Division by Estimation

Date: 11/4/96 at 10:23:38
From: James Humphrey
Subject: Help?

I am homeschooling my boys for the first time and math is a 
weakness for me.  My son, who is 11, had a pretty basic math problem, 
but I am not certain of the answer.

Could you please explain the following problem?

"Estimate" the quotient when 716 is divided by 12

I know what the answer is because I have the book, but my son 
originally got it wrong. Then when I tried to explain, thinking 
rounding off the quotient, the answer was different.  

Then we rounded 716 to 700 and divided it by 10.  Is this correct?

Date: 11/17/96 at 23:10:38
From: Doctor Lynn
Subject: Re: Help?

There are lots of ways of estimating divisions.  Personally, I look at 
716, see that it's being divided by twelve, and try dividing 720 by 
twelve, because I know that 72 = 12 x 6, to get the answer 60.  
Unfortunately, this is a rather specific method, of the kind you pick 
up over time.  A less accurate but probably equally acceptable method 
would be the one you suggest, where both numbers are rounded up or 
down to something easier to handle.

In fact, I wouldn't worry too much about this.  I did my GCSE exams 
when I was sixteen, and there were no problems of this sort harder 
than this one, so your eleven-year old probably isn't expected to get 
a very accurate answer.

As a rule of thumb, if you just round both numbers up or down to the 
first digit it will be sufficient, so 712 becomes 700 and 12 becomes 
10 as you have done.  

To give another example, 678 divided by 34 is approximately 700 
divided by 30.  This is still quite hard, so we can say that's 600 
divided by 30 = 20.

Alternatively, you could have changed the 34 to 35 and the 678 to 700 
and said that it was about 20, because 70 is twice 35.

As an aside, remember that a small change to a small number makes 
for a big change in a big number.  In other words, you have to be 
careful when you round the smaller number.  You can't, for example, 
say that 19 is about 10.  If you do, you will get almost twice the 
correct answer.  For example, 320 divided by 19 is about 320 divided 
by 20 = 16.  If you rounded 19 down to ten, you would get the answer 
320 divided by 10 = 32...

As I say, if your son sticks to the rule of thumb I gave earlier, he 
will probably be fine. 

Thanks for your question, I hope that helps.  If you have any other 
questions, please write back.

-Doctor Lynn,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/