We are asked how we decide which power of 10 we multiply an equation by when writing a decimal with repeating digits as a fraction. Let's take a look at an example of rewriting a repeating decimal as a fraction. We will start with 0.25. First, we will assign a variable x to our repeating decimal.
x = 0.25Our next step is to multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal. In our example, d=2 because there are two repeating digits in our repeating decimal, 2 and 5. Let's multiply both sides of our equation by 10^2, which is 100.
100x = 25.25
Now we will subtract equivalent expressions of the variable and the repeating decimal from each side of the equation. From the left-hand side we will subtract x and from the right-hand side we will subtract 0.25. Then we will solve the equation for x.
As we can see, we multiplied the original equation by 10^2 because there were two repeating digits in our repeating decimal. We choose the power of 10 by looking at the number of repeating digits in the repeating decimal.
