**High School Number Sense Lesson 82: Subtracting 3 Fractions in the Form 1/a - 1/ab - 1/abb**

Today's concept and explanation is courtesy of my student

**Riki**, who teaches me at least as much as I teach him. I appreciate his enthusiasm and humility. We covered a similar-looking concept last month:

**AddF1/n(n+1)**. This one, however, covers subtraction, is limited to 3 terms, and its denominators follow a different pattern. Today's concept has shown up

**9 times**so far this year on high school tests, with a median placement at

**question # 11**.

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Let's start off by showing an example of the problem:

**1/4 - 1/8 - 1/16**

**lowest common denominator**, which in this case would be

**16**. Then we would convert each fraction to one with that denominator--so

**4/16 - 2/16 - 1/16**. Notice that the answer would be the same as the last term being subtracted:

**1/16**.

But we don't want to do things the traditional way; we want to do things the

**number sense**way, or what I call the

**efficient way**. (Some people would call it the "lazy way," but they're just jealous).

**The Pattern:**

- There must be 3 fractions, with the 2nd and 3rd being subtracted from the first.
- The numerators must be equal.
- The 2nd & 3rd denominators must be increasing by the same
**factor**. (This factor is key to solving the problems quickly).

**How to Solve:**

- Determine the
**factor**being used to increase the denominators. - Multiply the denominator's
**factor**by its lower neighbor (itself minus 1), and then subtract 1. This is your numerator's**multiplier**. - Multiply this result by your numerator. This will be the numerator in your answer.
- Use the largest (in other words, the third) denominator as your denominator.
- Reduce if possible.

**Example 1: 1/3 - 1/6 - 1/12 = ___**

- Note that each denominator is increasing by a factor of
**2**. - Multiply 2 by (itself minus 1), and then subtract 1. 2 x 1 - 1 =
**1**. This is your multiplier. - Multiply 1 by the numerator (1), to get
**1**--your numerator. - Use the 3rd denominator:
**12**. - The answer is
**1/12**, which doesn't reduce.

**Example 2: 5/8 - 5/16 - 5/32 = ___**

- Note that each denominator is increasing by a factor of
**2**. - 2 x 1 - 1 =
**1**. This is your multiplier. - Multiply 1 by the numerator (5) to get
**5**--your numerator. - Use the 3rd denominator:
**32**. - The answer is
**5/32**, which doesn't reduce.

**Example 3: 2/7 - 2/21 - 2/63 = ___**

- Note that each denominator is increasing by a factor of
**3**. - 3 x 2 - 1 =
**5**. This is your multiplier. - Multiply 5 by the numerator (2) to get
**10**--your numerator. - Use the 3rd denominator:
**63**. - The answer is
**10/63**, which doesn't reduce.

**Example 4: 4/5 - 4/15 - 4/45 = ___**

- Each denominator is increasing by a factor of
**3**. - 3 x 2 - 1 =
**5**. This is your multiplier. - Multiply 5 by the numerator (4) to get
**20**--your numerator. - Use the 3rd denominator:
**45**. - The answer is
**20/45**, which reduces to**4/9**.

**Example 5: 3/4 - 3/16 - 3/64 = ___**

- Each denominator is increasing by a factor of
**4**. - 4 x 3 - 1 =
**11**. This is your multiplier. - Multiply 11 by the numerator (3) to get
**33**--your numerator. - Use the 3rd denominator:
**64**. - The answer is
**33/64**, which doesn't reduce.

__: I would recommend memorizing the first several__

**Note****multipliers**to make these problems much faster.

- If the denominators increase by a factor of
**2**, your numerator's multiplier is (2 x 1 - 1) =**1**. - If the denominators increase by a factor of
**3**, your numerator's multiplier is (3 x 2 - 1) =**5**. - If the denominators increase by a factor of
**4**, your numerator's multiplier is (4 x 3 - 1) =**11**. - If the denominators increase by a factor of
**5**, your numerator's multiplier is (5 x 4 - 1) =**19**. - And so on.

**Here's a free worksheet to help you practice Sub3F1/a-1/ab:**

sub3f1a-1ab.pdf |

**Up Next for High School: Reciprocal2**