+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 \(\frac{560}{123}=\frac{0.\overline{56}}{0.1\overline{24}}\)
.+2446 | \(18x-4=5\) | Combine like terms. In this step, the solver combined the \(5x\) and \(13x\) from the previous step. Like terms have to be both the same variable raised to the same power. This is indeed the case here. |
| \(18x=9\) | Addition Property of Equality. This step added 4 to both sides, which is allowed based on the addition property. |
| \(x=\frac{1}{2}\) | Division Property of Equality. This step divided 18 from both sides. This is a valid operation because of the division property of equality. |
