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

What is the fractional equivalent of the repeating decimal n = 0.1515 is 5/33

1: 1 and 5

2:To identify the power of 10 to multiply by, we need to look at the number of decimal places in the repeating pattern. In this case, the repeating pattern is "15". As there are two decimal places in the repeating pattern, we can multiply n by 100 (10^2) to eliminate the decimal part and convert it into a fraction.

Therefore, the power of 10 to multiply by is 10^2 (which is 100).

3:The equation can be written as:

10^n = 0.1515... × 10^n

Simplifying the right side by multiplying 0.1515... with 10^n, we have:

10^n = 0.1515... × 10^n

To solve this equation, we can consider that multiplying any number by 10^n is equivalent to shifting the decimal point n places to the right of the number.

So, the equation simplifies to:

10^n = 0.1515... × 10^n = 0.1515...[n decimal places]

Therefore, n decimal places after the decimal point in the number 0.1515... is equal to 10^n.

4:Original equation: n = 0.1515...

Subtracting the equations:

n - 0.1515...

= n

- (0.1515...)

--------------

= (n - 0.1515...)

Therefore, the result of subtracting the equations is (n - 0.1515...).

Multiply both sides of the equation by 100 to eliminate the decimal point:

100x = 15.1515...

Subtract the original equation (step 1) from the new equation (step 2) to eliminate the repeating part:

100x - x = 15.1515... - 0.1515...

Simplifying the equation gives:

99x = 15

Divide both sides of the equation by 99 to solve for x:

x = 15/99

To ensure the fraction is in simplest form, we can find the greatest common divisor (GCD) of the numerator and denominator, which is 3 in this case.

Dividing both the numerator and denominator by 3:

x = (15/3) / (99/3)

= 5/33

Therefore, the fraction representation of the repeating decimal 0.1515... in simplest form is 5/33