What you need to know
Long Multiplication is a form of repeated addition, i.e. multiplying numbers amounts to adding several copies of a number together. Division is the opposite of multiplication, it amounts to finding out how many lots of one number go into another number. Both of these operations can get quite difficult when we have to deal with bigger numbers, so we’re going to look at 2 methods of multiplication and 2 methods of division for dealing with trickier situations.
The two multiplication methods we will see are the grid method and long multiplication.
Example: Work out 23\times 281 using the grid method and long multiplication.
The grid method involves splitting each number into 1s, 10s, 100s and so on, and writing each component of one number along the top of a grid, and each component of the other number down the left-hand side, as seen on the left.
Then, we fill all the squares of the grid by multiplying each bit of one number by each bit of the other. So, the top left square will be 20\times 200=4000, and the bottom left one will be 200\times 3=600, and so on. Carrying this on with the rest of the multiplications and you get the filled grid on the right. Now, what remains is to add together all the numbers in blue. Along the top row, we get: 4000+1600+20= 5,620, and along the bottom row we get: 600+240+3= 843. Then, by whichever method you prefer, add together these two values to get
This is precisely the result of the multiplication, i.e. 23 \times 281 = 6,463.
Long multiplication involves the same idea of multiplying each component at a time, but it’s structure is a little more compact. To start it, write the bigger number over the smaller one, making sure that the 1s are above each other, the 10s are above each other and so on. Keeping everything in the right column matters a lot here.
Then, we want to multiply each component of 281 by the “3” part of 23 and write the results of the multiplications under the grey line. 1\times 3=3, so we write a 3 in the 1s column under the grey line. Then, 8\times 3=24, but we can’t write “24” in one column. What we do is write the 4 in the 10s column and carry the two over to the 100s column. Then, 2\times3=6, but we carried over an extra 2, so we write 6+2=8 in the 100s hundreds column (see: right).
Now, we do everything we just did but this time, multiply each component of 281 by the “2” part of 23. The only difference is because the 2 represents a 20, everything is shifted one space to the left and a zero is put in the 1s column. For the completed step, using same methods as before, see: left.
Finally, we add together the blue numbers and write the final answer underneath the second grey line. The answer (as we already know) is 5,620+843= 6,463. For the completed process, see: right.
Have a go at both of these methods of long multiplication and decide which you prefer – it’s completely up to you which one you use.
1) Work out 45 \times 619.
Here we’re going to use the grid method, but the long multiplication method is completely fine too (as long as you get the answer right of course). So, we construct a grid with 600, 10, and 9 across the top and 40 and 5 down the side. Then, fill in the gaps in the grid by multiplying each component of 619 by each component of 45. This looks like:
Now, we add the blue numbers together. Adding the top row, we get
Adding the bottom row, we get
Therefore, the result of the multiplication is
2) Calculate 52 \times 31.
Now, we add the numbers together.
Answer = 1612