How you would convert the repeating, nonterminating decimal to a fraction? Explain the process as you solve the problem. 0.1515 . . .

To convert the repeating, nonterminating decimal 0.1515.... to a fraction, follow these steps:

Step 1: Let's assume the repeating part of the decimal is represented by the variable x. In this case, we can write the decimal as 0.15x.

Step 2: To remove the repeating part and create an equation, we need to multiply the decimal by a power of 10 that will shift the repeating part directly after the decimal point. Since there are two digits in the repeating part, we can multiply by 100. So, we have 100 * (0.15x) = 15.x.

Step 3: Now, subtract the original decimal from the multiplied decimal to eliminate the repeating part. We get: 100 * (0.15x) - (0.15x) = 15.x - 0.15x.

Step 4: Simplify the equation: 100 * (0.15x) - (0.15x) = 15.x - 0.15x.
This can be simplified to: 15x - 0.15x = 15.x - 0.15x.
This further simplifies to: 14.85x = 15.x - 0.15x.

Step 5: Combine like terms on the right-hand side of the equation: 14.85x = 14.85x.

Step 6: Divide both sides of the equation by 14.85 to solve for x: x = 14.85x / 14.85.

Step 7: Simplify the equation: x = x / 1.

The solution to the equation is x = x / 1. This means that the repeating part x is equal to itself. Therefore, we can conclude that x = 0.1515... is equivalent to x = 14.85x / 14.85.

Step 8: To convert this equation into a fraction, multiply both sides of the equation by a suitable power of 10 to eliminate the decimal. Since there are two repeating digits in x, we can multiply by 100. We have 100x = 1485x / 14.85.

Step 9: Simplify the equation by multiplying both sides by 14.85 to remove the fraction: 100x = 1485x.

Step 10: Rearrange the equation to isolate x on one side: 100x - 1485x = 0.

Step 11: Combine like terms: -1385x = 0.

Step 12: Divide both sides of the equation by -1385 to solve for x: x = 0.

Therefore, the repeating decimal 0.1515... can be converted to the fraction 0/1, and simplifying further, we get x = 0.

To convert a repeating, nonterminating decimal to a fraction, we need to understand that the repeating pattern continues indefinitely. In this case, the decimal 0.1515... has a repeating pattern of "15". Here's how we can convert it to a fraction:

Step 1: Assign variable x to the repeating decimal. In this case, x = 0.1515...

Step 2: Multiply both sides of the equation x = 0.1515... by 100 to shift the decimal two places to the right. This eliminates the repeating part, as shown: 100x = 15.1515...

Step 3: Now, let's subtract the original equation (x = 0.1515...) from the equation obtained in step 2 (100x = 15.1515...). This eliminates the non-repeating part: 100x - x = 15.1515... - 0.1515...

Simplifying this equation, we get: 99x = 15

Step 4: To solve for x, we divide both sides of the equation by 99: x = 15/99

Step 5: The fraction 15/99 is not in simplest form. We can reduce it by dividing both the numerator and denominator by their greatest common divisor. In this case, the greatest common divisor of 15 and 99 is 3. Dividing both numerator and denominator by 3, we get: x = (15/3)/(99/3) = 5/33

Therefore, the repeating nonterminating decimal 0.1515... can be converted to a fraction, which is 5/33.

converting n, a repeating decimal...

100n-n=15.000000
but 100n-n=99n

so then think this out
.1515=15/99

let x = .151515

the length of the repeating period is 2 digits, so multiply by 10^2 or 100
100x = 15.151515...
x = .151515..
subtract them:
99x = 15
x = 15/99

If you have decimal which contains a repeating sequence after some decimals that don't repeat ...
e.g. 0.12456456456..
let x= .12456456456..
the length of the repeat is 3, so multiply by 1000
1000x = 124.56456456..
x = .12456456456
subtract:
999x =124.440000..
x = 124.44/999
=12444/99900
= 1037/8325