into the equations, we find that:

y = 10

z = 38

Therefore, there are 5 triangular pieces, 10 pentagonal pieces, and 38 hexagonal pieces in the box.

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