Question

Repeating decimals are rational numbers! This means they can be written as ratios of integers. For the repeating decimal 0.035, write it as an infinite series, and then write it as a fraction.

Answer 1

The repeating decimal 0.035 is represented as the fraction 707/19980 after recognizing the pattern, setting up an equation, and factoring out the repeating sequence, ultimately showing it is a rational number.

Step-by-step explanation:

The repeating decimal 0.035 can be written as an infinite series by recognizing the pattern of repetition. To do this, let's set x as the given decimal:

x = 0.0353535...

Now, if we multiply x by 1000, we move the decimal point three places to the right:

1000x = 35.353535...

Now, by subtracting the original x from 1000x, we eliminate the repeating part:

1000x - x = 35.353535... - 0.0353535...

999x = 35.3181818...

Solving for x gives:

x = 35.3181818... / 999

Which simplifies to:

x = 3535 / 99900

And we can reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 5:

x = 707 / 19980

Thus, the repeating decimal 0.035 can be written as the ratio of two integers, 707 over 19980, which is a rational number.