using alegra prove that 0.565656... / 0.124242424... is equal in value to 560/123
0.565656... / 0.124242424..... Multiply both top and bottom by 990.
[990 x 0.56565656.... / 990 x 0.124242424.....] =560/123
+2446 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 \(\frac{560}{123}=\frac{0.\overline{56}}{0.1\overline{24}}\)
