Arithmetic Operations with Fractions and Mixed Numbers

Hello. In the previous lesson, you evaluated integer expressions by preserving signs, rewriting subtraction as addition of the opposite, and following the order of operations. Fractions extend those habits: notation must remain precise, each operation has its own rule, and simplification is part of communicating a final answer clearly.

This lesson develops a complete working system for adding, subtracting, multiplying, and dividing fractions and mixed numbers. These operations will soon support percentages, ratio, algebraic fractions, probability, and later the numerical calculations used in programming and data work.

---

Fractions represent quantities in equal-sized parts

A fraction

$$\frac{a}{b}$$

has a numerator $a$, which counts parts, and a denominator $b$, which states how many equal parts make one whole. The denominator cannot be zero.

The denominator is not merely a number at the bottom. It specifies the unit size. For example, $\frac{3}{4}$ means three pieces where each piece is one quarter; $\frac{2}{3}$ means two pieces where each is one third. Because quarters and thirds are different-sized pieces, they cannot be directly added or subtracted until both are expressed in the same unit.

Two fractions can look different while representing the same quantity:

$$\frac{1}{2}=\frac{2}{4}=\frac{3}{6}.$$

This works because multiplying numerator and denominator by the same nonzero number is equivalent to multiplying by $1$:

$$\frac{a}{b}=\frac{a}{b}\times \frac{k}{k}=\frac{ak}{bk}.$$

This is the equivalent-fractions principle. It is the key to addition and subtraction with unlike denominators.

A mixed number combines a whole number and a proper fraction:

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

The plus sign is normally omitted, but it is still there conceptually. This is why mixed-number addition can often be understood as “combine wholes with wholes and fractional parts with fractional parts.”

To convert a positive mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the denominator:

$$2\frac{3}{5}=\frac{2(5)+3}{5}=\frac{13}{5}.$$

To convert an improper fraction back, divide the numerator by the denominator:

$$\frac{13}{5}=2\frac{3}{5}.$$

For signed mixed quantities, avoid ambiguous notation. Write the sign outside parentheses:

$$-\left(2\frac{3}{5}\right)=-\frac{13}{5}.$$

The sign rules from the preceding lesson apply to fractions just as they do to integers.

---

Addition and subtraction: first make the units match

When fractions have the same denominator, add or subtract only the numerators:

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

and

$$\frac{7}{10}-\frac{3}{10}=\frac{4}{10}=\frac{2}{5}.$$

The denominator remains $8$ in the first example because the answer is still measured in eighths. It would be incorrect to add denominators:

$$\frac{3}{8}+\frac{2}{8}\ne \frac{5}{16}.$$

For unlike denominators, find a least common denominator (LCD), which is the least common multiple of the denominators. Then rewrite both fractions as equivalent fractions with that denominator.

For example:

$$\frac{5}{6}-\frac{1}{4}.$$

The LCD of $6$ and $4$ is $12$. Rewrite each fraction in twelfths:

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

Now subtraction is meaningful because both quantities use the same-sized part:

$$\frac{10}{12}-\frac{3}{12}=\frac{7}{12}.$$

The full procedure is:

1. Find the LCD of the denominators.
2. Rewrite each fraction as an equivalent fraction with that LCD.
3. Add or subtract the numerators.
4. Simplify the answer.

Adding mixed numbers

Consider:

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

Add the whole-number parts and fractional parts, but first make the fractional units match. The LCD of $4$ and $6$ is $12$:

$$2\frac{3}{4}+1\frac{5}{6}=2+1+\frac{9}{12}+\frac{10}{12}.$$

Then:

$$=3+\frac{19}{12}.$$

Since $\frac{19}{12}=1\frac{7}{12}$,

$$2\frac{3}{4}+1\frac{5}{6}=4\frac{7}{12}.$$

A fractional answer greater than or equal to $1$ must be regrouped into an additional whole.

Subtracting mixed numbers: regrouping, not a mysterious “borrow”

Now consider:

$$5\frac{1}{6}-2\frac{5}{6}.$$

You cannot remove $\frac{5}{6}$ from $\frac{1}{6}$. But one whole is exactly $\frac{6}{6}$, so regroup one whole from the $5$:

$$5\frac{1}{6}=4+\frac{6}{6}+\frac{1}{6}=4\frac{7}{6}.$$

Now subtract:

$$4\frac{7}{6}-2\frac{5}{6}=2\frac{2}{6}=2\frac{1}{3}.$$

The operation has not changed the value of the first mixed number. It has only rewritten it in a form that contains enough sixths to perform the subtraction.

For difficult subtraction, especially when the final result may be negative, converting both mixed numbers to improper fractions is often safer. For instance,

$$1\frac{1}{4}-2\frac{2}{3}=\frac{5}{4}-\frac{8}{3}.$$

Using LCD $12$,

$$\frac{5}{4}-\frac{8}{3}=\frac{15}{12}-\frac{32}{12}=-\frac{17}{12}=-1\frac{5}{12}.$$

---

Multiplication: take a fraction of a quantity

Multiplication does not require a common denominator. To multiply fractions, multiply numerators together and denominators together:

$$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}.$$

For example:

$$\frac{3}{4}\times \frac{10}{21}.$$

Before multiplying, simplify common factors across a numerator and denominator:

$$\frac{3}{4}\times \frac{10}{21}=\frac{1}{2}\times \frac{5}{7}=\frac{5}{14}.$$

This is valid because

$$\frac{3}{21}=\frac{1}{7}$$

and

$$\frac{10}{4}=\frac{5}{2}.$$

Simplifying before multiplication keeps numbers small and makes errors less likely.

A crucial restriction: cross-cancellation works only with factors in a multiplication expression. It cannot be used across addition or subtraction. For example, in

$$\frac{2+3}{5},$$

the $2+3$ is a sum, not a factor of $5$. Nothing can be cancelled.

Multiplication can also be understood as scaling. Since $\frac{3}{4}$ is less than $1$,

$$\frac{3}{4}\times 20=15$$

finds three quarters of $20$, so the answer is smaller than $20$. By contrast, multiplying by a fraction greater than $1$, such as $\frac{5}{4}$, enlarges a positive quantity.

For mixed numbers, first convert to improper fractions:

$$2\frac{1}{3}\times 1\frac{1}{2}=\frac{7}{3}\times \frac{3}{2}.$$

Cancel the common factor $3$:

$$\frac{7}{3}\times \frac{3}{2}=\frac{7}{2}=3\frac{1}{2}.$$

---

Division: multiply by the reciprocal

To divide by a fraction is to ask how many groups of that fractional size fit into a quantity. For example:

$$\frac{3}{4}÷\frac{1}{8}=6$$

because six eighths make six eighths, and

$$\frac{6}{8}=\frac{3}{4}.$$

The general rule is:

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}.$$

The fraction $\frac{d}{c}$ is called the reciprocal of $\frac{c}{d}$. Reciprocals multiply to $1$:

$$\frac{c}{d}\times \frac{d}{c}=1.$$

So division by $\frac{c}{d}$ is undone by multiplication by $\frac{d}{c}$.

For example:

$$2\frac{1}{4}÷\frac{3}{5}.$$

First convert the mixed number:

$$\frac{9}{4}÷\frac{3}{5}.$$

Then multiply by the reciprocal of the divisor:

$$\frac{9}{4}\times \frac{5}{3}.$$

Simplify before multiplying:

$$\frac{9}{4}\times \frac{5}{3}=\frac{3}{4}\times 5=\frac{15}{4}=3\frac{3}{4}.$$

The familiar mnemonic “keep, change, flip” can remind you of the procedure, but state it precisely:

1. Keep the first fraction unchanged.
2. Change division to multiplication.
3. Replace the second fraction, the divisor, with its reciprocal.
4. Multiply and simplify.

The divisor must not be zero. Division by zero is undefined:

$$\frac{3}{5}÷0$$

has no value. However, zero divided by a nonzero fraction is zero:

$$0÷\frac{3}{5}=0.$$

---

Choose the operation before choosing a method

A reliable first question is: what operation is written between the quantities? The operation determines the method.

Operation	Core action	Common denominator needed?
Addition	Rewrite in equal-sized parts, then add numerators	Yes, unless denominators already match
Subtraction	Rewrite in equal-sized parts, then subtract numerators	Yes, unless denominators already match
Multiplication	Multiply numerators and denominators; simplify factors	No
Division	Multiply by the reciprocal of the divisor	No

When several operations appear in one expression, use the order of operations from the previous lesson. Multiplication and division occur before addition and subtraction.

For example:

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

Multiply first:

$$\frac{3}{4}\times \frac{2}{3}=\frac{1}{2}.$$

Then add:

$$\frac{1}{2}+\frac{1}{2}=1.$$

Do not decide to find a common denominator merely because fractions appear. A common denominator is needed for addition and subtraction, not for multiplication or division.

For your notebook, use a four-column error log: expression, my incorrect step, error type, and corrected principle. Useful error labels include:

• added or subtracted denominators;
• used a common denominator unnecessarily;
• changed only one part of an equivalent fraction;
• failed to convert a mixed number before multiplication or division;
• inverted the first fraction rather than the divisor;
• cancelled terms instead of factors;
• left an improper result unsimplified.

A short correction written in your own words is more useful than simply copying a correct answer.

---

Key takeaways

Fractions demand attention to the meaning of the denominator: it identifies the unit being counted.

• For addition and subtraction, first use the LCD to create equal-sized fractional units.
• A mixed number is a sum, such as

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

• In mixed-number subtraction, regrouping one whole means replacing it with an equivalent fraction such as $\frac{6}{6}$.
• For multiplication, multiply numerators and denominators, simplifying factors where possible.
• For division, multiply by the reciprocal of the divisor.
• Convert mixed numbers to improper fractions before multiplying or dividing.
• Simplify final answers, and convert positive improper fractions to mixed numbers when that form is requested.
• Preserve signs and use the order of operations exactly as you did with integers.

Next, you will connect fractions to their decimal and percentage forms. That translation is essential for financial calculations, data interpretation, probability, and programming contexts where numbers are often displayed as decimals rather than fractions.