Skill: Analyze and correct errors

To analyze and correct an error:

1. Read each step carefully.
2. Find the first step that is not valid.
3. Explain why that step fails.
4. Redo the work from that point, keeping the quantities equivalent.

Example

A student solves:

$$\frac{2}{3}+\frac{1}{4} =\frac{2+1}{3+4} =\frac{3}{7}$$

Step 1: Find the first error.

The first line is the problem, and the next line changes the fractions by adding both the numerators and denominators:

$$\frac{2+1}{3+4}$$

This is not a valid rule. The denominator tells the size of the parts. Thirds and fourths are different-sized parts, so they cannot be combined until they are changed into the same-sized parts.

Step 2: Use a common denominator.

The least common denominator of $3$ and $4$ is $12$.

Change each fraction into an equivalent fraction with denominator $12$:

$$\frac{2}{3}=\frac{8}{12} \qquad\text{and}\qquad \frac{1}{4}=\frac{3}{12}$$

Now the parts are the same size, so add the numerators:

$$\frac{8}{12}+\frac{3}{12}=\frac{11}{12}$$

Therefore,

$$\boxed{\frac{2}{3}+\frac{1}{4}=\frac{11}{12}}$$

As a quick check, $\frac{2}{3}$ is about $0.67$ and $\frac{1}{4}$ is $0.25$, so the sum should be close to $0.92$. The answer $\frac{11}{12}$ is about $0.92$, while $\frac{3}{7}$ is only about $0.43$. This confirms that $\frac{3}{7}$ is not reasonable.