**Middle School Number Sense Lesson 11: Adding Fractions with Different Denominators**

Fractions show up all over number sense tests. One of the most basic concepts dealing with fractions is addition, which is pretty simple when the denominators are the same. Today we will cover how to handle adding fractions when the denominators are different. This concept showed up

**12 times**this year--almost always at

**question**

**#16**.

**Number Dojo Level: 39**

I was taught to add fractions with different denominators by finding the

**lowest common denominator**(or LCD). By definition, the LCD would be the smallest integer which divides both denominators without a remainder. For example, if I add 1/3 + 1/4, the LCD would be 12, because 12 is the smallest number divisible by both 3 and 4.

**The Old Way: 1/3 + 1/4**

- Find the lowest common denominator. In this case, the LCD is
**12**. - Convert each fraction to an "un-reduced" fraction with 12 as the denominator. 1/3 =
**4/12**and 1/4 =**3/12**. - Add the new fractions together. 4/12 + 3/12 =
**7/12**. - Reduce the fractional sum if possible.
**7/12**is already reduced, so it is our answer.

**The New Way: Cross-Multiplying**

- Note: this is easier to demonstrate when the fractions are written vertically.

**1 1**

-- + --

3

-- + --

3

**4**

- Multiply diagonally: the numerator on the left by the denominator on the right (in
**red**above). 1 x 4 =**4**. - Multiply diagonally: the numerator on the right by the denominator on the left (in
**green**above). 1 x 3 =**3**. - Add the products in steps 1 and 2 to get the numerator of your answer. 4 + 3 =
**7**. - Multiply the two denominators to get the denominator of your answer. 3 x 4 =
**12**. - Reduce the resulting fraction if possible.
**7/12**cannot be reduced, so it is our answer.

**Example 1: 5/6 + 1/4**

- Multiple diagonally (when written horizontally, these are the numbers on the outside): 5 x 4 =
**20**. - Multiple diagonally (when written horizontally, these are the numbers on the inside): 6 x 1 =
**6**. - Add the products in steps 1 and 2 to get your numerator: 20 + 6 =
**26**. - Multiply the denominators to get your denominator: 6 x 4 =
**24**. - Reduce the resulting fraction (
**26/24**) if possible. Divide each number by 2 to get**13/12**.

**Example 2: 2/7 + 3/5**

- Combining steps 1 & 2 above, cross-multiply: 2 x 5 =
**10**and 7 x 3 =**21**. - Add the products: 10 + 21 =
**31**. - Multiply the denominators: 7 x 5 =
**35**. - Reduce the result (
**31/35**) if possible. This fraction does not reduce.

**Example 3: 1/5 + 3/10**

- Cross-multiply: 1 x 10 =
**10**and 5 x 3 =**15**. - Add the products: 10 + 15 =
**25**. - Multiply the denominators: 5 x 10 =
**50**. - Reduce the result (
**25/50**) if possible. This reduces to**1/2**, which is your answer.

**Note**: Sometimes it may be quicker to find the lowest common denominator, as in Example 3. Let's try the same question the old way:

- The LCD would be 10.
- 1/5 = 2/10.
- 2/10 + 3/10 = 5/10.
- 5/10 reduces to
**1/2**.

**Here's a free worksheet to help you practice AddFracDiff:**

addfracdiff.pdf |

**Up Next for Middle School: DivDouble1st**