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Lesson 6

# HOW TO ADD WHOLE NUMBERS AND DECIMALS

## Addition with Regrouping

The student should by now have mastered elementary addition.

In this Lesson, we will answer the following:

We can only add things that have the same name, which we call the units.  That is why, when we add a column of numbers, we align the ones in a column, the tens in a column, the hundreds, and so on for each unit.

 2,364 = 2 thousands + 3 hundreds + 6 tens + 4 ones + 5,132 = 5 thousands + 1 hundred  + 3 tens + 2 ones 7,496 = 7 thousands + 4 hundreds + 9 tens + 6 ones

Then, starting with the ones on the right, we add each column.

Now, that is the simplest example because no column adds to more than 9. But consider this example where there are more than 9 ones. There are 12:

 38 + 4

8 + 4 is 12 -- but there is no digit for 12. The largest digit we can write is 9.  Therefore we must group 10 ones of 12 into 1 ten --

-- and regroup it with the 3 tens:

We now have

4 tens + 2 ones = 42.

In practice, simply write 2 under the line --

-- and "carry" the 1 onto the tens column.  Say:

"8 plus 4 is 12."  (Write 2, carry 1.)  "3 plus 1 is 4."

That is the procedure for adding in columns. It's called addition with regrouping.

 1. How do we add whole numbers in writing? 4,674 1,422 5,533 3,840 15,469 Write the numbers under one another, taking care to align the same units; that is, align the ones, the tens, etc.; and draw a line. Then, starting with the ones on the right, add each column. When the sum of a column is 9 or less, write that sum under the line. But when that sum is more than 9, write the ones of that sum and regroup -- "carry" -- the tens digit onto the next column. For, we may compose 10 of a lower unit into 1 of the next higher unit.

 4,674 1,422 5,533 3,840 15,469
 2 1 4 thousands + 6 hundreds + 7 tens + 4 ones 1 thousands + 4 hundreds + 2 tens + 2 ones 5 thousands + 5 hundreds + 3 tens + 3 ones 3 thousands + 8 hundreds + 4 tens + 0 ones 15 thousands + 4 hundreds + 6 tens + 9 ones

On adding the ones column, we get 9.  But on adding the tens column, we get 16.

Write 6, and carry the 1 onto the hundreds column -- because 10 tens are equal to 1 hundred.  Therefore when we add the hundreds, we have

1 + 6 + 4 + 5 + 8 = 24 hundreds.

(See the previous Lesson, Example 3.)

Write 4, and carry the 2 onto the thousands column -- because 20 hundreds are equal to 2 thousands. (Lesson 2, Example 3c.)

When we add the thousands, then, we get

2 + 4 + 1 + 5 + 3 = 15.

The sum of those numbers is 15,469.

 Example 1. 9545 + 8982 18527

"5 + 2 is 7."  Write 7.

"4 + 8 is 12."  Write 2 -- carry 1.

"5 + 9 is 14, plus 1 is 15."  Write 5 -- carry 1.

"9 + 8 is 17, plus 1 is 18."  Write 18.

 2. How do we add decimals? Add decimals the same way as whole numbers, taking care to align the decimal points. Equivalently, align the ones because every number has ones.

Example 2.   Write in a column and add:  4,785 + 9 + 2.307

Solution..  The ones are shown:

4,785 + 9 + 2.307

(Lesson 3, Question 5.)  Therefore, align as follows:

 4,785 9 +     2 .307 4,796 .307

Example 3.   Write in a column and add:  .58 + 5.8 + 58

Solution.  Here are the ones:

_.58 + 5.8 + 58

As for .58, the ones are at the first place to the left of the decimal point.

Align as follows:

 .58 5 .8 + 58 64 .38

When the decimal points are aligned, the decimal point in the answer will fall in the same place.  (But that is true only in addition and subtraction, not in multiplication.   Lesson 10.)

As for a whole number such as 58, to help with alignment we may imagine a decimal point after the 8.

58 = 58.

Whole numbers, however, are normally written without a decimal point, because the decimal point means "and." Here come the fractions

Example 4.    .5 + .5 + .5

Choose the correct answer:

a)  15

b)  .15

c)  1.5

d)  .015

For, if we aligned them and wrote .5 as 0.5 --

 1 0.5 0.5 + 0.5 1.5

-- we would see that the 1 (of 15 tenths) carries over into the next column.

The point is:

will have the same number of decimal digits as the numbers themselves.

Example 5.   .007 + .003 + .004

The numbers being added have three decimal digits.  Therefore the answer also will have three decimal digits.

Example 6.   Perimeters.   The perimeter of a plane (flat) figure is its boundary.

This figure is a rectangle, which is a four-sided figure in which all the angles are right angles.  In a rectangle, the opposite sides are equal. Therefore the perimeter of that rectangle is:

 12 + 12 + 6.5 + 6.5 = 24 + 13 = 37 in.

 2 0.83 7 0.49 6 0.26 +  8 0.58

Technique.  Do not rewrite the problem by writing each column as a separate sum.  To add the column on the right, do not say -- or write -- "3 plus 9 is 12.  12 plus 6 is 18."  And so on.  Rather, let your eye go down the column and say each partial sum. Do not write it. (See Lesson 5, Question 2, Example 3.)

To add the column on the right, say

 2 0.83 7 0.49 "12" 6 0.26 "18" +  8 0.58 "26" 6

Write 6, carry 2.   To add the middle column, say

 2 2 0.83 "10" 7 0.49 "14" 6 0.26 "16" +  8 0.58 "21" 0.16

Write 1, carry 2.   To add the last column, say

 2 "4"  2 .83 "11"  7 .49 "17"  6 .26 "25"  8 .58 25 .16

Write 25.

You may write the decimal point in the answer when you come to it; that is, after adding the middle column.

Please "turn" the page and do some Problems.

or

Continue on to the next Lesson.

1st Lesson on Adding Whole Numbers