Area of Rectangles with Mixed Number Side Lengths

Method 1: split the rectangle into parts. The length $2\frac{1}{2}$ is $2$ units and then $\frac{1}{2}$ unit more. The width $1\frac{1}{2}$ is $1$ unit and then $\frac{1}{2}$ unit more.

Picture the rectangle with its length running from left to right. Make one straight cut from top to bottom, $2$ units from the left edge. Make another straight cut from left to right, $1$ unit up from the bottom edge. The two cuts make four parts. Each part is a smaller rectangle.

Every part has side lengths that are whole numbers or fractions, so you can find its area by multiplying. The four parts cover the whole rectangle with no gaps and no overlaps. So their areas add up to the area of the whole rectangle. The table lists the four parts.

Now add the four areas from the table. Add the whole numbers first: $2 + 1 = 3$. Then add the fractions. Write $\frac{1}{2}$ as $\frac{2}{4}$ so that both fractions are in fourths: $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$.

So the area of the whole rectangle is $2 + \frac{1}{2} + 1 + \frac{1}{4} = 3\frac{3}{4}$ square units.

Method 2: multiply improper fractions. In an improper fraction, the numerator (top number) is equal to or greater than the denominator (bottom number). One example is $\frac{5}{2}$. You can write a mixed number as an improper fraction by counting all of its equal parts. For $2\frac{1}{2}$, count halves. Each whole is $2$ halves, so $2$ wholes are $4$ halves. One more half makes $5$ halves, or $\frac{5}{2}$. In the same way, $1\frac{1}{2}$ is $3$ halves, or $\frac{3}{2}$.

Here is what that means for the same rectangle. Picture small squares that are $\frac{1}{2}$ unit on each side. The rectangle is $5$ small squares long and $3$ small squares wide, so it holds $5 \times 3 = 15$ small squares. Each small square has an area of $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ square unit. So the area is $15$ fourths of a square unit, or $\frac{15}{4}$ square units.

Why do the two methods give the same area? They measure the same rectangle. Method 1 cuts it into four parts. Cutting does not add any space or take any away, so the four areas add up to the area of the whole rectangle. Method 2 finds that whole area at once, by multiplying the whole length by the whole width.

You can check this by counting. Use the small squares from Method 2. Each one is $\frac{1}{4}$ square unit. The next table shows how many of them fit inside each part from Method 1. Together the four parts hold $15$ small squares. Those are the same $15$ small squares that Method 2 counts as $5 \times 3$. Method 1 counts them in four groups. Method 2 counts them all at once.

Word problems. A story may not use the word area. Look for a rectangle and a question about how much flat surface it covers, such as part of a wall, a floor or the ground. That amount is the area. Then follow three steps:

1. Find the length and the width in the story.
2. Multiply them. A side that is a fraction is ready to multiply. For a side that is a mixed number, split it into parts or write it as an improper fraction.
3. Give the answer in square units.

Why square units? Area tells how many squares, and parts of squares, cover a surface. When both sides are in feet, those squares are $1$ foot on each side. Each one is a square foot. So a length in feet times a width in feet gives an area in square feet. This works only when both sides are in the same unit. In the problems here, they are.

Takeaway. The area of a rectangle is its length times its width, with both sides in the same unit. This is true even when the side lengths are mixed numbers. You can split the rectangle into parts and add every part. Or you can write the side lengths as improper fractions and multiply. Both methods measure the same rectangle, so they give the same area. In a word problem, find the two side lengths, multiply them, and give your answer in square units.