**High School Number Sense Lesson 45: Solving for X (in the Denominator)**

This concept is a specific subset of our

**SolveX**questions. It appeared

**9 times**last year (on 6/7 of the UIL tests and only 3/14 of the TMSCA tests). The median placement was at

**question # 46**.

**Number Dojo Level: 239**

These questions usually show up in the form of:

**1/A + 1/B = 1/x**

or

**1/A - 1/B = 1/x**,

**A**and

**B**are given, and

**x**needs to be found. When the question is asked in the above format, there are quick ways to solve them...but first, let's look at the long way.

**Addition:**

- 1/A + 1/B = 1/x
- 1/A(B/B) + 1/B(A/A) = 1/x (finding a common denominator)
- B/AB + A/AB = 1/x (multiplying the fractions)
- (B + A)/AB = 1/x (adding the fractions)
- AB/(B + A) = x (taking the reciprocal)
- SO: x =
**the product of the denominators (AB) divided by their sum (B + A)**

**Subtraction:**

- 1/A - 1/B = 1/x
- 1/A(B/B) - 1/B(A/A) = 1/x (finding a common denominator)
- B/AB - A/AB = 1/x (multiplying the fractions)
- (B - A)/AB = 1/x (adding the fractions)
- AB/(B - A) = x (taking the reciprocal)
- SO: x =
**the product of the denominators (AB) divided by their difference (B - A)**

**Example 1: If 1/2 + 1/3 = 1/x, then x = ___**

- We are adding. Divide the
**product**of the denominators (AB) by their**sum**(B + A). So**AB/(B+A)**. - (2 x 3)/(3 + 2) =
**6/5**

**Example 2: If 1/12 + 1/3 = 1/x, then x = ___**

- We are adding. Divide the
**product**of the denominators (AB) by their**sum**(B + A). So**AB/(B+A)**. - (12 x 3)/(12 + 3) = 36/15 =
**12/5 or 2.4**

**Example 3: If 1/2 - 1/6 = 1/x, then x = ___**

- We are subtracting. Divide the
**product**of the denominators (AB) by their**difference**(B - A). So**AB/(B-A)**. - (2 x 6)/(6 - 2) = 12/4 =
**3**.

**Example 4: If 1/9 - 1/11 = 1/x, then x = ___**

- We are subtracting. Divide the
**product**of the denominators (AB) by their**difference**(B - A). So**AB/(B-A)**. - (9 x 11)(11 - 9) =
**99/2 or 49.5**

**Caveats:**

- Sometimes these problems are written with negative exponents instead of denominators.
- If the numerators are anything but 1, it will probably be faster to use another method.
- If the unknown is not isolated on the right of the equation (such as 1/2 + 1/x = 2/3), it will probably be faster to use another method.

**Example 5: If 2/5 - 1/x = 3/10, then x = ___**

- Move x to the right side of the equation:
**2/5 = 3/10 + 1/x** - Subtract 3/10 from both sides:
**2/5 - 3/10 = 1/x** - Find a common denominator by changing 2/5 to 4/10:
**4/10 - 3/10 = 1/x** - Subtract:
**1/10 = 1/x**. - Take the reciprocal of both sides: x =
**10**.

**Example 6: If 2/3 + 4/5 = 1/x, then x = ___**

- Find a common denominator:
**2/3(5/5) + 4/5(3/3) = 1/x** - 10/15 + 12/15 = 1/x
- 22/15 = 1/x
**15/22**= x

**Example 7:**

**If 3**

^{-2}+ x^{-1}= 6^{-1}then x = ___1. Change this mentally to a familiar format:

**1/9 + 1/x = 1/6**

2. Subtract 1/9 from both sides:

**1/x = 1/6 - 1/9**

3. Find a common denominator (18) and convert the fractions:

**1/x = 3/18 - 2/18 = 1/18**

4. x =

**18**

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

solvexdenom.pdf |

**Up Next for High School: Polyhedron**