Sub3F1/a-1/ab

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

The traditional way to solve this is to look for the 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:

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

How to Solve:

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

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

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

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

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

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

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

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

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

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

1. Each denominator is increasing by a factor of 4.
2. 4 x 3 - 1 = 11.  This is your multiplier.
3. Multiply 11 by the numerator (3) to get 33--your numerator.
4. Use the 3rd denominator: 64.
5. The answer is 33/64, which doesn't reduce.

Note: I would recommend memorizing the first several 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
File Size:	354 kb
File Type:	pdf

Download File

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Up Next for High School: Reciprocal2