Why multiply?
The easiest number to multiply by is 10. Just “add 0.”
3 x 10 = 30 140 x 10 = 1400
Isn't that easy? This “trick” is really just using our number system. 3 means “3 ones.” Move 3 once to the left and you get 30, which means, “3 tens.” See how our numbers cycle in tens? Whenever you move the digits once to the left, that's the same as multiplying by 10.
And that's the quick way to multiply by 10. Move each digit once to the left. Fill the last place with a 0. Easy.
See the pattern? When we multiply a singledigit number times 9:
This makes it easy to multiply any single digit times 9.
Suppose you want to multiply 5 times 9. First, subtract 1 from the original number to get the tens digit.
5  1 = 4 tens digit of answer
Then subtract this tens digit from 9 to get the ones digit.
9  4 = 5 ones digit of answer
So the answer is 45. Let's double check. Do the digits add up to 9?
4 + 5 = 9
Yes! Isn't this a great trick?
Remember, it only works for single digits. Don't try it on 13 x 9 or 6,425 x 9!
Is there a trick to multiply by 5? Let's look at a few facts:
2 x 5 = 10 3 x 5 = 15
4 x 5 = 20 5 x 5 = 25
6 x 5 = 30 7 x 5 = 35
See a pattern? If we multiply by an even number, the ones digit is 0. If the number's odd, the ones digit is 5.
So what's the shortcut? Look at the tens digit. If we multiply by an even number, the tens digit is half that number. The ones digit is always 0.
2 x 5 = 10 2 / 2 = 1
4 x 5 = 20 4 / 2 = 2
6 x 5 = 30 6 / 2 = 3
What if we multiply by an odd number? First subtract 1 from that number. Then take half the answer, and that's the tens digit. The ones digit is always 5.
3 x 5 = 15 3 – 1 = 2 2 / 2 = 1
5 x 5 = 25 5 – 1 = 4 4 / 2 = 2
7 x 5 = 35 7 – 1 = 6 6 / 2 = 3
So here's the shortcut:
To multiply 5 by an even number: Get the tens digit by dividing the number by 2. The ones digit is 0.
To multiply 5 by an odd number: Subtract 1 from the number. Get the tens digit by dividing that answer by 2. The ones digit is 5.
You can use this to check your work, too. What if you multiply 5 by 3 and get 20? Since 3 is an odd number, your answer should end in 5, not 0. You know you made a mistake.
Remember the Amazing Facts of Nine? When you multiply by 9, the digits of the answer eventually add up to 9.
8 x 9 = 72 7 + 2 = 9
So how about that number 9? It's 3 times 3, isn't it? Let's see if 3 has any special properties.
4 x 3 = 12 1 + 2 = 3
5 x 3 = 15 1 + 5 = 6
6 x 3 = 18 1 + 8 = 9
7 x 3 = 21 2 + 1 = 3
Whoa! When you multiply by 3, the digits of the answer add up to 3, 6, or 9.
You can't really use this to multiply faster. But it is a quick way to check your work. Say you multiply 11 by 3 and get 34. Well, 3 + 4 = 7. Oops. You must have made a mistake. The right answer is 33. And 3 + 3 = 6.
Of course, the trick can only show whether you're wrong. It can't prove you're right. Let's say you multiply 11 x 3 and get 36. Well, 3 plus 6 does equal 9, but 36 is still wrong.
Still, this is a neat trick. If you multiply any number by 3, the digits of the answer add up to 3, 6, or 9. Even big numbers.
524 x 3 = 1572
1 + 5 + 7 + 2 = 15 and 1 + 5 = 6
91,317 x 3 = 273,951
2 + 7 + 3 + 9 + 5 + 1 = 27 and 2 + 7 = 9
You can see that math has patterns. Thanks to patterns and cycles, the digits of your answer have to add up to three. Math works from every angle. That's what's so cool about it.
It's easy to multiply by 10, isn't it? Which problem would you rather do?
20 x 7 = ? 19 x 7 = ?
The first, right? Maybe you figured it out just looking at it: 140. You forget about the 0 and multiply 2 by 7. Easy. But the second problem is a real problem. You'll have to use a pencil and paper to get the answer: 133. Rather hard.
Well, you can use the easy problem as a shortcut to the harder problem.
Twenty sevens is 140. Nineteen sevens is one less seven than 140. You don't have to figure out 19 x 7. You can jump from 20 x 7 to 140. Then go back to 19 x 7 by subtracting 7 from 140.
19 x 7 = ? Ugh.
20 x 7 = 140 This is a close, easier answer.
140 – 7 = 133 Now subtract the extra seven...
19 x 7 = 133 And you're at the right place.
So you can use an easier problem as a shortcut, then add or subtract the difference.
A similar trick is to multiply by “chunks.” First multiply the tens, then multiply the ones, then add these sums together. This is all you really do on paper, but you might not realize it.
32 x 8 = ? Yikes.
30 x 8 = 240 Multiply the tens.
2 x 8 = 16 Multiply the ones.
240+16 = 256 Add the two answers.
32 x 8 = 256 And we have our final answer.
If you break a problem into smaller chunks, you can often do it more quickly than if you try to do the whole thing at once.
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