Final answer:
x represents the repeating decimal 0.01515..., and by multiplying x by 1000 and subtracting the original x from the result, we can show algebraically that x is equal to 1/66.
Step-by-step explanation:
To prove algebraically that x = 0.015 recurring can be written as 1/66, we will use a common method for converting repeating decimals to fractions. Let's first represent 0.015 recurring as x:
x = 0.0151515…
Now let's multiply x by 1000 to shift the decimal point three places to the right:
1000x = 15.151515…
Next, we subtract the original x from this new 1000x to eliminate the repeating part:
1000x - x = 15.151515… - 0.0151515
The right side of the equation simplifies to:
1000x - x = 15.1363636…
Now simplifying the left side:
999x = 15.1363636…
To solve for x, divide both sides by 999:
x = 15.1363636… / 999
The result is:
x = 1/66
This algebraic manipulation shows that the repeating decimal 0.015 recurring is indeed equivalent to the fraction 1/66.