Question

What is the fractional equivalent of the repeating decimal n = 0.1515... ?
Answer the questions to find out.

1. How many repeating digits does the number represented by n have? (2 points)


2. You need to multiply n by a power of 10 to help you find the fraction. Decide on the power of 10 to multiply by, and tell how you identified that number. (2 points)

3. Write an equation where the left side is your power of 10 times n and the right side is the result of multiplying 0.1515... by that power. (2 points)

4. Write the original equation, n = 0.1515... underneath your equation from question 3. Then subtract the equations. Show your work. (2 points)

5. Write n as a fraction in simplest form. Show your work. (2 points)

Answer 1

Answer:

$n=\frac{5}{33}$

Step-by-step explanation:

Question 1:

The decimal number n has two repeating digits (1 and 5). 0.151515... repeats forever.

Question 2:

In order to find the fractional equivalent of n, we can multiply n by 10 to the power of the number of repeating digits. We identified that quantity to be 2, in the previous question, so we can conclude that we must multiply n by $10^2$, which is 100

Question 3:

We can set up an equation that sets $10^2n$ equal to our decimal times $10^2$:

$10^2*n=10^2*0.151515...\\100n=100*0.151515...\\100n=15.1515...$

Note that when you multiply a number by 100, you can just move the decimal point two places to the right

Question 4:

Now, we can set up a system of equations using the one we found in question 4 and the one given to us at the start of the problem:

$100n=15.1515...\\n=0.1515...$

Question 5:

We can subtract the bottom equation from the top by subtracting the components on the left and subtracting the components on the right. After we do this, we can just solve for n:

$100n-n=15.1515...-0.1515...\\99n=15\\n=\frac{15}{99}=\frac{5}{33}$

Note that, when you subtract the two decimal numbers, the repeating parts go away since they are both technically the same quantity (in that they are the same numbers and the repeat infinite times)

That being said, our final fraction is 5/33 after simplication