How to do long multiplication

What is long multiplication?

Put simply, it's "multiplying two long numbers together". So 342 x 8 isn't long multiplication - the 342 is a long(ish) number, but the 8 isn't! Basically, you're doing long multiplication when each of the two numbers you're multiplying has at least two digits, e.g. 34 x 86 or 780 x 53.

If the thought of long multiplication without a calculator is a scary one, don't worry. There are several methods for doing long multiplication and one of them is bound to strike a chord - it's really a question of picking the method that suits you best. In this hub, I will talk about three methods of long multiplication: the grid method (sometimes also known as the boxes method), the lattice method and the so-called "traditional" method.

(1) The grid method

Before explaining the grid method in detail it's worth revisiting "short" multiplication.  Suppose you want to multiply 26 by 7.  You can lay it out the "standard" way, like this:

      26
      x7
    182

But another way to do it is to break it down as follows:

26 x 7   =   (6 x 7)  +  (20 x 7)  =   42 + 140   =   182

In other words, take the number that has more than one digit (the 26) and break it down into units and tens.  Thus there are 6 units and 2 tens (2 x 10 = 20).  Then multiply each component by 7 and add the two answers together.

Try another one.  How about the one I mentioned earlier, i.e. 342 x 8?

The 342 breaks down into 2 units, 4 tens and 3 hundreds, i.e. 2 + (4 x 10) + (3 x 100), or 2, 40 and 300.

So

2 x 8 = 16
40 x 8 = 320
300 x 8 = 2400 

Add the answers together:

16 + 320 + 2400 = 2736

Thus 342 x 8 = 2736

Long multiplication via the grid method uses the same principle.  So if you're doing 34 x 86, you split each number into tens and units, and then multiply the separate components together.  In other words, the 34 becomes 4 units (4) and 3 tens (30), and the 86 becomes 6 units (6) and 8 tens (80).  You end up doing the following calculations:

4 x 6
30 x 6
4 x 80
30 x 80

…and then adding them all together.

Laying it all out in a grid would actually be a lot simpler, wouldn't it?  (Or maybe you don't need a grid, but let's assume you do LOL.)

In the above grid, I've put the 86 down the side and the 34 at the top, but actually it doesn't matter which way round you do it.

In the appropriate squares, you then need to write down the answers to each of the four calculations I listed above, to give:

Then you need to start adding your figures.  I've added the figures in each of the two columns and then added the results to give my answer, which is in italics on the bottom right:

Thus 34 x 86 = 2924

You can also add across the rows and do a total - it should get you the same result!  (Try it.)

Suppose you want to multiply 134 by 32.  Split both these numbers into hundreds, tens and units.  So:

134 becomes 100 (hundreds), 30 (tens) and 4 (units)
32 becomes 30 (tens) and 2 (units).

Draw a grid with an appropriate number of rows and columns (in this example I've done three columns and two rows, but again, you could just as easily do a grid with two columns and three rows):

Then multiply the numbers and put each answer in the appropriate cell:

Finally, add each column together and then do a total, bottom right:

Thus 134 x 32 = 4288


(2) Lattice method

This method looks a bit weird to start with but is actually easier than the grid method once you get used to it. You draw exactly the same type of grid, but each of the inner "cells" needs to be sliced in half with a diagonal line, as shown below.

Read down from the columns and across from the rows to multiply the sets of numbers. If the answer is a single digit number, put it in the bottom right hand triangle in the appropriate cell. If it’s a two-digit number, use the top left triangle as well:

Then add diagonally as shown below, starting from the right and working your way leftwards (carry numbers over if necessary).  For example, the "2" in the answer of 4288 is obtained by adding the 1, 9 and 2 in the appropriate diagonal, which of course gives 12 - the "1" is carried over to the square on the left and added to the 3 to give 4. 

(3) Traditional method

I've left this method till last but it's probably the one that's most commonly taught, or at least it was when I was at school! 

In this method, you basically split the smaller number (the 32) into tens and units, which of course gives you 30 and 2. Row (a) is what you get when you multiply 134 by 2 and row (b) is what you get when you multiply 134 by 30. Because 134 x 30 is 134 x 3 with a zero added to the answer, I've added a zero first (shown in red). Adding row (a) and row (b) together gives you your answer of 4288.

© Empress Felicity May 2010