Joe Doe (joe_doe_zero) wrote in 0_arithmetic,
Joe Doe
joe_doe_zero
0_arithmetic

General adding - lesson 2

Adding numbers with more than one digit



We know to a good extent how to add numbers containing only one digit. But let's say we want to add numbers with more than one digit.

We want to add: 10 + 10

While we could look it in the adding chart and find the answer (it's there!), it would be too impractical to have an adding chart for every single number. Therefore we need a more general method to add numbers rather than looking it up in an adding chart.

10 + 10 = 20.

Does this problem resemble the problem 1 + 1 = 2? Let's take a closer look and compare the two problems:

1 + 1 = 2
10 + 10 = 20

Instead of adding by ones, we added by tens in the last problem. Let's try another:

10 + 20 = ?

We find [in our adding chart] that 1 + 2 = 3. Using the same type of logic we find that

10 + 20 = 30

What about 50 + 30 = ?

5 + 3 = 8
Therefore 50 + 30 = 80.

Could we use the same with bigger numbers? Yes, we can.

100 + 300 = ?

From the adding chart, 1 + 3 = 4

Therefore 100 + 300 = 400.

What if we wanted to add 100 + 30?

Can we use 1 + 3 = 4 for this problem, too? No, we cannot. 30 is a smaller number than 300. Therefore

100 + 30 is NOT equal to 100 + 300

If we cross out a zero from each numeral.

100 + 30

We get 10 + 3

10 + 3 = 13

Now add back a zero to the answer.

100 + 30 = 130

Let's try another: 20 + 7 = ?

Remember our "breaking" technique from the previous lesson? 20 can be broken into 10 + 10

10 + 10 + 7
10 + 7 = 17
17 + 10 = 27

How exactly did we add 17 and 10? This problem is actually two addition problems in one:

17 + 10 = ?

1 + 1 = 2
7 + 0 = 7

17 + 10 = 27

1 + 1 = 2 (each 1 is actually a 10)
7 + 0 = 7

Another problem: 26 + 20 = ?

2 + 2 = 4 (each 2 is actually a 20)
6 + 0 = 6

26 + 20 = 46

Let's try a more difficult one: 140 + 45 = ?

Think of the number 45 as 045. The number is not changed, you're merely adding an imaginary zero to the front of it.

So the problem becomes 140 + 045 = ?
1 + 0 = 1 (each digit is actually a hundred)
4 + 4 = 8 (each digit is actually a ten)
0 + 5 = 5 (each digit is a one)
140 + 045 = 185
140 + 45 = 185.

Let's do an even more difficult one: 132 + 66
66 becomes 066

So the problem becomes 132 + 066 = ?
1 + 0 = 1 (each digit is a hundred)
3 + 6 = 9 (each digit is a ten)
2 + 6 = 8 (each digit is a one)
132 + 66 = 198.

Another one: 133 + 256 = ?
1 + 2 = 3
3 + 5 = 8
3 + 6 = 9
133 + 256 = 389

Problems in real life are not always as easy (or as pretty) as the ones presented above.

Let's try: 356 + 289 = ?

3 + 2 = 5
5 + 8 = 13
6 + 9 = 15
But the answer is not 51315. Let's go back to an earlier problem we worked:

133 + 256 = ?
1 + 2 = 3 (each digit is a hundred)
3 + 5 = 8 (each digit is a ten)
3 + 6 = 9 (each digit is a one)

What does it mean when each digit is a hundred? When we were adding 1 + 2, in essence we were adding 100 + 200 and got 300. When we were adding 3 + 5, we were actually adding 30 + 50 to get 80. We added the 300 + 80 + 9 to get 389.

133 + 256 = 389

Now let's take a look at this problem:
3 + 2 = 5
5 + 8 = 13
6 + 9 = 15

The 5 is actually a 500; the 13 is actually a 130; the 15 is a 15. So we have

500 + 130 + 15 = ?
Turn the 15 into 015 and remember how we add more than two numbers.
5 + 1 + 0 = 6
0 + 3 + 1 = 4
0 + 0 + 5 = 5

Since the 6 is actually a 600; 4 is a 40; 5 is a 5, we get 645 as our answer.

356 + 289 = 645.

Another, just for more practice: 781 + 998

7 + 9 = 16
8 + 9 = 17
1 + 8 = 9

16 is really 1600; 17 is really 170; 9 is really a 9.

1600 + 170 + 9
Translate: 1600 + 0170 + 0009

1 + 0 + 0 = 1
6 + 1 + 0 = 7
0 + 7 + 0 = 7
0 + 0 + 9 = 9

781 + 998 = 1779.

The technique we just outlined is called "carrying the 1". If we have a sum in a column that is greater than ten (usually between 10 and 19, inclusive) we only count the lowest digit and add a one to the column on the left.

Say we want to add: 25 + 37

We simply add 5 + 7 to get 12 and then "carry the 1" from the 12 and then add the 1 to 2 + 3.

The ones digit will be 2 and the tens digit will be computed by 1 + 2 + 3 = 6

Thus giving us an answer of 62.

Finally, let's go for something that is sure to get our minds going:

34,768 + 8,051*

34768 + 08051

3 + 0 = 3
4 + 8 = 12
7 + 0 = 7
6 + 5 = 11
8 + 1 = 9

3 = 30,000; 12 = 12,000; 7 = 700; 11 = 110; 9 = 9.

30,000 + 12,000 = 42,000
700 + 110 = 810
9 = 9
42,000 + 810 + 9

810 + 9 = 819
42,000 + 819 = 42,819

34,768 + 8,051 = 42,819

And now you know how to add all whole numbers.

*When numerals have more than three digits a comma is placed between every three digits. Thus 10000000000 would be written as 10,000,000,000.
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