Imagine Numbers as Friends Holding Hands:
- Think of each digit in a number as a friend. When you multiply by 11, it’s like these friends are holding hands and making new friends in between!
The Two-Digit Trick:
- Let’s say you have the number 23.
- In this number, 2 and 3 are friends.
- When you multiply by 11, something magical happens! You take these two friends, and they reach out to hold hands with their new friend in the middle.
- But who is this new friend? It’s the sum of their friendship! You add 2 and 3 together, which makes 5.
- So, 2 and 3 stay on the outside, and their new friend 5 comes to join them in the middle. Now you have 253!
The Three-Digit Trick:
- Now, let’s look at a number with three digits, like 314.
- Here, you have three friends: 3, 1, and 4.
- When multiplying by 11, each pair of friends makes a new friend by adding their digits together.
- 3 and 1 add up to make 4.
- 1 and 4 add up to make 5.
- So, your new number is 3454. The original friends stay on the outside, and the new friends from adding digits are placed in the middle!
What About Carries?
- Sometimes, when two friends add up their numbers, they get more than 9. Imagine they made a really big new friend!
- For example, with 89, the digits are 8 and 9.
- When they add together, they get 17.
- Here, the big friend 17 can’t fit in just one space, so the 7 stays in the middle, and the 1 goes to the next space on the left, helping out the 8.
- So, 89 × 11 becomes 979!
Why It Works:
- The reason this trick works is that when you multiply by 11, you’re really just making each digit a little bit more by adding its neighbor. It’s like every digit is saying, “I want to share some of my number with my friend next door!”
- That’s why you add the digits together to get the middle number. It’s just the digits helping each other out!
Rationale Behind the Shortcut:
When you multiply a number by 11, you’re essentially multiplying the number by (10 + 1). This can be broken down into two steps:
Multiply by 10: This shifts all the digits of the number one place to the left. For example, multiplying 23 by 10 gives you 230.
Add the Original Number: After shifting the digits by multiplying by 10, you add the original number back. So, for 23, you’d add 23 to 230, resulting in 253.
Mathematically, for any two-digit number ab (where a is the first digit and b is the second digit):
- ab × 10 gives you ab10.
- Then you add the original number ab to ab10.
This process gives you:
- The first digit stays the same (from the multiplication by 10).
- The middle digit is the sum of the two digits a and b (from adding the original number).
- The last digit is the original second digit b.
For example, with 23:
- 23 × 10 = 230
- 230 + 23 = 253
Thus, the middle digit is the sum of 2 + 3 = 5, resulting in 253.
For Three-Digit Numbers:
The process is similar, but you’re adding neighboring digits together to create the middle parts:
- For a three-digit number like abc, where a is the hundreds digit, b is the tens digit, and c is the ones digit:
- You first shift everything left by multiplying by 10: abc10.
- Then, you add the original number abc.
- The result is that the first digit stays the same, the next digit is the sum of a + b, and the following digit is b + c.
For example, with 314:
- 314 × 10 = 3140
- 3140 + 314 = 3454