**Middle School Number Sense Lesson 35: Adding a Fraction to its Reciprocal**

Early last week we covered the

**Reciprocal**concept, so you know how to find it. Today we'll cover how to add it. This concept appeared on average more than once per test...

**21 times**this year, with a median spot at

**question # 42**.

**Number Dojo Level: 180**

This concept will usually appear like this:

**a/b + b/a = ___ (mixed number)**

**a**and

**b**are unique integers.

**How to Solve:**

- Subtract a from b (or b from a) and square this difference to find the
**numerator**of your fraction. - Multiply the two denominators to find the
**denominator**of your fraction. - Write a
**2**in front of the resulting fraction. You're done. (But I'll show you some examples of an exception.)

**Example 1: 3/4 + 4/3 = ___ (mixed number)**

- 4 - 3 =
**1**. 1 squared =**1**. This is the numerator. - 3 x 4 =
**12**. This is the denominator. - Write a
**2**in front. Your answer is**2 1/12**.

**Example 2: 5/8 + 8/5 = ___ (mixed number)**

- 8 - 5 =
**3**. 3 squared =**9**. This is the numerator. - 5 x 8 =
**40**. This is the denominator. - Write a
**2**in front. Your answer is**2 9/40**.

**Example 3: 7/9 + 9/7 = ___ (mixed number)**

- 9 - 7 =
**2**. 2 squared =**4**. This is the numerator. - 9 x 7 =
**63**. This is the denominator. - Write a
**2**in front. Your answer is**2 4/63**.

**Example 4: 7/13 + 13/7 = ___ (mixed number)**

- 13 - 7 =
**6**. 6 squared =**36**. This is the numerator. - 7 x 13 =
**91**. This is the denominator. - Write a
**2**in front. Your answer is**2 36/91**.

**Example 5: 4/11 + 11/4 = ___ (mixed number)**

- 11 - 4 =
**7**. 7 squared =**49**. This is the numerator. - 4 x 11 =
**44**. This is the denominator. But WAIT!**49 > 44**, so we have to change this to**1 5/44**. Write the**5/44**and regroup the**1**. - Write a 2 (plus the regrouped 1 =
**3**) in front. Your answer is**3 5/44**.

**Example 6: 8/3 + 3/8 = ___ (mixed number)**

- 8 - 3 =
**5**. 5 squared =**25**. This is the numerator. - 8 x 3 =
**24**. This is the denominator. But WAIT!**25**, so we have to change this to**1 1/24**. Write the**1/24**and regroup the**1**. - Write a 2 (plus the regrouped 1 =
**3**) in front. Your answer is**3 1/24**.

**Example 7: 3 - (3/4 + 4/3) = ___ (fraction)**

- Solve within the parentheses first. For the numerator (4 - 3) =
**1**. 1 squared =**1**. - For the denominator: 4 x 3 =
**12**. Place a**2**in front (mentally) to get**2 1/12**. Now subtract this from 3. - 3 - (2 1/12) =
**11/12**.

**Example 8: If a/b + b/a = 2 9/40, where a and b are relatively prime, then the larger of a and b is ___**

**This is AddFracRecip at its best.**They give you the sum, and you have to figure out the fractions.- You know that the numerator (9) is the square of the difference between
**a & b**, so**a - b**has to be**3**. - You know that the denominator is the product of
**a & b**, so think of two integers (3 apart) whose product is 40. They are**5 & 8**. - The question asked for the larger one, so your answer is
**8**.

**Example 9: If a/b + b/a = 2 25/104, where a and b are relatively prime, then a + b = ___**

- You know that the numerator (25) is the square of the difference between
**a & b**, so**a - b**has to be**5**. - You know that the denominator is the product of
**a & b**, so think of two integers (5 apart) whose product is 104. This is a little tougher...but they are**8 & 13**. - The question asked for
**a + b**, so your answer is**21**.

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

addfracrecip.pdf |

**Up Next for Middle School: DiffSquares2**