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Lesson 3

MULTIPLYING AND DIVIDING

BY POWERS OF 10

The Meaning of Percent


Thanks to our system of positional numeration, it is a simple matter to multiply or divide by a power of 10. It never calls for the methods we use with other numbers. And it certainly does not call for a calculator. For the remainder of this course, multiplying and dividing by powers of 10 will be a fundamental skill.


In this Lesson, we will answer the following:

  1. How do we multiply a whole number by a power of 10?
  2. How do we multiply a decimal by a power of 10?
  3. How do we divide a decimal by a power of 10?
  4. How do we divide a whole number by a power of 10?

    Section 2: The Meaning of Percent

  5. What does "percent" mean?
  6. How can we take 1% of a number?
  7. How can we take 10% of a number?
  8. How do we change a percent to a number?
  9. How do we change a number to a percent?

    Section 3:

  10. When is a number written in "scientific notation"?
  11. How do we convert units in the metric system?


 1.   How do we multiply a whole number by a power of 10?
36 × 10
 
  Add on as many 0's as appear in the power.

  Examples.    36 × 10  =  360     Add on one 0.
  36 × 100  =  3600     Add on two 0's.
  36 × 1000  =  36,000     Add on three 0's.

Doing that is an example of skill in arithmetic, which is to be able to do a problem as quickly and as 28 × 100 vertically easily as possible.  Traditional multiplication would eventually give the right answer (we hope).  But it is not skillful, and it does not take advantage of positional numeration.  The student should let go of that written method immediately.

 2.   How do we multiply a decimal by a power of 10?
 
7.32 × 10
 
  Move the decimal point right as many digits
as there are 0's in the power. If there are not enough digits, add on 0's.

Examples.   

7.32 × 10 = 73.2 Move the decimal point one place right.
7.32 × 100 = 732 Move the point two digits right:  732.  However, since all the digits fall to the left of the decimal point, the answer is a whole number, 732, which we write without a decimal point.
7.32 × 1000 = 7,320  Move the point three digits right.  To do this, we must add on a 0.
Again, the answer is a whole number.

Problem.   If 5 pounds of sugar cost $2.79, how much will 50 pounds cost?

Answer.  Since 50 pounds are ten times 5 pounds, they will cost ten times more.  Move the decimal point one place right:  $27.90.  Since money has two decimal digits, we added on a 0. (Lesson 3, Question 8)

 3.   How do we divide a decimal by a power of 10?
 
63.4 ÷ 10
 
  Move the decimal point left as many digits as there are 0's in the power. If there are not enough digits, add on 0's.

Examples.    63.4 ÷ 10 = 6.34 Move the point one place left.
  63.4 ÷ 100 = .634   Move the point two digits left.
  63.4 ÷ 1000 = .0634    Move the point three digits left.  To
do this, add on a 0.

These example illustrate that, whenever we multiply or divide by a power of 10, the digits do not change  We simply move the decimal point or add on 0's.

Finally, we must see how to divide a whole number by a power of 10. Now in Lesson 1 we saw that when a whole number ends in 0's, we simply take off 0's. (Lesson 1, Question 11)

265,000 ÷ 100 = 2,650

But when a whole number does not end in 0's -- as 265 -- then there are no 0's to chop off  We will see that we must place a decimal point to separate digits on the right.

 4.   How do we divide a whole number by a power of 10?
 
265 ÷ 10
 
  Starting from the right of the whole number, separate as many decimal digits as there are 0's in the power. If there are not enough digits, add on 0's.

  Examples.   265 ÷ 10   =  26.5 Starting from the right of 265,
separate one decimal digit.
 
   265 ÷ 100   =  2.65 Separate two decimal digits.
 
   265 ÷ 1000   =  .265 Separate three decimal digits.

Again, as in Lesson 1, consider this array:

Place values

As we move down the list -- as we push the digits one place left -- the number has been multiplied by 10, because each next place is worth 10 times more.  (As we move from 2.658 to 26.58, we go from 2 ones to 2 tens.)  It appears, though, as if the decimal point has shifted one place right, or, with whole numbers, that a 0 has been added on.

As we move up the list -- as we push the digits to the right -- each number has been divided by 10.

And so we can easily multiply or divide by a power of 10 because of the written system itself.  Each place belongs to the next power of 10.


At this point, please "turn" the page and do some Problems.

or

Continue on to the Section 2:  The meaning of percent.


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