Add & Subtract Rational Expressions Calculator

Enter Calculation

Numerator a

Denominator b

Operation Add (+) Subtract (−)

Numerator c

Denominator d

Results

Simplified Result

5 / 6

≈ 0.833333

Combined fraction (before simplifying)	5 / 6
Simplified	5 / 6

What this calculator does

This tool adds or subtracts two rational expressions (fractions) of the form \(a/b \pm c/d\). It builds a common denominator, combines the numerators, simplifies the resulting fraction by its greatest common factor (GCF), and also reports the value as a decimal. It works for any whole-number numerators and denominators, positive or negative.

How to use it

Enter the numerator a and denominator b of the first fraction, choose Add or Subtract, then enter the numerator c and denominator d of the second fraction. Click calculate. You will see the combined fraction before simplifying, the fully simplified fraction, and the decimal equivalent.

The formula explained

To combine two fractions you need a common denominator. The simplest common denominator is the product \(b \cdot d\). Rewriting each fraction over \(b \cdot d\) gives \(a \cdot d\) over \(b \cdot d\) and \(c \cdot b\) over \(b \cdot d\), so $$\frac{a}{b} \pm \frac{c}{d} = \frac{a \cdot d \pm c \cdot b}{b \cdot d}$$ To simplify, find the GCF of the new numerator and denominator and divide both by it. The denominator is kept positive by convention.

Cross-multiplying gives a common denominator \(b \cdot d\) and combines the numerators.

Worked example

Add \(1/4\) and \(1/6\). Here \(a=1\), \(b=4\), \(c=1\), \(d=6\). The combined numerator is $$a \cdot d + c \cdot b = 1 \cdot 6 + 1 \cdot 4 = 10$$ and the combined denominator is $$b \cdot d = 4 \cdot 6 = 24$$ giving \(10/24\). The GCF of 10 and 24 is 2, so dividing both by 2 yields \(5/12 \approx 0.4167\).

After combining, divide the numerator and denominator by their GCD to simplify.

How to Add or Subtract Fractions by Hand

To combine \(\dfrac{a}{b} \pm \dfrac{c}{d}\), follow these steps in order:

1. Find a common denominator. The simplest choice is the product \(b \cdot d\). For smaller numbers, use the least common denominator (LCD) — the least common multiple of \(b\) and \(d\).
2. Rewrite each numerator. Scale each fraction to the common denominator: the first becomes \(a \cdot d\) and the second becomes \(c \cdot b\), both over \(b \cdot d\).
3. Add or subtract the numerators. Keep the common denominator: \(\dfrac{a \cdot d \pm c \cdot b}{b \cdot d}\). The denominator does not change during this step.
4. Find the GCF. Compute the greatest common factor of the resulting numerator and denominator.
5. Divide both by the GCF. This reduces the fraction to lowest terms. If the GCF is 1, the fraction is already simplified.
6. Keep the denominator positive. If the denominator came out negative, multiply both numerator and denominator by \(-1\) so the sign lives on the numerator (e.g. write \(\tfrac{-1}{15}\), not \(\tfrac{1}{-15}\)).
7. Convert to a decimal (optional). Divide the simplified numerator by the denominator. Repeating decimals (like \(0.8\overline{3}\)) are exact only in fraction form.

Key Terms

Numerator

The top number of a fraction, such as the \(a\) in \(\tfrac{a}{b}\); it counts how many equal parts are taken.

Denominator

The bottom number of a fraction, such as the \(b\) in \(\tfrac{a}{b}\); it states how many equal parts make one whole. It can never be zero.

Rational expression / fraction

A quotient of two quantities, \(\tfrac{a}{b}\), where the denominator is not zero. A numeric fraction is the simplest kind of rational expression.

Common denominator

A shared denominator for two or more fractions, allowing their numerators to be added or subtracted directly. The product \(b \cdot d\) always works.

Least common denominator (LCD)

The smallest common denominator — the least common multiple of the original denominators. Using the LCD keeps the numbers as small as possible.

Greatest common factor (GCF / GCD)

The largest integer that divides both the numerator and denominator exactly. Dividing both by the GCF simplifies the fraction in one step.

Simplified / lowest terms

A fraction in which the numerator and denominator share no common factor other than 1, so it cannot be reduced further.

Decimal equivalent

The value of the fraction written in base-ten form, found by dividing the numerator by the denominator (e.g. \(\tfrac{7}{12} = 0.58\overline{3}\)).

FAQ

Can I use negative numbers? Yes. Negative numerators or denominators are allowed; the result keeps a positive denominator.

What if the answer is a whole number? If the simplified denominator is 1, the fraction equals that whole number — for example \(2/1\) means 2.

Why is my combined fraction not the smallest form? Using \(b \cdot d\) as the common denominator can give larger numbers, which is exactly why the calculator then divides by the GCF to show the simplest form.