Question

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

Answer 1

42 pentagons and 18 octagons

Explanation:

We will call the number of pentagons '
`x`' and the amount of octagons '
`y`'.

If the sum of both is 60 figures:

`x + y = 60`

The previous one will be equation 1.

In addition, the problem mentions that there are 354 sides in total, for this we will have to add the amount of sides corresponding to the pentagons: 5x, because the pentagon has 5 sides and we have x pentagons, plus the number of sides corresponding to the octagons: 8y, because the octagon has 8 sides and we have 'y' octagons.

so the equation 2 will be:

`5x + 8y = 354`

Now we must solve the system of two equations we have.

For this we clear 'x' from equation 1, and substitute in equation 2.

Clearing for 'x' in equation 1:

`x + y = 60\\x = 60 -y`

Substituting in eqution 2:

`5x + 8y = 354\\5(60 - y) + 8y = 354`

And solving for 'y'

`300 -5y + 8y =354\\3y = 354 -300\\3y = 54\\y=54/3\\y=18`

This means there are 18 octagons.

And now to find the number of pentagons, we substitute the value of y = 18 in any of the two equations, i will do it in the equation 1:

`x + y = 60`

`x + 18 = 60`

`x = 60 -18`

`x = 42`

There are 42 pentagons

Answer 2

Answer: There are 42 pentagons and 18 octagons in the set.

Explanation:

Let the number of pentagons be x and the number of octagons be y.

Then
`x+y=60`

`\\\Rightarrow\ y=60-x`....(1)

We know that a pentagon has 5 sides and an octagon has 8 sides, thus the total number of sides in the set is written as

`5x+8y=354`....(2)

Substitute the value of y from (1) in (2), we get

`5x+8(60-x)=354\\\Rightarrow\ 5x+480-8x=354\\\Rightarrow\ -3x=354-480\\\Rightarrow\ -3x=-126\\\Rightarrow\ x=42\\\\\Rightarrow\ y=60-42=18`

Hence, there are 42 pentagons and 18 octagons in the set.