Mixed Numbers to Improper Fractions

A mixed number contains whole units and fractional parts.

To rewrite it as an improper fraction, replace every whole with the number of unit fractions that make that whole, then count all the parts.

This makes sense of the familiar calculation whole number × denominator + numerator: the multiplication counts the fractional parts hidden inside the wholes.

Let the red rod represent one whole. White is one-half of red:

2w = r

Build 2½:

r + r + w

Now replace each red whole with two white halves.

r + r + w = 2w + 2w + w = 5w

So:

2½ = 5⁄2

Why do we calculate 2 × 2? Because there are two wholes and each whole contains two halves. That gives four halves, and the extra half makes five.

2 × 2 + 1 = 5

Now let the light green rod represent one whole. White is one-third of light green:

3w = g

Build 2⅔:

g + g + w + w

Replace each whole with three thirds:

g + g + 2w = 3w + 3w + 2w = 8w

Therefore:

2⅔ = 8⁄3

The formal calculation records exactly what happened:

2 × 3 + 2 = 8

The denominator tells us how many unit fractions are hidden inside each whole.