 Most students learn to add fractions with like denominators in 3rd or 4th grade. In 5th grade, you'll learn to add, subtract and multiply fractions with different denominators. To do this, you need some background information. Read on for an explanation! You probably already know how to add and subtract fractions with common denominators, like 3/4 and 1/4. You simply add or subtract the numerators, and leave the denominators the same. In 5th grade, you'll have to add and subtract fractions that have different denominators, like 3/8 and 1/5. To do this, you need to understand equivalent fractions, and you must know how to find the least common multiple (LCM) of two numbers.

### Equivalent Fractions

Imagine that you cut a pizza into two halves, and you take one. This means that you've taken 1/2 of the pizza. If you cut that same pizza into four slices, and take two, then you've taken 2/4 of the pizza. Either way, you've taken the same amount of the pizza. That's because 1/2 and 2/4 are equivalent fractions.

The relationship between their numerators and denominators is the same. One is half of two, and two is half of four. Other fractions that are equivalent to 1/2 and 2/4 are 3/6 and 4/8, because three is half of six and four is half of eight. Here are some more examples of equivalent fractions:

• 1/6 = 2/12 = 3/18
• 3/5 = 6/10 = 12/20
• 2/7 = 4/14 = 6/21

To generate these equivalent fractions, you just multiply both the numerator and the denominator of the original fraction by the same number. For example, we found that 3/18 was equivalent to 1/6 by multiplying both one and six by three. To find that 4/14 was equal to 2/7, we multiplied two and seven each by two.

### Least Common Multiples

The multiples of a number are what you get when you multiply it by other numbers. For instance, some multiples of four are eight, 12, 16 and 20. Multiples of three include six, nine, 12 and 15.

The least common multiple (LCM) of two numbers is the smallest number that's a multiple of both. For example, the LCM of four and three is 12, because it's the smallest number that's a multiple of both of them.

Now that you've mastered equivalent fractions and least common multiples, you're ready to add and subtract fractions with different denominators. Since you can't directly add and subtract fractions like this, you have to find equivalent fractions with common denominators to represent the fractions that you're trying to add or subtract. Here's how you do it:

1. First, find the LCM of the two denominators of your fractions. For instance, if you're trying to add 2/5 and 1/4, you need to find the LCM of five and four. If you list out their multiples and search for the smallest number that's a multiple for both, you'll find that the LCM of five and four is 20.
2. Next, figure out what number you need to multiply the denominator of each fraction by to get the LCM. For 1/4, it would be five, because 4 x 5 = 20. For 2/5, the number would be four, because 5 x 4 = 20.
3. Once you've found this number for each fraction, multiply the denominator and the numerator by it. For 2/5, multiply two and five both by four to get 8/20. Then, for 1/4, multiply both one and four by five to get 5/20.
4. Now you have two fractions, 8/20 and 5/20, that are equivalent to the two original fractions. Even better, they have common denominators, so you can add them. Just add up the numerators and leave the denominators the same, and you get 13/20 as your answer.

For subtraction, it works the same way. Just subtract the numerators at the end, instead of adding them together. Using the same example, 8/20 - 5/20 = 3/20.

### Multiplying Fractions

Multiplying fractions is a lot less complex than adding and subtracting, because the denominators don't have to be the same. You just multiply the two denominators together, then multiply the two numerators together, and you have your answer.

Let's look at the example of 1/4 times 1/3. The product of one and one is one, so that's the numerator of the answer. The product of four and three is 12, so that's the denominator of the answer, which is 1/12. Notice that 1/12 is smaller than 1/4 and 1/3. When you multiply proper fractions, your answer should be less than the original fractions that you multiplied.

### Improper Fractions and Mixed Numbers

Improper fractions are fractions where the numerator is larger than the denominator. For example, 11/5 is an improper fraction. Improper fractions represent numbers greater than one.

Mixed fractions also represent numbers greater than one, but they include an integer and a fraction. For instance, 3 2/5 is a mixed fraction. Improper fractions can be added, subtracted and multiplied, just like proper fractions. To do these things to mixed fractions, you have to convert them into improper fractions first.

### Converting Between Mixed Numbers and Improper Fractions

To convert an improper fraction into a mixed number, divide the numerator by the denominator. The integer that results is the integer in the mixed number. The remainder becomes the numerator of the fraction in the mixed number, and the denominator from the improper fraction becomes the denominator of the fraction in the mixed number.

To turn a mixed number into an improper fraction, multiply the denominator in the mixed number by the integer. Then add the numerator of the mixed number's fraction to that to get the numerator of the improper fraction. The denominator remains the same as the one in the fraction of the mixed number. Here are some examples:

• 16/7 = 2 2/7
• 13/4 = 3 1/4
• 10/9 = 1 1/9

If the numerator of an improper fraction can be divided evenly by the denominator, then it will just become an integer, not a mixed number. For example, 8/2 = 4 and 12/6 = 2. An integer over one always equals itself. For instance, 3/1 = 3 and 17/1 = 17.

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