Interpret a fraction's components. The top number is called a numerator and represents the part of a whole unit. The bottom number is the denominator and represents the whole unit.
Simplify fractions by reducing them to their lowest terms. Do this by determining the largest number that will divide into the numerator and denominator evenly. For example, 2/5 is in its simplest form. On the other hand 10/15 is not, since both numbers are divided evenly by a common number. This common number is called the "greatest common factor." Find the greatest common factor by listing the factors of each number and choosing the largest shared factor. In this example, the factors of 10 are 2 and 5. The factors of 15 are 3 and 5. Therefore, the greatest common factor is 5. Divide the numerator and denominator by 5 to get 2/3 as your simplified or "reduced" fraction.
Compare fractions by finding a common denominator. A common denominator is a number into which both denominators will divide evenly. You can find this by listing multiples of both denominators and choosing a common one between the two. For many problems concerning fractions, you'll need to find the "least common denominator," which is the lowest number into which both denominators will divide. For instance, if you have 3/4 and 5/6, list the first few multiples of 4 (4 x 1 = 4, 4 x 2 = 8, 4 x 3 = 12) and the first few multiples of 6 (6 x 1 = 6, 6 x 2 = 12, 6 x 3 = 18). The least common denominator is 12. Rewrite the fractions by placing the least common denominator in the denominator of both fractions. Divide the original denominator into the least common denominator and multiply the quotient by the numerator. In this case, 12 divided by 4 is 3. Therefore, multiply 3 by the original numerator of the first fraction (3) to get 9. This fraction is 9/12. Do the same for the second fraction to get 10/12. Compare the numerators to see which one is greater or lesser.
Add fractions by finding a common denominator (if the original denominators are not alike) and then add the numerators. For example, if you add 1/4 and 4/6, convert the fractions to 3/12 and 8/12. Add 3 and 8 to get the sum: 11/12.
Subtract fractions by finding a common denominator and subtracting the numerators. For example, consider 3/5 - 1/4. Rewrite the equation as 12/20 - 5/20. Subtract the numerators to get the answer: 7/20.
Multiply fractions by multiplying the numerators and the denominators. For example, in the problem 2/7 x 1/4, multiply 2 and 1 to get 2 and 7 and 4 to get 28. This new fraction is 2/28, which reduces to 1/14.
Divide fractions by flipping the second fraction to form a reciprocal and multiplying. For instance, if you have 9/10 / 3/7, first create a multiplication sentence with the reciprocal of 3/7: 9/10 * 7/3. This answer is 63/30. This is an improper fraction, since the numerator is larger than the denominator. Therefore, to simplify, divide the denominator into the numerator to get 2 with 3 left over. Write the remainder as the numerator over the original denominator: 2 3/30. Reduce again to get 2 1/10. This is known as a mixed number.