Fraction Word Problems with Bar Models

Draw the whole first

A fraction always refers to some whole. The bar model starts there.

1. Draw a bar. That is one whole.
2. Split it into the number of parts the denominator names.
3. Shade what the problem gives you.

Ana eats $\tfrac{3}{8}$ of a pizza.

The bar is the pizza. The eight parts are the slices. Three are shaded.

Everything else in the problem is drawn on the same bar.

Adding parts of the same size

Ana eats $\tfrac{3}{8}$ of the pizza and Ben eats $\tfrac{2}{8}$. How much is gone?

Both fractions count eighths of the same pizza. So count all the eighths:

$$\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$$

Add the top numbers. Keep the bottom number unchanged, because the slices did not change size. There are more of them shaded, and that is all.

Subtraction works the same way:

$$\frac{7}{8} - \frac{2}{8} = \frac{5}{8}$$

Why the whole has to be the same

Ana eats $\tfrac{1}{2}$ of a large pizza. Ben eats $\tfrac{1}{2}$ of a small one. Together, have they eaten a whole pizza?

No. The two halves are different amounts, because the two wholes are different.

$$\frac{1}{2} + \frac{1}{2} = 1 \quad\text{only when both halves are halves of the same thing}$$

A fraction is meaningless until you name the whole. In a word problem this is the first thing to settle, and it is where wrong answers usually start.

Drawing one bar rather than two keeps you honest about it. If two amounts will not fit on the same bar, they do not add.

Repeating the same fraction

A glass holds $\tfrac{2}{5}$ of a liter. You fill $4$ glasses. How much in all?

You could add:

$$\frac{2}{5} + \frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \frac{8}{5}$$

Four lots of the same thing is a multiplication, so write it that way:

$$4 \times \frac{2}{5} = \frac{8}{5}$$

Multiply the top number. The bottom stays at $5$, because every glass is measured in the same fifths.

$$n \times \frac{a}{b} = \frac{n \times a}{b}$$

An answer over one whole is normal here. $\tfrac{8}{5}$ is $1$ whole and $\tfrac{3}{5}$, which is what four glasses of nearly half a liter should come to.

Checking the answer is sensible

Every answer describes something real, so hold it against the story.

Warning sign	What it usually means
more than the whole, when some was eaten	added instead of subtracting
a smaller total after adding	the denominators were added too
the denominator changed	the parts were resized by mistake
the answer is under one for many servings	multiplied the bottom instead of the top

In the glasses problem, $\tfrac{8}{5}$ is more than one liter. Four glasses of $\tfrac{2}{5}$ each should pass a liter, so the size is right.

Worked examples

Common mistakes

Practice problems