Maybe you were snoozing in math class and feel like brushing up on your basic arithmetic. Or could it be that the math your kid is doing is beyond what you can remember now, and you need a little touch-up on your skills to help with homework? Whatever your reason to pick up long multiplication today we’ve got you sorted. We’ll discuss the basic formula and some cheats we like to use to make things a little tad easier.
Long multiplication, like long division, is dreaded by many math learners as it can be tedious, time-consuming and the risk of making simple errors is high. We suggest being in a quiet environment where you can concentrate and writing ALL your steps down. Yes, you might be able to keep a couple in your head but as soon as something interrupts your train of thought you’ll lose all that hard brain crunching.
This is all, of course, presuming that you for whatever reason, cannot pick up a trusty calculator. In each section, we will discuss the method, go through an example, and give you a few problems to work out for yourself. You can check your answers with a calculator once you’ve tried the exercises, but no cheating! You’re here to learn.
We’ll build up to multiplying three-digit numbers by working with one- and two-digit numbers first. Let’s get to it!
Multiplying a Three Digit Number by One Digit
In this section we will learn the basic principles of long multiplication using progressively more difficult examples.
Example 1: Easy
In Example 1 and the first set of exercises we will look at multiplying a single digit by a number rounded to the nearest hundred:
1a: 200 x 8 =
Solution 200 x 8 is the same time as adding 200 up eight times, we multiply 8 x 2 = 16 and then multiply by 100. Multiplying by 100 is easy – its just adding two zeros to the end hence;
200 x 8 = 100 x 2 x 8 = 100 x 16 = 1600
1b: 400 x 2 =
Similarly, 2×4=8 then we add the two zeros, so 400 x 2 = 800
1c: 700 x 7 =
Similarly, to above we carry out this operation 7 x 7 = 49 and add the zeros so
700 x 7= 4900
Here are some example problems for you to practice. Remember you can check your solutions with a calculator but only after you’ve had a go!
1d: 100 x 9 =
1e: 900 x 1 =
1f: 600 x 4 =
1g: 400 x 6 =
1h: 700 x 5 =
Example 2: Increasing Difficulty
In the first example we looked at the simplest case of multiplying a three-digit number by a single digit number. In example two we will look at three-digit numbers with a zero at the end multiplied by a single digit number.
2a: 250 x 5 =
Here we have to carry out two multiplication operations and add up the results as such:
250 x 5 = (200 x 5) + (50 x 5) + (0 x 5) =
0 x 5 = 0
50 x 5 = 250
200 x 5 = 1000
Therefore 250 x 5 = 1000 + 250 = 1250
Usually when we write long multiplication it is often done on a vertical axis to allow you to keep track of adding the extra zeros like below:
0 In this row we multiply the 5 by the zero
+250 This row shows the result of multiplying the 5 by 50
1000 This row is the result of multiplying 5 by 200
1250 Here we can easily do the sum as we have already lined up the numbers
The extra zeros are highlighted in bold text
2b: 320 x 4 =
This time lets use the vertical method straight away to calculate this example
0 (4 x 0) = 0
+ 80 (4 x 20) = 80
1200 (4 x 300) = 1200
1280 The sum of the multiplications
Here are some numbers for you to crunch. We recommend practicing the vertical method.
2c: 150 x 3 =
2d: 420 x 2 =
2e: 640 x 7 =
2f: 830 x 3 =
2g: 910 x 9 =
You’re now starting to get a grasp of long multiplication. Remember this is all a lead up to multiplying two three-digit numbers but we need to know the basics well in order to do that.
3a: 123 x 3
9 (3 x 3) =9
+ 60 (20 x 3) = 60
300 (100 x 3) = 300
Example 3a is the easiest type of multiplying a single number by a three-digit number without any zeros. Apart from keeping track of the zeros correctly, there is not any carrying over required in adding up each component in this example. Let’s look at an example where the multiplication of each component requires a little more finesse.
3b: 467 x 6 =
Luckily the final addition is very easy if you line up the numbers properly. We’ve again highlighted the extra zeros in bold and suggest you write these first as you do your multiplication.
Here are a few examples for you to solve:
3c: 112 x 3 =
3d: 234 x 2 =
3e: 467 x 2 =
3f: 793 x 7 =
Multiplying Two-Digit Numbers by Three-Digit Numbers
If you have got the previous step down and can multiply any three-digit number by a single digit number, then you’re actually about 80% there for getting to three digits by three digits. The difficulty increases but the method is relatively similar.
Here we’ll go through two digits by three digits first to ease you in.
4a: 323 x 22 =
We first need to explain a change in notation here. Above we broke down each multiplication step onto its own row, however now we will need to be more conscience and so each row will have to contain a larger fraction of the operation as such; the first row will contain 2 x 323 and the second row will have the answer for 20 x 323 as such:
646 this row = 2 x 323 = (2 x 3) + (2×20) + (2 x 300)
+ 6460 this row = 20 x 323 = 10 x 2 x 232 = 10 x (2 x 323)*
*By taking the 10 out of the bracket and putting it in as the extra zero (in bold) this multiplication becomes identical to 2 x 323. This is the secret to long multiplication. Breaking down the components and making sure to preserve zeros.
We are now ready to multiply three-digit numbers but first here are some examples for your practice:
4b: 242 x 12 =
4c: 535 x 72 =
4d: 783 x 83 =
Multiplying three-digit numbers
You’ve made it this far, well done, it’s not easy to say concentrated this long while on the internet. Your reward is that you have now nearly mastered long multiplication. We will go through some examples of multiplying three-digit numbers now. Hopefully this will be relatively easy for you if you’ve followed the examples and completed the exercises. If you were in our classroom, you’d definitely have earned a few gold stars!
Here we go!
Let’s start easy shall we?
5a: 200 x 400= 400
80000 since the first two multiplications = zero, we have two extra zeros and multiply 2 x 400
Et viola! You have multiplied your first three-digit numbers.
Here are a few for you to practice:
5c: 300 x 300 =
5d: 600 x 200 =
5e: 700 x 800 =
Nearly there. let’s try some more difficult ones:
6a: 320 x 240 = 320
12800 this row is the 40 x 320, however we simplify by carrying over the zero
+ 64000 adding the two extra zeros, we multiply 200 x 320
Note that each operation might need to be taken apart in multiple steps, as we have previously explained 320 x 240 = (320 x 40) + (320 x 200). However, you may need to write out your multiplications to each section/step before adding it all up if you’re new to this.
The more you practice the easier it’ll get.
Here are your penultimate set of exercises for this tutorial:
6b: 430 x 520 =
6c: 730 x 490 =
6d 234 x 200 =
6e: 123 x 320 =
If you managed all the above exercises you are now ready to multiply two three-digit numbers with non-zero elements. Congratulations!
7a: 367 x 245 367
1835 this row = 5 x 367
+ 14680 this row = 40 x 367
73400 this row = 200 x 367
As you can see each multiplication is difficult in its own right when multiplying numbers of this type. Writing out and lining up the extra zeros will save you a lot of mistakes. As with everything practice makes perfect.
Well done for making it to the end of the tutorial. Here is the last problem set to practice.
7b: 123 x 321 =
7c: 250 x 535 =
7d: 789 x 999 =
We find that simplifying arithmetic as much as possible helps out in real life. If your problem can be made easier then use the easy way unless it’s an exam!
Here is an example: say we want to solve this problem:
896 x 483 Multiplying this beast would be very time consuming, however;
896 x 483 = (900 x 483) – (4 x 483)
Although we have to make two multiplications now, they are both substantially easier than the original one, and subtracting at the end shouldn’t be too difficult.
Good luck with all your long multiplication we hope we have been of help. Honestly if you’ve made it this far you deserve a pat on the back and a treat of your choice. Math is tons of fun, but long multiplication is on the tedious spectrum.