# Multiplication

Multiplying means adding the same number to itself a number of times:-

4 × 3 = 12 (4 times 3 is 12): this means 3 + 3 + 3 + 3 = 12.

4 × 4 = 16 (4 times 4 is 16): this means 4 + 4 + 4 + 4 = 16.

7 × 9 = 63 (7 times 9 is 63): this means 9 + 9 + 9 + 9 + 9 + 9 + 9 = 63.

You can also say four threes are 12, four fours are 16 and seven nines are 63.

Or you can even say 4 multiplied by 3 is 12, 4 multiplied by 4 is 16 and 7 multiplied by 9 is 63.

If you want to buy five toys, each costing four pounds, you can work out the total cost either by adding four fives together or five fours: the answer will be the same either way (20). Adding a number to itself many times is something we do a lot of, but because we're lazy, we use lots of simple tricks to find all the answers so that we don't have to bother adding them up. Look at the table of numbers below. You can find the answer to the sum 4 × 3 simply by finding the 4 in the top row and the 3 in the first column, and then you run one finger down from the 4 and another finger along from the 3 until they meet at the answer (so the answer to 4 × 3 is 12):-

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

If you look up 7 × 9 and 9 × 7 you will see that they both give the same answer as each other: 63. This happens with any pair of numbers that you are multiplying together, so it doesn't matter which of them you look up in the top row and which of them you look up at the side.

Let me remind you: if you're reading this on a computer with a small screen you'll be able to see much more of the page at a time by pressing the F11 key. Pressing that same key again will return things to normal afterwards.

Now, ideally you would memorise the whole table, and most people eventually manage to do just that, but there is no need to put any effort into doing this because there are lots of tricks you can use to work out the answers quickly, and you will eventually memorise all the answers by accident, just in the course of doing lots of sums.

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

Look at the row beginning with 9, or the column topped by a 9. All these orange numbers can be worked out without any effort at all: hold your ten fingers up in front of you; put one finger down (any one at all); count the fingers to the left of it; and then count the fingers to the right of it. Can you see how this fits with the table? If you want to multiply 5 × 9, hold down the fifth finger. There will be four fingers to the left of it and five fingers to the right: so that's 4 and 5. The answer is 45. Try another one: for 8 × 9 you hold down the eighth finger; count the fingers to the left of it (7) and to the right (2), and so the answer is 72. So multiplying by nine is dead easy!

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

Multiplying by 10 is even easier: you just write a zero onto the end of the other number, so 4 × 10 = 40 and 7 × 10 = 70.

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

Multiplying by 1 is even easier: any number multiplied by 1 will just stay exactly as it is, so 1 × 6 = 6.

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

Multiplying by two always means that the other number is simply added onto itself, so to do the sum 2 × 5 you can simply add 5 + 5 instead, and for 2 × 9 you can just add 9 + 9.

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

The trick with multiplying by five is to learn to count up in fives: 5, 10, 15, 20, 25, and so on. If you want to know what 7 × 5 is, you can hold up seven fingers and then count up in fives as you look along them: think 5 when you look at the first, 10 when you look at the second, 15 when you look at the third, and so on until you have reached the seventh finger (the last one), by which time your count will have got to 35, and this is the right answer (7 × 5 = 35). If you find it hard to count up in fives, you will have to practise it over and over again until you get the hang of it.

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

Multiplying by four is a bit harder than multiplying by 5, but it's still quite easy: the pattern repeats after 20 (if you look at the second digit of each number in the sequence). To do 6 × 4, hold up six fingers and count up in fours as you go along them: 4, 8, 12, 16, 20, 24. So the answer to 6 × 4 is 24. Again you must practise this until you get the hang of it. Read out the row of blue numbers, then look away and try to say them all again, then look at it and read them out again, then look away and try to say them all again, and keep doing this until you can do it with ease. You want to memorise this sequence of numbers so that you can run through it effortlessly without having to think.

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

Now we've reached the hardest one, but it's actually not too difficult: multiplying by three is again just a matter of counting up in threes. You should try to memorise the coloured row of numbers so that you can run through it without any effort. To do 3 × 7, you must hold up seven fingers, then work your way along the line of fingers and count up in threes: 3, 6, 9, 12, 15, 18, 21. So the answer to 3 × 7 is 21.

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30
36 42 48 54 60
7 14 21 28 35
42 49 56 63 70
8 16 24 32 40
48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

You now know easy ways to work out all the blue answers in the table. We now only have to find ways to deal with the remaining square of black numbers. Look at the table below:-

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

Whenever 6 is multiplied by an even number, the answer will have that same even number as its second digit, while its first digit will be half of that value. So 6 × 2 will have an answer with 2 as its second digit and half of 2 (which is 1) as its first digit: 12. The same happens for 6 × 4: the second digit of the answer must be 4, while the first digit must be half that (2), so the answer is 24. See how it works for 6 × 6: the second digit of the answer must be 6, the first digit must be half that (3), so the answer is 36. And 6 × 8: the second digit of the answer must be 8, the first digit must be half that (4), so the answer is 48. so we have a rule: six times any even number = half that number as the first digit and the whole of that number as the second digit.
indentThe only other difficult number to multiply by 6 is 7. The easiest way to remember the answer for this is to start with the smaller of those two numbers, the 6, and then count down two from there to get the first digit of the answer, and then count down another two to get the second digit: so you start from 6, going down two gives you 4, then going down two more gives you 2, so the answer is 42. Memorise this: 6 × 7 =... (start at 6, go down two to get 4, go down two more to get 2)... 42.

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

The coloured numbers here are the answers to 1 × 1, 2 × 2, 3 × 3, 4 × 4 and so on. It is a good idea to learn this sequence of numbers by memory (1, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100), though you don't have to if you don't want to. You can see that 10 × 10 = 100. It is interesting that 5 × 5 = 25, because that means that half of 10 times half of 10 equals a quarter of 100. Now look at the answer to 7 × 7. It's very nearly 50, but it just falls short with 49. So, a good way to remember the answer to 7 × 7 is that it tries its best to be 50, but it just can't quite make it. 7 × 7 = 49 (which is nearly 50).
indent8 × 8 = 64 can be solved in the same way as 6 × 7. Its first digit is 2 less than 8, and its second digit is 2 less than that, so that gives you a 6 and a 4. Memorise this: 8 × 8 =... (start at 8, go down two to get 6, go down two more to get 4)... 64.
indentThe only one left in the table now is 7 × 8, but it just so happens that there is a simple way to remember that the answer is 56: just think 5 6 7 8. There's the answer followed by the question. So, 7 × 8 =... (just think 5, 6, 7, 8)... 56.

There is one final rule to learn: any number × 0 = 0 (for example, no fives are none).

You have now seen how all the answers in the table can be worked out quickly, but you need to practise using these tricks. The best way to do this is with a game, and the game I've put together is a race against the clock. The quicker you complete the game, the higher your mental age for maths will be. Every time you play the game you will be reminded of the tricks you need to learn, so you'll find that the game rapidly becomes easier and your mental age will shoot up dramatically to years above your actual age. To play the game, click on this link.

Once you feel that you're getting good at the game, you are ready to learn how to do multiplication with bigger numbers, though you might still want to copy out the following table on a piece of paper so that you can look it up easily:-

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

Multiplication sums are written out in much the same way as adding and taking-away sums. The sum below shows you how to solve 4 × 12:-

12
× 4

Click on these buttons to solve the sum: 4 × 2 = 8, 4 × 1 = 4. The 4 has to be multiplied by each of the digits of the bigger number in turn, and the answers must always be placed in the same column as the digit being multiplied by the 4. Press this button to reset the sum, then click the first two buttons again to have another look at what happened.

Let's do another one, but with a bigger top number (2543 × 3). Watch what happens with the overflowing digits when the answers are too big to fit in the space:-

2543
×  3

Click these four buttons: 3 × 3 = 9,  3 × 4 = 12,  3 × 5 (=15) + 1 = 16,  3 × 2 (=6) + 1 = 7. Press this button to reset the sum, then click the four buttons again to have another look at what happened.
indentSo overflow digits are dealt with in the same way as with adding: they have to be added onto the answer to the next column. Click the reset button and go through it all again, making sure you understand what happens as you press each button.

Here's another one for you to try for yourself. See if you can work out the answer for each column before you press the button for it. If you click the wrong button, just click the reset button and start again:-

9964
×   5

5 × 4 = 20,  5 × 6 (=30) + 2 = 32,  5 × 9 (=45) + 3 = 48,  5 × 9 (=45) + 4 = 49. To start again, click: . Click through the sum several times to make sure you understand it properly.

You can now practise what you have just learned by playing another game (click on this link), though you should also make sure you can do multiplication sums on paper with a pencil.

The way to multiply big numbers by big numbers (eg. 34 × 62) will be looked at later.