Five Multiplication Tricks

You studied your multiplication tables way back in the third grade. You can quickly rattle off the product of 7 and 9 or 8 and 6 (can't you?). Well, it never too late to discover some new patterns and tricks to help you with products and processes — shortcuts that save you time. You're also more apt to get the right answer when using these tips (not that you ever have the wrong answer, of course). And think of how you can amaze your friends and colleagues by pulling seemingly tough answers out of thin air.

Squaring numbers that end in 5

You know that 5 squared is 25. But what about 15², 25², 35², and so on? Don't have your calculator on you? No worries. To square a number that ends in 5, follow these steps:

1. Write down the last digits of the answer: 25.

The squared form of a number that ends in 5 always ends with 25.

2. Take the digit or digits in front of the original 5 and multiply them by the next bigger number.

3. Put the product of Step 2 in front of the 25, and you have the square of the number you want.

To square 35, for example, you write down the last two digits, 25. Now you multiply the 3 times the next bigger number, 4, to get 12. Put the 12 in front of the 25, and you have your answer: 35² = 1,225. Squaring 65, you know that 6 x 7 = 42, so 65² = 4,225.

This trick even works with three-digit numbers. You find the square of 105 by multiplying 10 x 11, giving you 110: 105² = 11,025.

Finding the next perfect square

Now you know how to find the squares of numbers that end in 5. But what about all the other squares? A nice property you can use deals with the next square in any list of the squares of whole numbers. The property states that you can get to the next square in the list by taking the square of the number you already have and adding the root plus the next number (the root of the square you want).

For example, if you know that 25² = 625,and you want 26², you just add 625 (the square of 25) + 25 (the root of 25) + 26 (the next highest number), which equals 650 + 26 = 676. To find the square of 81, you take 80² = 6,400 and then add up the ingredients: 6,400 + 80 + 81 = 6,400 + 161 = 6,561.

Multiplying by 11

Multiplying single digits by 11 is simple, in-your-head math. You just take the single digit, make two of them, and you're done. Multiplying a larger number times 11 is a bit trickier. However, you can make it just as easy by bookending the value with zeroes and adding adjacent digits.

For instance, when multiplying 142,327 x 11, you put a zero in front of the number and behind the number, and double every digit of the original number giving you 01144223322770. Now you add each pair of adjacent digits:

0 + 1, 1 + 4, 4 + 2, 2 + 3, 3 + 2, 2 + 7, 7 + 0

The sums are 1, 5, 6, 5, 5, 9, and 7, so the product of 142,327 x 11 is 1,565,597.

In the previous product, you see no carry over. If one or more of the sums you find is greater than nine, you carry the tens digit over to the sum to the left of the digit in question.

For instance, when multiplying 56,429 x 11, you add the following:

0 + 5, 5 + 6, 6 + 4, 4 + 2, 2 + 9, 9 + 0

The sums are 5, 11, 10, 6, 11, and 9. Starting at the right side of the answer, you see that the last digit is 9. Now follow these steps:

1. Write down the 1 from the ones place of the 11 in front of the 9, and carry the other 1 over to the 6 to make it 7. Write down the 7 in front of the 1.

You now have 719.

2. Write down the 0 in front of the 7, and carry the 1 over to the left, adding it to the 11.

Now you have 0719.

Adding the 1 to the 11 gives you 12.

3. Write down the 2 in front of the 0, and carry the 1 in the tens place over to the 5 to make it 6. Write the 6 in front of the 2.

You now have 620,719.

So, 56,429 x 11 = 620,719.

Multiplying by 5

To multiply any number by five in your head, you can just halve the number you want to multiply and add a zero to the end of the number.

To multiply 14 x 5, for example, you take half of 14, which is 7, and put a 0 after the 7 — 14 x 5 = 70.

But what if the number you want to multiply is odd and halving gives you a decimal? In this case, you don't add a zero to the end of the halved number; you just drop the decimal point.

For example, if you want to multiply 43 x 5, you take half of 43, which is 21.5. Dropping the decimal point, you get 215 — 43 x 5 = 215.

Multiplying two-digit numbers

To multiply two 2-digit numbers in your head, you can use the FOIL method (First, Outer, Inner, Last).

For instance, to multiply 23 x 12, do the Last portion first by multiplying the 3 times the 2. Put a 6 in the right-most answer position. Now you cross-multiply. The 2 in 23 times the 2 in 12 is 4. You add that value to the value for 3 times 1, and you get 7, the Outer and Inner portion. Put the 7 in front of the 6 for the answer. Now you multiply the 2 in 23 by the 1 in 12. Put the 2 in front of the 7 and 6 for a complete answer: 276.

If any of the products you find are greater than nine, you carry over the number in the tens place and add it to the next product or cross product.