by Wacks Formula
I’m sure you’ve seen this type of problem:
Find 3 consecutive numbers whose sum is 12.
And I’m sure you’ve heard this old solution:
Long Method: Use algebra. (Don’t do this, it’s too slow.)
x = 1st number
x+1 = 2nd number
x+2 = 3rd number
Sum = 12
1st + 2nd + 3rd = Sum
(x) + (x+1) + (x+2) = 12
3x + 3 = 12
3x = 9
x = 3
1st number = x = 3
2nd number = x+1 = 3+1 = 4
3rd number = x+2 = 3+2 = 5
You are not allowed to do this in early school but using your fingers can buy you some time. Presenting:
Short cut #2:
THE WACKS MIDDLE FINGER TECHNIQUE
(For Consecutive Number Problems)
STEP 1: Divide the sum by the number of items. The answer becomes the middle number.
STEP 3: Find the rest by counting forward and backwards.
- Find 3 consecutive numbers whose sum is 12.
STEP 1: Divide 12 / 3 = 4 –> the middle number.
STEP 2: The 3 numbers are: __, 4, __.
Answer: 3, 4, 5
The method is usually done mentally without a pencil and paper. You use your fingers instead. In the example above, use your 3 fingers to represent the numbers. You look at your fingers to guide you while solving in your mind. The middle finger represents the middle number. Thus, the “middle finger” technique. Optionally you can move your fingers while solving. 😀
Trial and error can also work, and it’s also one of the faster methods. The Wacks Middle Finger Technique does not compete with trial and error. Rather, it helps give direction to your many trials and errors, so it narrows down your number of trials into just one trial (and zero error).
NOW, IT’S YOUR TURN!
1. Find 3 consecutive numbers whose sum is 15.
2. Find 5 consecutive numbers whose sum is 20.
3. There are 3 consecutive numbers whose sum is 99. Find the biggest number only.
*CHALLENGE: Find 4 consecutive numbers whose sum is 10.
Further thoughts for the Pro (if you are BRAVE enough!)
- What if there is no middle number? (4 or 6 or 8 consecutive numbers)
- What if the answer does not become a whole number? (1.5, 2.5, etc.)
- Can we use the shortcut if we are looking for consecutive odd numbers? (1, 3, 5, 7, 9 ..)
- Can we use the shortcut if we are looking for consecutive even numbers? (2, 4, 6, 8..)
- Can we use the shortcut if we are looking for numbers that are multiples of 3? (3, 6, 9, 12..)
See you next episode! 🙂
Disclaimer: The name of the formula is purely for fun, and does not discredit others who might have discovered the same formula independently.