Adding and Subtracting Fractions with Positive and Negative Integers

Hello, and welcome. This short module builds a reliable method for expressions that mix fractions, positive integers, and negative integers. By the end of this lesson, you will be able to evaluate expressions with several additions and subtractions without losing track of signs.

The main idea is simple but powerful: turn every subtraction into addition of the opposite. Once an expression is made entirely of additions, you can group convenient terms, find common denominators for the fractional terms, and simplify.

---

1. Signed numbers: movement and opposites

A positive number represents movement to the right on a number line; a negative number represents movement to the left. Adding a positive moves right, while adding a negative moves left.

Subtracting is best understood as adding the opposite:

$$a-b=a+(-b)$$

This rule works for integers and fractions alike.

For example:

$$5-3=5+(-3)$$ $$5-(-3)=5+3$$

The second example is especially important: subtracting a negative increases the value because you are adding its opposite.

The same applies to fractions:

$$-\frac{3}{4}-\frac{1}{6}=-\frac{3}{4}+\left(-\frac{1}{6}\right)$$ $$-\frac{3}{4}-\left(-\frac{1}{6}\right)=-\frac{3}{4}+\frac{1}{6}$$

A negative fraction can be written in several equivalent ways:

$$-\frac{3}{5}=\frac{-3}{5}=\frac{3}{-5}$$

For clear work, it is usually easiest to place the negative sign in front of the fraction or in the numerator.

---

2. The method for multi-term expressions

Use this four-step method every time.

1. Rewrite subtraction as addition of the opposite.
   Keep parentheses around negative quantities until the signs are clear.

2. Group convenient terms.
   You may group integer terms together and fraction terms together because addition can be reordered and regrouped.

3. Add or subtract the fractions.
   Find a common denominator, rewrite each fraction as an equivalent fraction, then combine the signed numerators.

4. Combine the integer and fractional results.
   If needed, write the integer using the common denominator. Simplify the final answer.

A useful caution: you may not casually rearrange subtraction. For instance,

$$8-3-2$$

does not mean that you can switch the $3$ and $2$ without thinking. First rewrite it:

$$8+(-3)+(-2)$$

Now every operation is addition, so grouping is safe.

The Commutative and Associative Properties of Addition justify this:

$$a+b=b+a$$ $$(a+b)+c=a+(b+c)$$

They apply to the addition form of the expression.

---

3. Worked example: integers, fractions, and a double negative

Evaluate:

$$-3+\frac{5}{6}-\left(-\frac{1}{4}\right)+1$$

Step 1: Rewrite subtraction

Subtracting a negative fraction means adding the positive fraction:

$$-3+\frac{5}{6}+\frac{1}{4}+1$$

Step 2: Group integers and fractions

$$(-3+1)+\left(\frac{5}{6}+\frac{1}{4}\right)$$

The integer part is:

$$-3+1=-2$$

For the fractions, the least common denominator of $6$ and $4$ is $12$:

$$\frac{5}{6}=\frac{10}{12}$$ $$\frac{1}{4}=\frac{3}{12}$$

So:

$$\frac{5}{6}+\frac{1}{4}=\frac{10}{12}+\frac{3}{12}=\frac{13}{12}$$

Now combine:

$$-2+\frac{13}{12}$$

Step 3: Write the integer with denominator $12$

$$-2=-\frac{24}{12}$$

Therefore:

$$-\frac{24}{12}+\frac{13}{12}=-\frac{11}{12}$$

So the final answer is:

$$\boxed{-\frac{11}{12}}$$

A quick reasonableness check helps: $-2+\frac{13}{12}$ is a little greater than $-1$, so $-\frac{11}{12}$ makes sense.

---

4. Adding fractions with different signs

When fractions have a common denominator, keep that denominator and combine the numerators, including their signs.

For example:

$$-\frac{7}{12}+\frac{5}{12}=\frac{-7+5}{12}=-\frac{2}{12}=-\frac{1}{6}$$

If the denominators differ, find a least common denominator first.

Here is the most common sign pattern to remember:

Original operation	Rewrite as addition
$a-b$	$a+(-b)$
$a-(-b)$	$a+b$
$-a-b$	$(-a)+(-b)$
$-a-(-b)$	$(-a)+b$

Do not treat two minus signs as automatically becoming a plus. They become a plus only when the second minus belongs to the number being subtracted, such as in:

$$7-\left(-\frac{2}{3}\right)$$

---

5. Worked example: several subtractions

Evaluate:

$$2-\frac{7}{8}-\frac{5}{12}-(-1)$$

Step 1: Convert all subtraction

$$2+\left(-\frac{7}{8}\right)+\left(-\frac{5}{12}\right)+1$$

Step 2: Group the integer terms and fractions

$$(2+1)+\left(-\frac{7}{8}-\frac{5}{12}\right)$$

The integers give:

$$2+1=3$$

The least common denominator of $8$ and $12$ is $24$:

$$-\frac{7}{8}=-\frac{21}{24}$$ $$-\frac{5}{12}=-\frac{10}{24}$$

Add the signed numerators:

$$-\frac{21}{24}-\frac{10}{24}=-\frac{31}{24}$$

Now combine:

$$3-\frac{31}{24}$$

Write $3$ in twenty-fourths:

$$3=\frac{72}{24}$$

Then:

$$\frac{72}{24}-\frac{31}{24}=\frac{41}{24}$$

As a mixed number:

$$\boxed{1\frac{17}{24}}$$

Notice that the final answer is positive. Although the expression contains several subtractions, the term $-(-1)$ contributes $+1$.

---

6. A clear written layout prevents sign errors

For expressions with many terms, write one transformation per line. Avoid doing several sign changes mentally at once.

For example, begin with:

$$-2+\frac{5}{6}-\frac{1}{4}-\left(-\frac{1}{3}\right)$$

First rewrite:

$$-2+\frac{5}{6}-\frac{1}{4}+\frac{1}{3}$$

Then find a common denominator for the fractions. The least common denominator of $6$, $4$, and $3$ is $12$:

$$-2+\frac{10}{12}-\frac{3}{12}+\frac{4}{12}$$

Combine the fractional terms:

$$-2+\frac{11}{12}$$

Rewrite the integer in twelfths:

$$-\frac{24}{12}+\frac{11}{12}=-\frac{13}{12}$$

So:

$$\boxed{-\frac{13}{12}=-1\frac{1}{12}}$$

The intermediate line is not “extra work”; it is how you make the signs visible and checkable.

---

7. Final checklist

Before accepting an answer, check these five points:

• Did I rewrite each subtraction as addition of the opposite?
• Did I keep negative numbers in parentheses while rewriting?
• Did I find a common denominator before combining unlike fractions?
• Did I add the numerators with their signs?
• Is the final fraction simplified?

The central habit is:

$$\text{rewrite signs first, then calculate}$$

This lesson established the full process for multi-term expressions involving integers and fractions: translate subtraction, group addition terms, use common denominators, and simplify. In the next work on this skill, focus on applying the same method efficiently and accurately across a wider range of practice expressions.