Solving a fraction means performing an operation on it: simplifying, adding, subtracting, multiplying, or dividing. Each operation follows a short set of rules, and once you learn those rules, every fraction problem uses the same steps. Let’s break down each one.
Parts of a Fraction
A fraction has two numbers separated by a line. The top number is the numerator, and the bottom number is the denominator. In 3/4, the denominator (4) tells you how many equal pieces something is divided into, and the numerator (3) tells you how many of those pieces you’re working with. Think of it as 3 out of 4 slices of a pie.
How to Simplify a Fraction
Simplifying means rewriting a fraction in its smallest form without changing its value. To do this, find the greatest common factor (GCF) of the numerator and denominator, then divide both by that number.
Take 8/12. The factors of 8 are 1, 2, 4, and 8. The factors of 12 are 1, 2, 3, 4, 6, and 12. The largest number they share is 4. Divide both parts by 4: 8 ÷ 4 = 2, and 12 ÷ 4 = 3. So 8/12 simplifies to 2/3.
If the only factor the numerator and denominator share is 1, the fraction is already in its simplest form.
Adding and Subtracting Fractions
You can only add or subtract fractions when their denominators match. If they already match, just add or subtract the numerators and keep the denominator the same. For example, 2/7 + 3/7 = 5/7.
When the denominators are different, you need a common denominator before you can combine them. The most efficient choice is the least common denominator (LCD), which is the smallest number that both denominators divide into evenly.
Say you want to add 1/3 + 1/6. List the multiples of each denominator:
- Multiples of 3: 3, 6, 9, 12 …
- Multiples of 6: 6, 12, 18 …
The smallest number that appears in both lists is 6, so 6 is your LCD. The fraction 1/6 already has 6 on the bottom. To convert 1/3, multiply the top and bottom by 2: 1/3 becomes 2/6. Now you can add: 2/6 + 1/6 = 3/6, which simplifies to 1/2.
The key rule: whatever you multiply the denominator by, you must also multiply the numerator by. This keeps the fraction’s value the same.
Subtraction works identically. Find the LCD, convert both fractions, then subtract the numerators.
Multiplying Fractions
Multiplication is the most straightforward fraction operation. Multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.
For 2/5 × 2/3: multiply 2 × 2 = 4 for the numerator, and 5 × 3 = 15 for the denominator. The answer is 4/15. No common denominator needed.
If you’re multiplying a fraction by a whole number, write the whole number as a fraction over 1. So 3 × 2/5 becomes 3/1 × 2/5 = 6/5, which you can convert to 1 1/5 as a mixed number.
After multiplying, always check whether the result can be simplified.
Dividing Fractions
Dividing by a fraction means multiplying by its reciprocal. The reciprocal is just the fraction flipped upside down. Some people remember this as “keep, change, flip”: keep the first fraction, change the division sign to multiplication, and flip the second fraction.
For example, 8 ÷ 7/5. Keep 8 (written as 8/1), change ÷ to ×, and flip 7/5 to 5/7. Now multiply: 8/1 × 5/7 = 40/7, which equals 5 5/7 as a mixed number.
This works because dividing by a number and multiplying by its reciprocal are mathematically the same thing. Once you flip the second fraction, the problem becomes a regular multiplication problem.
Converting Mixed Numbers and Improper Fractions
A mixed number like 2 1/3 combines a whole number with a fraction. An improper fraction like 7/3 has a numerator larger than its denominator. Many fraction problems require you to convert between these two forms.
Mixed Number to Improper Fraction
Multiply the whole number by the denominator, then add the numerator. That gives you the new numerator, and the denominator stays the same. For 2 1/3: multiply 2 × 3 = 6, then add 1 = 7. The improper fraction is 7/3.
Another example: 3 5/8. Multiply 3 × 8 = 24, then add 5 = 29. The result is 29/8.
Improper Fraction to Mixed Number
Divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the new numerator over the same denominator. For 13/6: 13 ÷ 6 = 2 with a remainder of 1. So 13/6 = 2 1/6.
Converting to improper fractions is especially useful before multiplying or dividing, since it’s easier to work with a single fraction than a mixed number.
Comparing Two Fractions
When you need to figure out which of two fractions is larger, cross-multiplication gives you a quick answer. Multiply the numerator of the first fraction by the denominator of the second, and write that number above the first fraction. Then multiply the numerator of the second fraction by the denominator of the first, and write that above the second.
Compare 2/3 and 4/5. Cross-multiply: 2 × 5 = 10, and 3 × 4 = 12. Since 10 is less than 12, 2/3 is less than 4/5. The larger cross-product sits above the larger fraction.
You can also compare fractions by finding a common denominator, just like you would for addition. Once both fractions share the same denominator, the one with the bigger numerator is the bigger fraction.
Putting It All Together
Most fraction problems combine these skills. A word problem might ask you to add two fractions and then simplify, or convert a mixed number before multiplying. The process is always the same: identify the operation, apply the rule for that operation, and simplify your answer at the end. Here’s a quick reference:
- Simplify: Divide numerator and denominator by their greatest common factor.
- Add or subtract: Find a common denominator, convert, then combine numerators.
- Multiply: Multiply straight across, numerator × numerator and denominator × denominator.
- Divide: Flip the second fraction and multiply.
Fractions are typically introduced in 3rd grade, with addition and subtraction of like denominators in 4th grade and unlike denominators in 5th grade. If any of these operations feel unfamiliar, that’s normal. Practice each one separately until the steps feel automatic, then combine them in more complex problems.