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

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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