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## Lesson 45:  Division with Remainders and Decimals

If you are in high school or college, you are probably allowed to use calculators on all of your exams.  However, it is important to understand how division really works.  This lesson explains the basics, and you'll learn more in later lessons.

Let's say you are asked to do 9 ÷ 4.  You can represent this problem using this notation for division shown at right.  In this problem, the 9 is called the dividend, and the 4 is called the divisor.  The answer that we get is called the quotient.  We know that 4 does not divide evenly into 9.  If we count by fours, we count 4, 8, 12, etc.  It only goes in twice, making 8.  Since it goes in twice, we can write a 2 above the 9.  We need to show that there is 1 left over, so we can write R1 next to the 2, which means that there is a remainder of 1.

The real world analogy for this problem would be if you had 9 cookies, and wanted to give an even amount to each of 4 children.  Each child would get 2 cookies, and then one cookie would be left over, since it can't be split up (at least not without cutting it into fractions, and we'll talk about how to do that later).

Here is another problem.  23 ÷ 3.  To do this, we first check to see if 3 can divide into 2.  It can't, since 2 is smaller.  So we just move to the right, and now see if 3 can divide into 23.  It can, 7 times.  3 x 7 is 21, so that means we have two left over, and we can write R2 next to the quotient.

Here is a harder example:  493 ÷ 5.  First, we can see that 5 does not go into 4, so we'll try seeing how many times it goes into 49.  Counting by 5's, we can see that it goes in 9 times, making 45, but we have 4 left over.  What we do is write a little 4 to the left of the next digit, making that digit a 43.  Now we see how many times 5 goes into 43.  The answer is 8, with a remainder of 3.  Our final answer is then 98 R 3.

Here is an example involving a fraction.  Very often we need to convert fractions into decimals.  When you do this, it becomes very easy to see which of two fractions is larger, and there are many other reasons why we need to do this.  Let's take the fraction 1/2 (one-half).  Remember that fractions are really just division problems, so what we really have is 1 ÷ 2.

This is what we do.  We can set up the problem as shown on the right.  Obviously 2 does not divide into 1, so what we can do is rewrite 1 as 1.0, which is the same thing.  We immediately put a decimal point in the quotient right above the one in the dividend.  Now we can ignore the decimal point for just a moment, and divide 2 into 10.  That's 5, and we can write that in the quotient as shown.  The answer then becomes 0.5.  That makes sense.  We've learned that 0.5 means 5/10 (five-tenths), which we know is one-half.

Here is a harder problem involving decimals:  Convert 3/8 into a decimal, which is the same as saying we want to do 3 ÷ 8.  Look at how we do this at right.  8 does not go into 3, so we write a 0 above it.  We'll change 3 into 3.0.  Now we'll divide 8 into 30, which goes in 3 times, with a remainder of 6.  As above, we'll need write a little 6 to the left of the next digit, but we don't have one, so we'll add a 0.  That is allowed.  3.00 is the same as 3.0.  Now we'll divide 8 into 60 to get 7, with a remainder of 4.  Repeating the process above by adding another 0, with a little 4 to the left of it, we can divide 8 into 40 which goes in 5 times exactly.  The answer is 0.375.

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