Skill: Represent equations with tape diagrams

A tape diagram shows how quantities fit together.

• A whole is the total.
• Parts placed side by side show addition.
• Equal-sized parts show repeated amounts, or multiplication.
• A variable labels an unknown part.

Suppose a problem says:

Three equal boxes and 4 extra items make 19 items altogether. How many items are in each box?

Let $x$ represent the number of items in one box.

The tape diagram is:

                 19 items
        ┌─────┬─────┬─────┬───┐
        │  x  │  x  │  x  │ 4 │
        └─────┴─────┴─────┴───┘

1. The entire tape is labeled $19$ because $19$ is the total.
2. There are three equal parts labeled $x$, so they represent $3x$.
3. The last part is labeled $4$.
4. The diagram translates to the equation

$$x+x+x+4=19$$

or, using multiplication,

$$3x+4=19.$$

To find $x$, first remove the extra $4$:

$$3x=15$$

Then divide the 15 items equally among the 3 equal boxes:

$$x=5.$$

Each box contains $5$ items. Checking the diagram:

$$5+5+5+4=19.$$

The three equal sections show the coefficient $3$, while $x$ labels the unknown amount in one section.