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Reduced fraction

Date: Wed, 14 Dec 1994 12:55:14 -0500 (EST)
From: Anthony DAuria
Subject: Can you solve this in 15 minutes???

1. Write as a reduced fraction: -------

2. Find the area of a triangle whose sides measure 17, 25, and 28.

3. Find the negative value of x for which log  (x{squared} - 2x) = log 2

Date: Wed, 14 Dec 1994 14:48:37 -0500
Subject: Re: Can you solve this in 15 minutes???

Hello there!

Ordinarily, we don't accept questions which are intended to challenge us
(as opposed to questions that students really need help on), but these
were pretty neat.  So here's what I'll tell you:

>1. Write as a reduced fraction: -------
>                                .428571

I'll assume that you mean .428571428571428571428571 and not 1/.428571 .
Usually people only put the bar over the digits that repeat, and not the
decimal point.  To be extra clear, you can write it with the < shift - > key,
like this: ______  
          .428571  .

So how do we find the fraction for this repeating decimal?  Well, there's a
standard trick that people do.  You take your number, multiply by 10 as many
times as there are repeating digits (in this case multiply by 10 six times,
which is 1000000), and subtract the original number.  Then you get
a number whose digits don't go on forever, and you can use that number to
give you the fraction you're looking for.  Here's your example:

   1000000 x   =  428571.428571428571428571428571428571.....
        -  x   =        .428571428571428571428571428571.....
    999999 x   =  428571

Now divide both sides by 999999:

        x = 428571/999999

We're almost there!  All we have to do now is reduce the fraction.  Since
the factors of 428571 are 3 3 3 3 11 13 and 37, and the factors 
of 999999 are             3 3 3 7 11 13 and 37, the only things that don't 
cancel are the 3 and the 7.  So the answer is 3/7.

>2. Find the area of a triangle whose sides measure 17, 25, and 28.

One of the more useful formulas for finding the area of a triangle is
Hero's formula:  given the side lengths of a triangle a,b,c, let
s=(a+b+c)/2.  Then the area of the triangle is Sqrt{(s)(s-a)(s-b)(s-c)}.

With your numbers, the area just pops right out:

s = (17+25+28)/2 = 35

A = Sqrt{(35)(35-17)(35-25)(35-28)} = Sqrt{7*5*3^4*2*5*7} = 
7*5*9*Sqrt{2} = 315*Sqrt{2}.

Note that this is one way to conjure up triangles with a given area.

>3. Find the negative value of x for which   log (x{squared} - 2x) = log 2
Since Log is a one-to-one function, we have

                    x^2 - 2x = 2

so by the quadratic formula, x = 1 +- Sqrt{3}.
The negative value is x = 1 - Sqrt{3}.

-Ken "Dr." Math
Associated Topics:
Middle School Fractions

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