Question

a geometry teacher has a set of 60 plastic pentagons and octagons. she happened to notice that all the figures together have a total of 354 sides. how many of each shape are there.

Answer 1

Let \( p \) be the number of pentagons and \( o \) be the number of octagons. We can set up the following system of equations based on the information provided:

1. The total number of shapes is 60:
\[
p + o = 60
\]

2. The total number of sides is 354:
\[
5p + 8o = 354
\]

We can solve this system of equations step by step.

First, from the first equation, we can express \( o \) in terms of \( p \):
\[
o = 60 - p
\]

Next, we substitute this expression for \( o \) into the second equation:
\[
5p + 8(60 - p) = 354
\]
Expanding this gives:
\[
5p + 480 - 8p = 354
\]
Combining like terms, we have:
\[
-3p + 480 = 354
\]
Subtracting 480 from both sides:
\[
-3p = 354 - 480
\]
\[
-3p = -126
\]
Dividing by -3:
\[
p = 42
\]

Now that we have \( p \), we can find \( o \) using the expression we found earlier:
\[
o = 60 - p = 60 - 42 = 18
\]

Thus, the teacher has \( \boxed{42} \) pentagons and \( \boxed{18} \) octagons.

Let's verify our solution:
- The total number of shapes is \( 42 + 18 = 60 \) (correct).
- The total number of sides is \( 5(42) + 8(18) = 210 + 144 = 354 \) (correct).

Everything checks out!