Operations on rational numbers

Take two rational numbers — say $-\dfrac{7}{12}$ and $\dfrac{5}{8}$ — and every calculation you do with them, whether adding, subtracting, multiplying or dividing, is really just careful bookkeeping of two things: a common denominator (for adding and subtracting) and the sign of the result (for all four operations). Rational numbers extend the fraction and integer arithmetic already familiar from earlier classes to numbers like $\dfrac{p}{q}$ where $p$ and $q$ can be negative, and the operations on them follow the same rules as on fractions, with sign rules borrowed from integers layered on top. This article works through addition, subtraction, multiplication and division of rational numbers, the standard-form convention every final answer must satisfy, and how the order of operations governs multi-step expressions that mix all four.

The core insight

Adding and subtracting rational numbers needs a common denominator, multiplying and dividing needs numerator-times-numerator over denominator-times-denominator (dividing first flips the second fraction), and every answer must be simplified to standard form with the correct sign.

The core idea

Standard form fixes how a rational number should look once you are done calculating: $\dfrac{p}{q}$ is in standard form when $q>0$ and the HCF of $|p|$ and $q$ is $1$ — no common factor left to cancel, and the negative sign (if any) sitting with the numerator, never the denominator.

Same denominator — add or subtract the numerators directly.

\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c}, \qquad \dfrac{a}{c}-\dfrac{b}{c}=\dfrac{a-b}{c}

where: $a$ and $b$ are the numerators being combined and $c\neq0$ is the shared denominator.

Different denominators — convert first, using the LCM of the denominators, then combine.

\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{a\times d + c\times b}{b\times d}

where: $b$ and $d$ are the two original denominators; multiplying straight across always works, but using the LCM of $b$ and $d$ instead of the plain product keeps the numbers smaller.

Multiplication — numerators together, denominators together.

\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{a\times c}{b\times d}

Division — multiply by the reciprocal of the divisor.

\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}=\dfrac{a\times d}{b\times c} \quad (c\neq0)

where: $\dfrac{d}{c}$ is the reciprocal of the divisor $\dfrac{c}{d}$ — flipping numerator and denominator, never dividing straight across.

Sign rules follow integer arithmetic exactly: two negatives multiplied or divided give a positive, one negative and one positive give a negative. By convention the minus sign of a negative rational number is written with the numerator, $-\dfrac{3}{7}$ , and never as $\dfrac{3}{-7}$ or with both signs negative as $\dfrac{-3}{-7}$ .

Order of operations applies unchanged: brackets first, then multiplication and division left to right, and finally addition and subtraction left to right. When an expression mixes all four operations it is easy to slip into adding before multiplying, so it helps to mark the multiplication/division parts of the expression first, finish those completely, and only then work through the remaining addition/subtraction. Addition and multiplication of rational numbers are commutative and associative, so terms can be reordered to simplify mental work — grouping additive-inverse pairs like $-\dfrac14$ and $\dfrac14$ to cancel to $0$ before finishing the rest of the sum. Subtraction and division are neither commutative nor associative, so their order must be preserved unless first rewritten as addition of the inverse or multiplication by the reciprocal. It is worth noticing that these four rules are not four unrelated facts to memorise but two ideas applied twice. Addition and subtraction both hinge on making the pieces being counted the same size (the common denominator); multiplication and division both hinge on the same numerator-times-numerator, denominator-times-denominator mechanics, with division simply preceded by one extra step — replacing the divisor by its reciprocal. Seeing the pattern this way cuts the amount to remember roughly in half.

Figure

A four panel diagram showing addition needs a common denominator, subtraction needs a common denominator, multiplication multiplies straight across, and division flips the second fraction before multiplying

Derivation ACombining three unlike fractions and reducing to standard form

The goal is to show, in full generality, why converting to a common denominator before combining numerators is not an arbitrary rule but the only way to make addition and subtraction of unlike fractions correspond to combining the same-sized pieces, and why the resulting fraction must then be reduced using the HCF of numerator and denominator to reach standard form.

Step 1. Recognise that $\dfrac{a}{b}$ and $\dfrac{c}{d}$ describe pieces of different sizes — one piece is $\dfrac{1}{b}$ of a whole, the other is $\dfrac{1}{d}$ of a whole — so their numerators cannot be added or subtracted directly; $2$ apples and $3$ pears do not combine into $5$ of either.

Step 2. Rewrite both fractions using a common denominator so the pieces being counted are identical in size. Multiplying $\dfrac{a}{b}$ by $\dfrac{d}{d}$ and $\dfrac{c}{d}$ by $\dfrac{b}{b}$ gives $\dfrac{a}{b}=\dfrac{ad}{bd}$ and $\dfrac{c}{d}=\dfrac{bc}{bd}$ — both are now counted in units of $\dfrac{1}{bd}$ , so their numerators, $ad$ and $bc$ , can validly be added or subtracted.

Step 3. Combine the numerators over the shared denominator: $\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}$ . Using the LCM of $b$ and $d$ in place of the plain product $bd$ gives the same value with smaller numbers, since the LCM is itself a common multiple of $b$ and $d$ .

Step 4. Reduce the result to standard form by dividing numerator and denominator by their HCF, and adjust the sign so the denominator is positive — this final cleanup is not optional, since two unreduced fractions that are numerically equal (like $\dfrac{-3}{24}$ and $\dfrac{-1}{8}$ ) are not considered the same standard-form answer.

Worked example

Simplify: $\dfrac{-7}{12} + \dfrac{5}{8} - \dfrac{1}{6}$

Step 1. Find the LCM of the denominators $12$ , $8$ and $6$ . Since $12 = 2^2\times3$ , $8=2^3$ and $6=2\times3$ , the LCM is $2^3\times3=24$ .

Step 2. Convert each fraction to denominator $24$ .

\dfrac{-7}{12}=\dfrac{-14}{24}, \qquad \dfrac{5}{8}=\dfrac{15}{24}, \qquad \dfrac{1}{6}=\dfrac{4}{24}

Step 3. Combine the numerators over the common denominator.

\dfrac{-14}{24}+\dfrac{15}{24}-\dfrac{4}{24} = \dfrac{-14+15-4}{24} = \dfrac{-3}{24}

Step 4. Reduce to standard form using the HCF of $3$ and $24$ , which is $3$ .

\dfrac{-3}{24} = \dfrac{-1}{8}

The final answer is $-\dfrac{1}{8}$ .

Competitive stretch

Extend the same expression with a multiplication and division: simplify $\left(\dfrac{-7}{12} + \dfrac{5}{8} - \dfrac{1}{6}\right) \times \dfrac{5}{6} \div \left(-\dfrac{5}{4}\right)$ .

Step 5. The bracket was already evaluated above as $-\dfrac{1}{8}$ , so the expression becomes $-\dfrac{1}{8} \times \dfrac{5}{6} \div \left(-\dfrac{5}{4}\right)$ . Multiply first, left to right: $-\dfrac{1}{8}\times\dfrac{5}{6} = \dfrac{-5}{48}$ .

Step 6. Divide by $-\dfrac{5}{4}$ by multiplying by its reciprocal $-\dfrac{4}{5}$ : $\dfrac{-5}{48} \div \left(-\dfrac{5}{4}\right) = \dfrac{-5}{48}\times\left(-\dfrac{4}{5}\right) = \dfrac{20}{240} = \dfrac{1}{12}$ . Two negatives give a positive, so the final answer is $\dfrac{1}{12}$ . Notice that the entire bracket from the first worked example was reused as a single value here rather than re-expanded — once a sub-expression has been fully simplified, treat its result as one number and carry it forward, instead of re-deriving it inside a longer calculation.

Common traps

Adding or subtracting numerators before finding a common denominator — this only ever works when the denominators already match.
Forgetting to flip the second fraction when dividing — dividing numerators and denominators straight across is not the same operation.
Sign slips when multiplying or dividing two negative rational numbers, where the correct result is positive.
Leaving the final answer unreduced, e.g. stopping at $\dfrac{-3}{24}$ instead of $-\dfrac{1}{8}$ , or writing the negative sign in the denominator instead of with the numerator.

Ask the tutor: Want to practise a longer chain of mixed operations, or get help spotting exactly where a sign error crept in?

Why can I not just add the numerators of two fractions with different denominators?

Because the two fractions are counting pieces of different sizes. In $\dfrac{1}{4}+\dfrac{1}{3}$ , one piece is a quarter and the other a third, and adding $1+1$ to get $\dfrac{2}{7}$ would be wrong — that is not even close to the true sum of about $0.58$ . Converting both fractions to a common denominator, such as $12$ , first rewrites them as $\dfrac{3}{12}$ and $\dfrac{4}{12}$ , pieces of the same size, whose numerators can then be safely combined to give $\dfrac{7}{12}$ .

Do I always need the LCM, or can I just multiply the denominators together?

Multiplying the two denominators together always gives a valid common denominator, so it always works. The LCM is preferred because it is usually smaller, which keeps the numbers in later steps manageable and reduces the simplifying you need to do at the end. For example, adding fractions with denominators $12$ and $8$ using their product $96$ instead of the LCM $24$ still gives the correct final answer, just with larger intermediate numbers to simplify away.

Why does the minus sign go with the numerator and not the denominator?

It is a convention that keeps notation consistent and unambiguous. $\dfrac{-3}{7}$ , $\dfrac{3}{-7}$ and $-\dfrac{3}{7}$ are all numerically equal, but writing the sign with the denominator or as a separate fraction bar makes it easy to lose track of during a multi-step calculation, especially when multiplying or dividing several rational numbers together. Standard form requires the denominator to be positive and any negative sign to sit with the numerator, so results are always written and compared the same way.

Operations on rational numbers

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads