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Long Division of Small Numbers

Date: 03/26/2014 at 23:23:35
From: Jay
Subject: 200/3500

I recently started learning and using long division again. I'm studying
for the Armed Services Vocational Aptitude Battery and I would need to do
long division on most of my problems, like

   200/3500

I tried to find out what would go into 200 to get me close to 3500. I got
200 x 17 = 3400. From the looks of the answer that the book gave me
(0.057), I assumed I was already wrong.

I'm lost on how they got that. Just where do you start?

Like I said, it's been a while since I worked with long division problems
like this. 

I'm lost on how to solve the answer. 



Date: 03/27/2014 at 08:13:11
From: Doctor Ian
Subject: Re: 200/3500

Hi Jay,

Note that a fraction is a division, and a division is a fraction. So one
thing you can do is simplify the fraction before doing the division:

     200      2
   ------ = ----
    3500     35

You can just divide 35 into 2, and you'll get the same answer.

You can also scale the numerator, so long as you scale the result in a
compensating way. That is,

     2     2 * 1000
   ---- = ---------- * 1/1000
    35        35

This lets you divide 35 into 2000, then multiply the result by 1/1000.

So let's try that:
       

           7 help
          50
       _____
   35 ) 2000
        1750
        ----
         250
         235
         ---
          15

We get approximately 57. Multiply that by 1/1000, and we get 57/1000, or
0.057.

Note that if you're in a multiple choice situation, or one where you just
need an estimate, you could take one more step beyond reducing to 2/35. 
You could multiply that by 3/3, to get 6/105, which is close to 6/100, or
0.06.

The basic idea is always the same: Try to change the problem you have into
one that's easier to deal with, but has (whether exactly or approximately)
the same answer.

Make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 03/27/2014 at 13:11:12
From: Jay
Subject: 200/3500

Why is the 7 placed above the 50? And where does it come from?



Date: 03/27/2014 at 16:26:55
From: Doctor Ian
Subject: Re: 200/3500

Hi Jay,

Let's first look at what it means to do a division. It's really just a way
of writing a multiplication.

For example, look at this division:

       18
      ___
   4 ) 72

This means the same thing as

   72 = 4 * 18

More generally, these are two ways of expressing exactly the same fact:

             quotient
            _________        AND      dividend = divisor * quotient
   divisor ) dividend

So far, so good?

Now, note that we could write those as 


              8
             10
            ___
         4 ) 72              AND      72 = 4 * (10 + 8)

That is, we're just using the distributive property to break these up.  

Still with me?

Okay, so how do we do this division a bit at a time? We bite off little
pieces, starting from here:

      ___
   4 ) 72

We can note that 40 is smaller than 72, while 80 is larger. So we can subtract 40 as 
a first step, keeping track of the fact that 40 is 4*10:

       10
      ___
   4 ) 72          72 = 4 * (10 + ?)
       40  
       --
       32

Now we might note that 20 is less than 32, and subtract 20 as a next 
step:

        5
       10
      ___
   4 ) 72          72 = 4 * (10 + 5 + ?)
       40  
       --
       32
       20
       --
       12

Now we might note that 12 is just 4*3, so we can finish up:

        3
        5
       10
      ___
   4 ) 72          72 = 4 * (10 + 5 + 3)
       40  
       --
       32
       20
       --
       12
       12
       --
        0
 
And what this division is telling us is that

        3
        5
       10
      ___
   4 ) 72          
       40   <- 4*10
       --
       32
       20   <- 4*5
       --
       12
       12   <- 4*3
       --      -----------------------------------------------
        0      4*10 + 4*5 + 4*3 = 4*(10 + 5 + 3) = 4*18 = 72

That is, you're looking for some numbers such that 

   72 = 4*(? + ? + ? + ... ?)

The 'long division' algorithm is designed to be very efficient, in terms
of doing the least amount of writing. But you don't need to be efficient!
You can bite off pieces of any size that you like:

         4
         4
         2
         5
         3
       ___
    4 ) 72
        12  <- 4 * 3
        --
        60
        20  <- 4 * 5
        --
        40
         8  <- 4 * 2
        --
        32
        16  <- 4 * 4
        --
        16
        16  <- 4 * 4
        --
         0

All together,

   72 = 4 * (3 + 5 + 2 + 4 + 4) 

      = 4 * 18

Can you answer your own question now? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 03/28/2014 at 00:30:29
From: Jay
Subject: Thank you (200/3500)

I got it! Thank you so much!

I figured that is where it came from! Just wanted to ask and make sure!

This is very helpful. I will continue to use this site. I'm so happy I
stumbled upon it! This will help me with my ASVAB substantially, allowing
me to score higher.

I really appreciate your service! 
Associated Topics:
Middle School Division
Middle School Fractions

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