The one idea behind all four rules
A fraction is a count of equal pieces. The denominator says how big each piece is (fifths, eighths) and the numerator says how many you have. Keep that picture in mind and the rules stop feeling arbitrary.
Adding and subtracting are about counting pieces, so the pieces must be the same size first. Multiplying and dividing are about scaling, so piece size doesn't need to match. That's why adding needs a common denominator and multiplying doesn't.
Adding and subtracting: match the pieces first
If the denominators already match, add or subtract the numerators and keep the denominator. 3/8 + 2/8 = 5/8. Three eighths and two eighths are five eighths.
If they don't match, find the least common denominator. List multiples of the larger denominator until the smaller one divides in. For 5/6 + 3/8, multiples of 8 are 8, 16, 24, and 6 goes into 24. Rewrite: 5/6 = 20/24 and 3/8 = 9/24. Add to get 29/24, or 1 5/24.
You can always use the product of the denominators instead (6 x 8 = 48). The answer is the same after you simplify, but the numbers get bigger and simplifying takes longer.
Multiplying: tops times tops, bottoms times bottoms
Multiply the numerators together and the denominators together. 2/3 x 4/5 = 8/15. Half of a half is a quarter: 1/2 x 1/2 = 1/4. Multiplying by a fraction less than 1 makes a positive number smaller, which is a good sense check.
Cancel before you multiply when you can. In 3/5 x 10/9, the 3 and 9 share a factor of 3, and the 10 and 5 share a factor of 5. After canceling you have 1/1 x 2/3 = 2/3, and you never had to simplify 30/45.
Dividing: flip the second fraction
Dividing by a fraction is the same as multiplying by its reciprocal, the fraction turned upside down. Keep the first fraction, change divide to multiply, flip the second. 7/8 divided by 1/4 becomes 7/8 x 4/1 = 28/8 = 7/2.
Why does flipping work? Asking "how many quarters fit in 7/8?" is the same as asking how many times 1/4 goes into 7/8. Each whole holds 4 quarters, so you multiply by 4. Check the answer by going backwards: 7/2 x 1/4 = 7/8.
Working with mixed numbers
Turn every mixed number into an improper fraction before multiplying or dividing. 2 3/4 divided by 1 1/2 is 11/4 divided by 3/2, which is 11/4 x 2/3 = 22/12 = 11/6, or 1 5/6.
For adding and subtracting, you can convert too, or work with the whole numbers and fractions separately. The separate method gets messy when you have to borrow. For 3 1/3 - 1 3/4, converting gives 10/3 - 7/4 = 40/12 - 21/12 = 19/12 = 1 7/12, with no borrowing needed.
Mistakes that cost the most marks
Most wrong answers come from mixing up which rule goes with which operation. Before you start, say the rule out loud. If you'd like to check your work afterward, our fraction calculator shows the LCD, the flipped fraction and every step.
- Adding denominators. 1/2 + 1/3 is 5/6, not 2/5.
- Finding a common denominator to multiply. It's not wrong, just wasted work.
- Flipping the wrong fraction when dividing. Only the second one flips.
- Multiplying mixed numbers part by part. 1 1/2 x 2 2/3 is 4, while 1 x 2 plus 1/2 x 2/3 gives 2 1/3.
- Leaving the answer unsimplified. 28/8 should end up as 7/2 or 3 1/2.
Frequently asked questions
Do you need a common denominator to multiply fractions?
No. Multiply straight across: numerator times numerator and denominator times denominator. Common denominators are only needed for adding, subtracting and some comparisons.
What is the reciprocal of a mixed number?
Convert it to an improper fraction, then flip it. 2 1/2 is 5/2, so its reciprocal is 2/5. Multiplying the two always gives 1.
Why does dividing by a fraction make the number bigger?
Dividing asks how many of that piece fit in. Smaller pieces fit more times, so dividing by a fraction under 1 gives a larger result. 3 divided by 1/2 is 6.
Is it better to use the LCD or just multiply the denominators?
Both give the right answer once simplified. The LCD keeps numbers smaller, which means fewer arithmetic slips and less simplifying at the end.
How do you add mixed numbers without converting them?
Add the whole numbers, then add the fractions. If the fractions add up to more than 1, carry the extra whole. 2 1/2 + 1 3/4 is 3 plus 5/4, which is 4 1/4.
References
These sources support the method or guidance used for Fraction Calculator. Verify time-sensitive rules at the source.
Try the calculator
Open Fraction Calculator, enter your scenario, and compare its supporting rows with this guide's method and checks.