word problem, algebra

Question

Emily counts the triangles and parallelograms in an art piece and determines that there are altogether 42 triangles and parallelograms. If there are a total of 150 sides, how many triangles and parallelograms are there?

Answer 1

Laura,

 

Word problems require concepts to be turned into equations.

 

There are two key concepts in this problem: 42 triangles and parallelograms combined; a total of 150 sides between them. These ideas need to be turned into equations. Start by replacing words with symbols. Make triangles T and parallelograms P.

 

Hint: If you get stuck, picture an actual mural with this combination of triangles and parallelograms. Imagine counting up all the shapes and all the sides. This will make the problem more concrete.

 

The first equation

 

The first key concept is that there are 42 triangles and parallelograms combined.

 

The word "combined" should make you think "when added together." When the number of triangles and parallelograms are added together, they equal 42.

 

T + P = 42

 

I've taken a shortcut and made T and P be the number of triangles and parallelograms, respectively. There's no obvious reason why that should be. But it works.

 

The second equation

 

The second key concept is that the number of sides combined is 150.

 

Think about how to find the number of triangle-sides given T. One triangle, 3 total sides. Two triangles, 6 sides. Three, 9 sides.

 

3T = total number of triangle sides.

 

What about converting the number of parallelograms to the number of sides? One parallelogram, four sides. Two, eight sides. And so on.

 

4P = total number of parallelogram sides.

 

Now find the number of sides between the triangles and parallelograms combined. Note the word "combined." I'll leave finding that equation up to you.

 

If you're having trouble, think about how many total sides there would be if there were only one triangle and one parallelogram. What about two triangles and two parallelograms? Three of each? Four of each? And so on. 

 

The solution

 

You now have two equations and two variables (T, P). Two variables and two equations means you can solve for T and P by linear combinations or substitution.

 

Good luck and leave a comment if you need help,

Bryce