Lesson 7

SUBTRACT WHOLE NUMBERS
SUBTRACT DECIMALS

To subtract means to find what number we must add to the smaller number. (Lesson 6.) Thanks to the calculator, to find that number by a written method is fast becoming a historical subject.  Nevertheless we will present two such methods. One is the familiar subtraction with regrouping ("borrowing").  The other is the less familiar but simpler, subtraction by compensating.   It is taught in many counties, and it has the virtue of exploiting the property stated in the previous Lesson:

We will present that method first.  The student should be clear about the place value of each digit (Lesson 1).

But we cannot take 7 tens from 4 tens.   Therefore, we add 10 tens to 4 tens, and take 7 from 14.   To compensate, we add 1 hundred to 8 in the bottom number -- because 1 hundred is equal to the 10 tens we added to 4.  (Lesson 1).

9 hundreds from 13 hundreds are 4 hundreds.  

6 thousands from 8 thousands are 2 thousands.  

Here is the rule:

"9 from 12 is 3."

"8 from 11 is 3."

"9 from 13 is 4."

"3 from 5 is 2."

The top digits  5312  never change.  That is the simplicity of this method.  Only a bottom digit might change.  

Finally, since subtracting is finding what number to add to the smaller number, the student should always check the answer by adding.

3 + 9 is 12.   3 + 7 is 10, plus 1 is 11.   And so on.  

Note that we must add 0's onto the right of 2.1 (Lesson 2).  Both numbers must have the same number of decimal places.

On checking:  .867 plus 1.233 is 2. 100.

Example 5.    Subtract  .698  from  5

Solution.  This problem means

5 − .698

.698 is the smaller number.  Write it below 5.000:

(Lesson 1, Question 8.)

We cannot take 6 ones from 2 ones.  Therefore, we must decompose 1 of the 4 tens -- break it up -- into 10 ones, and regroup them onto the ones column.

We now have 3 tens and 12 ones.  "6 ones from 12 ones are 6 ones."

Next, we cannot take 9 tens from 3 tens.  So we decompose 1 of the 5 hundreds into 10 tens, and again regroup.  We now have 4 hundreds and 13 tens:

"9 tens from 13 tens are 4 tens."

Finally,

"1 hundred from 4 hundred is 3 hundred."

We can always check the difference  346  by adding it to the smaller number.  "6 plus 6 is 12.  4 plus 9 is 13, plus 1 is 14." And so on.

Now, this is the concept of subtraction with regrouping.  In actual practice, the student should try not to write the crossings out or the regrouped numbers.  The student should practice those steps mentally.  The answer can always be checked by adding.

In other words, with practice the student should be able to say:

"6 from 12 is 6."

"9 from 13 is 4."

"1 from 4 is 3."

We will do the following example completely in the expanded form.

We cannot take 4 ones from 0 ones, and we cannot decompose 0 tens.  We must move left and start decomposing at the first place that is not 0, namely 6.

6 is in the thousands place.  We will decompose 1 of those thousands into 10 hundreds, and regroup.  We now have 5 thousands and 10 hundreds:

Now we can decompose 1 of the 10 hundreds into 10 tens. That leaves 9 hundreds:

Finally, we can decompose 1 of the 10 tens into 10 ones. That leaves 9 tens:

That completes the decomposing and regrouping. The subtraction easily follows:

The answer is 4,766.

The problem will appear, of course, like this:

And with practice, the student should be able to say,

"4 from 10 is 6."

"3 from 9 is 6."

"2 from 9 is 7."

"1 from 5 is 4."

Again, we can always check the answer by adding.  (Please compare the simplicity of compensating.)

Example 8.  Alignment.   68.273

 Answer.   3 is 3 ones.  We must subtract it from 8 ones.  The answer is 65.27.

As in addition, we must align the digits that have the same place value.

Example 9.   Your checking account has a balance of $310.12, and you write a check for $45.24.  Then you make a deposit of 126.75, and you write another check for $22.50.  What is your present balance?

 Answer.   Each time you write a check, you must subtract.  And each time you make a deposit, you add.  Here then is the sequence of subtractions and additions:

310.12		Opening balance
45.24		Check
264.88
+ 126.75		Deposit
391.63
22.50		Check
369.13		Present balance

Please "turn" the page and do some Problems.

or

Continue on to the next Lesson.

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