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 #1
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Nov 12, 2017
 #2
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Nov 12, 2017
 #2
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In order to prove algabraically that 0.5656...0.12424...=560123, let's try to convert the interminable decimals to fractions.

 

I'll start with 0.5656...

 

1. Set the Repeating Decimal equal to a Variable!

 

This is a farily simple step. 0.¯56=x. Now, you're good to go!

 

2. Multiply Both Sides by 10 such until the Repeating Portion is the Whole Number

 

In this case, if I multiply both sides by 100, which is 10^2, then the repeating portion will be to the left of the decimal point.

 

56.¯56=100x

 

3. Subtract your 2 Equations.

 

56.¯56 =100x
0.¯56 =x
56 =99x

 

4. Solve for x

 

56=99x Divide by 99 on both sides.
5699=x=0.¯56  
   

 

Great! Now, let's convert the next one.

 

0.1¯24=y

 

Now, multiply by multiples of ten to get the repeating portion to the left until the repeating part lines up.

 

12.4¯24=100y

 

Now, subtract the two equations.

 

12.4¯24 =100y
0.1¯24 =y
12.3 =99y

 

Now, solve for y.

 

12.3=99y Multiply by 10 on both sides to make the left hand side a whole number.
123=990y Divide by 990 to isolate y.
123990=y  

 

Now, let's calculate what x/y is.

 

xy=560123

Let's see if this is true.

5699123990 Multiply by 990/123 to eliminate the complex fraction.
5699990123 Notice that 990 and 99 can be simplified before any multiplication takes place.
56110123 Simplify from here.
560123  
   

Therefore, we have proven algabraically that 560123=0.¯560.1¯24

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Nov 12, 2017
 #3
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Nov 12, 2017

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