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using alegra prove that 0.565656... / 0.124242424... is equal in value to 560/123

 Nov 12, 2017
 #1
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 0.565656... / 0.124242424.....   Multiply both top and bottom by 990.

[990 x 0.56565656.... / 990 x 0.124242424.....] =560/123

 Nov 12, 2017
 #2
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In order to prove algabraically that \(\frac{0.5656...}{0.12424...}=\frac{560}{123}\), 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. \(0.\overline{56}=x\). 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.

 

\(56.\overline{56}=100x\)

 

3. Subtract your 2 Equations.

 

\(56.\overline{56}\) \(=100x\)
\(\hspace{2mm}0.\overline{56}\) \(=\hspace{7mm}x\)
\(56\) \(=\hspace{1mm}99x\)

 

4. Solve for x

 

\(56=99x\) Divide by 99 on both sides.
\(\frac{56}{99}=x=0.\overline{56}\)  
   

 

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

 

\(0.1\overline{24}=y\)

 

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

 

\(12.4\overline{24}=100y\)

 

Now, subtract the two equations.

 

\(12.4\overline{24}\) \(=100y\)
\(\hspace{3mm}0.1\overline{24}\) \(=\hspace{7mm}y\)
\(\hspace{1mm}12.3\) \(=\hspace{2mm}99y\)

 

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.
\(\frac{123}{990}=y\)  

 

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

 

\(\frac{x}{y}=\frac{560}{123}\)

Let's see if this is true.

\(\frac{\frac{56}{99}}{\frac{123}{990}}\) Multiply by 990/123 to eliminate the complex fraction.
\(\frac{56}{99}*\frac{990}{123}\) Notice that 990 and 99 can be simplified before any multiplication takes place.
\(\frac{56}{1}*\frac{10}{123}\) Simplify from here.
\(\frac{560}{123}\)  
   

Therefore, we have proven algabraically that \(\frac{560}{123}=\frac{0.\overline{56}}{0.1\overline{24}}\)

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 Nov 12, 2017

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