# Fractions

**What are Fractions?**

Fractions are one of several ways to express parts of a whole number. For example, if I had a pizza, and I cut it into eight slices, each slice could be expressed as 1/8. This means it is one of 8 parts of a pizza. An entire pizza would be 8/8 or 1 pizza. The top number of a fraction, showing the number of parts is called the numerator. The bottom number, showing how many parts are in each whole number, is called the denominator.

**Equivalent Fractions**

Fractions that are equal can be written in different ways. For example, consider the pizza above, cut into 8 slices. If you had two slices, that could be 2/8. If cut another pizza in four slices, one slice would be the same size as two slices of the first pizza, so 2/8 is the same as 1/4. That fraction could also be written as 4/16 or 8/32.

**Reducing Fractions**

In most cases, the correct answer to a problem will be the fraction in reduced form. This means with the smallest possible number in the numerator and denominator positions. If you need to reduce a fraction and have access to the calculator, enter the fraction and hit enter and it will be reduced. If you do not have the calculator, find a number that you can divide both the numerator and denominator by. For example, for the fraction 2/8, you can tell that both the numerator and denominator are even numbers, which means that they can be divided by 2.

**Mixed Numbers**

Sometimes fractions are needed to represent an amount that is more than one whole. For example, if we had the pizza cut into 8 slices, and we had one whole pizza and one extra slice we could write that as 1 1/8.

When a fraction has a larger number in the numerator than in the denominator, it is an improper fraction, which also means that is an amount more than one whole. An improper fraction can be turned into a mixed number by dividing the numerator by the denominator, the writing the result as the whole number and the remainder as the numerator. The denominator stays the same. For example, if you have the improper fraction of 29/8, divide 29 by 8. You will get 3 with a remainder of 5, so the mixed number is 3 5/8.

Mixed numbers can be turned into improper fractions by multiplying the denominator by the whole number, then adding in the numerator. For example, in the mixed number 3 5/8, you would multiply 8 by 3 to get 24, and then add in the 5 to get 29. Place that over the denominator to get 29/8.

**Comparing Fractions**

When comparing two fractions to see which is larger, you have a couple of options. One way would be to reduce or enlarge the fractions until they have the same denominator. If you were to compare 3/8 and 1/4, you could enlarge 1/4 by multiplying both the numerator and denominator by two and getting 2/8 which is an equivalent fraction to 1/4. Since 3 is greater than 2, 3/8 is the greater fraction.

The other option, and perhaps the best for more complicated numbers, is to cross multiply. If you want to know which is larger of 3/7 and 5/11, multiply the numbers that are diagonal from each other. Multiply 11 × 3 and write the answer above 3/7: 33. Then multiply 7 × 5 and write the answer above 5/11: 35. Since 35 is more than 33, 5/11 is the larger fraction.

**Multiplying and Dividing Fractions**

When multiplying two fractions, simply multiply the two numerators and make that the numerator of the new fraction. Then, multiply the denominators and make it the denominator of the new fraction. For example, if I multiply 1/4 by 2/3, multiply 1 × 2 to get 2, then multiply 4 × 3 to get 12. Then write your answer as 2/12, which can be reduced to 1/6.

For division, flip the second fraction upside-down, which is called finding the inverse. Then, multiply across as we did above. For example, if we were to divide 1/4 by 2/3, flip 2/3 to 3/2, then multiply across. 1 × 3 is 3 and 4 × 2 is 8, so our answer is 3/8

Remember, if you have the option to use the calculator, fraction operations can be done on the calculator. See the calculator section for more information.

**Factoring and Least Common Multiple**

To factor a number means to find all the numbers that can be multiplied together to get that number until you get down to all prime numbers. Prime numbers are numbers that can only be divided evenly by themselves and one. The factors of 6 would be 2 and 3, because they multiply together to get 6 and are both prime numbers. If you were asked to factor 28, you could see that 4 × 7 is 28, but 4 is not a prime number. 2 × 2 = 4, so the prime factors of 28 are 7, 2, and 2. Knowing the factors of numbers can be helpful in finding the least common multiple and later on in algebra.

Sometimes you need to get two fractions with different denominators to have the same denominator. You can do this by enlarging to equivalent fractions until the denominators are the same. One way to think of this is to think of what number both of the denominators are factors. You could always multiply the two denominators together, or you could count off by that number on the multiplication chart and see where you get a repeat. For example if one of the denominators is 4 and the other is 6, your multiples of 4 are 4, 8, 12, 16 etc. and your multiples of 6 are 6, 12, 18, 24 etc. The least common multiple from both lists is 12, so that will be your new denominator.

**Adding and Subtracting Fractions**

Adding and subtracting fractions is easy once the denominators are the same. You simply add or subtract the numerators and leave the denominator alone. If the denominators are different, find equivalent fractions using least common multiple to make the denominators the same. To keep your new fractions equivalent, remember that you must multiply the numerators by the same number as you did the denominator. For example, if you were to add 1/4 and 2/3, you would need to find the least common multiple of 4 and 3, which is 12. Now you will have two fractions with 12 in the denominator. Since you multiplied the 4 in the first fraction by 3 to get 12, you must also multiply the numerator, 1, by 3 to keep the fraction equivalent, which would give you 3/12. Do the same thing to the other fraction. Since you multiplied the denominator of 3 by 4 to get 12, you must multiply the numerator 2 by 4, which would give you an equivalent fraction of 8/12. Then add the numerators of your two fractions to get the answer: 3/12 + 8/12 = 11/12.

Again, if you have the option to use the calculator, fraction operations can be done on the calculator. See the calculator section for more information.