How to Add, Reduce, and Convert Fractions (With Worked Examples)

Fractions trip up more people than any other piece of grade-school math, and the reason is almost never the multiplication. It's the rule-switching. Add fractions and you need a common denominator. Multiply them and you don't. Divide them and you flip the second one. The four operations each follow a different recipe, and forgetting which recipe applies is how you end up writing that 1/2 + 1/3 = 2/5.

This guide walks through all four operations, plus reducing to lowest terms, mixed numbers, and decimal conversion. Every step is something you can verify yourself with the Fraction Calculator, which shows the working instead of just spitting out a number.

Adding and subtracting: the common denominator is everything

You cannot add numerators and denominators straight across. The denominator is the size of the piece, and you can only add pieces that are the same size. So before you add anything, you rewrite both fractions over a shared denominator.

Here is the full worked example for 2/3 + 1/4:

Find a common denominator. The denominators are 3 and 4. The least common multiple of 3 and 4 is 12, so 12 becomes the shared denominator.
Rewrite each fraction. 2/3 = 8/12 (multiply top and bottom by 4). 1/4 = 3/12 (multiply top and bottom by 3).
Add the numerators only. 8/12 + 3/12 = 11/12. The denominator stays 12.
Reduce. The greatest common divisor of 11 and 12 is 1, so 11/12 is already in lowest terms.

Final answer: 2/3 + 1/4 = 11/12.

Subtraction works identically, you just subtract the numerators in step 3. The mistake to watch for is skipping step 1 and adding straight across. If you ever get a denominator smaller than the ones you started with, you did exactly that, and the answer is wrong.

Multiplying and dividing: no common denominator needed

Multiplication is the easy one, which surprises people who just fought through addition. Multiply the numerators, multiply the denominators, reduce. That's it.

3/4 × 2/5 = (3 × 2) / (4 × 5) = 6/20 = 3/10 after reducing.

Division is multiplication with one extra move: flip the second fraction (take its reciprocal) and then multiply.

1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2.

The single most common division error is flipping the wrong fraction. If you flip the first one instead of the second, you get the reciprocal of the correct answer, which often looks plausible. Flip the second one, every time. A useful mental check: dividing by a fraction smaller than 1 should make the answer bigger, which is exactly why 1/2 ÷ 1/4 lands on 2 and not on 1/8.

Reducing to lowest terms

A fraction is in lowest terms when the numerator and denominator share no common factor other than 1. To reduce, find the greatest common divisor (GCD) of the two numbers and divide both by it.

6/8: the GCD of 6 and 8 is 2, so 6/8 = 3/4. 4/2: the GCD is 2, so 4/2 = 2/1, which is just 2.

Teachers care about this step because "5/6" and "10/12" are the same value but only one is the expected answer on a worksheet. According to the U.S. Common Core State Standards for Mathematics, students are formally introduced to equivalent fractions and reduction in grades 3 through 5, which is why this skill shows up on so much homework. (Source: Common Core State Standards Initiative, Mathematics Standards, Number and Operations — Fractions.)

Mixed numbers and improper fractions

A mixed number like 1 1/2 means "one whole plus one half." An improper fraction like 3/2 means the same value written as a single ratio. You convert between them constantly.

Mixed to improper: multiply the whole number by the denominator, add the numerator, keep the denominator. 1 1/2 becomes (1 × 2 + 1) / 2 = 3/2.

Improper to mixed: divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator. 5/2 = 2 remainder 1 = 2 1/2.

The typing trap here is the space. 1 1/2 is one and a half. 11/2 with no space is eleven halves, which equals 5.5, a completely different number. If a result comes out roughly ten times too large, a missing space is almost always why.

Converting fractions to exact decimals

Every fraction is a division problem in disguise, so a/b is just a divided by b. Some divide cleanly and some repeat forever.

1/4 = 0.25, a terminating decimal. 1/3 = 0.3333... which repeats, written 0.3 with an overline on the 3. 1/7 = 0.142857142857... where the whole block 142857 repeats.

This matters when a spreadsheet shows you 0.3333333 and you want to know whether that's truly 1/3 or some other ratio that got rounded. A proper long-division conversion catches the repeating remainder and marks it, so you can tell an exact recurring decimal apart from a number that was simply truncated for display.

How I actually use this

When I'm helping with a kid's homework, I never trust myself to add fractions in my head anymore, because the fast wrong answer is too tempting. I type the problem into the Fraction Calculator, read the step where it builds the common denominator, and use that line to explain the why rather than just handing over a number. The first time a kid sees 1/2 + 1/3 turn into 3/6 + 2/6 before it becomes 5/6, the whole "why isn't it 2/5" confusion evaporates. That visible middle step is worth more than the answer.

For anything past basic fractions, the same site has neighbours that pick up where this leaves off. The Scientific Calculator handles the exponents and roots that show up once fractions get raised to powers, and the Percentage Calculator is the natural next stop, since every percentage is just a fraction over 100.

Quick reference

Add / subtract: common denominator first, then combine numerators, then reduce.
Multiply: tops times tops, bottoms times bottoms, reduce.
Divide: flip the second fraction, then multiply.
Reduce: divide both numbers by their GCD.
Mixed to improper: whole × denominator + numerator, over the same denominator.
To decimal: divide; watch for a repeating remainder.

Master those six lines and fractions stop being a guessing game. The arithmetic was never the hard part — knowing which recipe to reach for is.

Made by Toolora · Updated 2026-06-13