We have seen how tape diagrams represent relationships between quantities. Because of the meaning and properties of addition and multiplication, more than one equation can often be used to represent a single tape diagram.

Let’s take a look at two tape diagrams.

We can represent this diagram with several different equations. Here are some of them:

• $26 + 4x=46$, because the parts add up to the whole.
• $4x+26=46$, because addition is commutative.
• $46=4x+26$, because if two quantities are equal, it doesn’t matter how we arrange them around the equal sign.
• $4x=46-26$, because one part (the part made up of 4 $x$’s) is the difference between the whole and the other part.

Here are some equations that represent this diagram:

• $4(x+9)=76$, because multiplication means having multiple groups of the same size.
• $(x+9)\boldcdot 4=76$, because multiplication is commutative.
• $76\div4=x+9$, because division tells us the size of each equal part.