Mental Arithmetic 003
Sep. 23rd, 2010 04:02 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
fiat_knoxSo here we are, then, with the next stage of the lessons - bringing together all of those exercises in halving digits, doubling them and adding 5 in the column is odd that we did the last few times.
First, some digit exercises to get you going. Here's a table of digits. First, go through each one, double it and say out loud what the doubled digit is. Don't just say "Twice 4 is eight," just see 4 and say "Eight."
Single Digits
Now go back to the above table, only now halve the digit. See 4 and say "2;" see 9 and say "4."
Next, the table below has a list of odd numbers. Halve each number and add 5 to it; when you see 3, don't just say "half of 3 is 1, add 5 to yield 6." See 1, say "5;" see 3, say "6;" see 5, say "7," and so on.
Now here's the rule for multiplying by 5.
When multiplying each column by 5, just write down half its neighbour.
Add 5 if the column is odd.
Let us try it with the following multiplication - 345 x 5.
The first digit is easy - half of 0 (the neighbour) is 0, and add 5 because the column is odd. The first figure is 5:-
With the second column, write down half the neighbour - 5, yielding 2.
The rest of the calculation proceeds with the same rules:-
The product is 1,725.
Lastly, here's the rule for multiplying by 7:-
When multiplying each column by 7, double each column and add half its neighbour.
Add 5 if the column is odd.
Let us try it with the following multiplication - 345 x 7.
Let's take the first column, 5. Double that - 10 - and add half its neighbour. Since the neighbour doesn't exist, take it as 0, so the sum is now 10. Finally, add 5 because the column is odd, yielding 15:-
Now move on to the rest of the number:-
The answer is 2,415.
Now try the above on on the following numbers:-
So now you have learned the techniques you'd need to do basic multiplications by 5, 6, 7, 11 and 12. Halving and doubling digits, adding the product to the neighbour, adding 5 if the column is odd; these are simple enough.
Multiplying by other numbers - 3, 4, 8 and 9 - is a little more involved, a process requiring complements of 9 and of 10. We'll go into that in the next installment.
But first, something I've been wanting to demonstrate since the start of these lessons. And I'll start with a question. Without a calculator, can you tell me what 75 x 75 is?
Not only will I be able to show you how to work it out in a split second; I will show you a neat tool to allow you to work out the answers to similar calculations such as 45 x 45, 95 x 95 and even 113 x 117.
All these await in the interlude.
First, some digit exercises to get you going. Here's a table of digits. First, go through each one, double it and say out loud what the doubled digit is. Don't just say "Twice 4 is eight," just see 4 and say "Eight."
| 6 | 0 | 2 | 5 | 1 | 3 | 8 | 0 | 4 | 7 |
| 5 | 3 | 9 | 0 | 6 | 5 | 2 | 1 | 4 | 7 |
| 6 | 7 | 4 | 0 | 8 | 3 | 2 | 9 | 5 | 1 |
| 8 | 0 | 9 | 7 | 6 | 3 | 4 | 1 | 8 | 2 |
| 3 | 5 | 1 | 8 | 6 | 9 | 7 | 9 | 2 | 4 |
Now go back to the above table, only now halve the digit. See 4 and say "2;" see 9 and say "4."
Next, the table below has a list of odd numbers. Halve each number and add 5 to it; when you see 3, don't just say "half of 3 is 1, add 5 to yield 6." See 1, say "5;" see 3, say "6;" see 5, say "7," and so on.
| 5 | 3 | 9 | 1 | 7 | 9 | 7 | 3 | 5 | 1 | 9 | 5 | 1 | 7 | 3 |
Now here's the rule for multiplying by 5.
Add 5 if the column is odd.
Let us try it with the following multiplication - 345 x 5.
| 0 | 3 | 4 | 5 | x 5 |
The first digit is easy - half of 0 (the neighbour) is 0, and add 5 because the column is odd. The first figure is 5:-
| 0 | 3 | 4 | 5 | x 5 |
| 5 |
With the second column, write down half the neighbour - 5, yielding 2.
| 0 | 3 | 4 | 5 | x 5 |
| 2 | 5 |
The rest of the calculation proceeds with the same rules:-
| 0 | 3 | 4 | 5 | x 5 |
| 7 | 2 | 5 |
| 0 | 3 | 4 | 5 | x 5 |
| 1 | 7 | 2 | 5 |
The product is 1,725.
Lastly, here's the rule for multiplying by 7:-
Add 5 if the column is odd.
Let us try it with the following multiplication - 345 x 7.
| 0 | 3 | 4 | 5 | x 7 |
Let's take the first column, 5. Double that - 10 - and add half its neighbour. Since the neighbour doesn't exist, take it as 0, so the sum is now 10. Finally, add 5 because the column is odd, yielding 15:-
| 0 | 3 | 4 | 5 | x 7 |
| 15 |
Now move on to the rest of the number:-
| 0 | 3 | 4 | 5 | x 7 |
| 11 | 15 |
| 0 | 3 | 4 | 5 | x 7 |
| 14 | 11 | 15 |
| 0 | 3 | 4 | 5 | x 7 |
| 2 | 14 | 11 | 15 |
The answer is 2,415.
Now try the above on on the following numbers:-
| 08295 | 09922 | 01666 | 08602 | 09132 | 06556 | 07078 | 09709 | 02930 | 08240 |
| 02737 | 07688 | 03580 | 05584 | 05142 | 04683 | 05616 | 03905 | 08808 | 05275 |
| 06757 | 09965 | 09897 | 03830 | 06388 | 05187 | 01451 | 02731 | 05777 | 03304 |
So now you have learned the techniques you'd need to do basic multiplications by 5, 6, 7, 11 and 12. Halving and doubling digits, adding the product to the neighbour, adding 5 if the column is odd; these are simple enough.
Multiplying by other numbers - 3, 4, 8 and 9 - is a little more involved, a process requiring complements of 9 and of 10. We'll go into that in the next installment.
But first, something I've been wanting to demonstrate since the start of these lessons. And I'll start with a question. Without a calculator, can you tell me what 75 x 75 is?
Not only will I be able to show you how to work it out in a split second; I will show you a neat tool to allow you to work out the answers to similar calculations such as 45 x 45, 95 x 95 and even 113 x 117.
All these await in the interlude.
