Express 0.314141414… as a ratio of two integers.

Verified step by step guidance

1. Identify the repeating decimal pattern in the number 0.314141414… Here, the digits '14' repeat indefinitely after the initial '3'.
2. Let \( x = 0.314141414\ldots \). To isolate the repeating part, multiply \( x \) by a power of 10 that moves the decimal point just before the repeating block. Since '14' has 2 digits, multiply by 100: \( 100x = 31.4141414\ldots \).
3. Next, multiply \( x \) by a power of 10 that moves the decimal point just before the first repeating block starts after the non-repeating part. Since the non-repeating part is '3' (one digit), multiply by 10: \( 10x = 3.14141414\ldots \).
4. Subtract the two equations to eliminate the repeating decimal: \( 100x - 10x = 31.4141414\ldots - 3.14141414\ldots \). This simplifies to \( 90x = 28.273\ldots \), but since the decimals after subtraction cancel out, you get \( 90x = 28.273\ldots \) with the repeating part removed.
5. Solve for \( x \) by dividing both sides by 90, then simplify the resulting fraction to express the original decimal as a ratio of two integers.

Key Concepts

Repeating Decimal Representation

A repeating decimal is a decimal number in which a digit or group of digits repeats infinitely. Recognizing the repeating part is essential to convert the decimal into a fraction. For example, in 0.314141414…, the digits '14' repeat indefinitely after the initial '3'.

Algebraic Method for Converting Repeating Decimals to Fractions

This method involves setting the repeating decimal equal to a variable, multiplying by powers of 10 to shift the decimal point, and subtracting to eliminate the repeating part. Solving the resulting equation yields the fraction form. This approach systematically converts infinite decimals into ratios of integers.

Simplification of Fractions

After finding the fraction equivalent of a repeating decimal, simplifying it by dividing numerator and denominator by their greatest common divisor ensures the fraction is in lowest terms. This step is important for expressing the ratio in its simplest and most understandable form.