Math Problem Statement

Express the repeating decimal 2.715 (with 715 repeating) as a fraction.

Solution

Let's work through part (c) where we need to express the repeating decimal 2.7152.\overline{715} as a fraction.

Steps to convert a repeating decimal to a fraction:

Let x=2.715x = 2.\overline{715}, meaning: x=2.715715715x = 2.715715715\ldots

Step 1: Eliminate the repeating part by multiplying both sides by 1000.

This shifts the decimal point three places: 1000x=2715.7157157151000x = 2715.715715715\ldots

Step 2: Subtract the original x=2.715715715x = 2.715715715\ldots from the equation.

1000xx=2715.7157157152.7157157151000x - x = 2715.715715715\ldots - 2.715715715\ldots 999x=2713999x = 2713

Step 3: Solve for xx:

x=2713999x = \frac{2713}{999}

Step 4: Simplify the fraction (if possible).

We now check if 2713999\frac{2713}{999} can be simplified. However, 2713 and 999 have no common factors, so the fraction is already in its simplest form.

Thus, the repeating decimal 2.7152.\overline{715} as a fraction is: 2713999\boxed{\frac{2713}{999}}

Let me know if you need more details or have any other questions!

Here are five related questions to expand on this:

  1. How do you convert a non-repeating decimal to a fraction?
  2. What is the general method to convert any repeating decimal to a fraction?
  3. Why is it important to simplify fractions?
  4. Can repeating decimals always be expressed as fractions?
  5. What are some real-world applications of converting repeating decimals to fractions?

Tip: When simplifying fractions, always check for the greatest common divisor (GCD) of the numerator and denominator to ensure the fraction is in its simplest form.

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Math Problem Analysis

Mathematical Concepts

Repeating Decimals
Fractions
Algebra

Formulas

x = repeating decimal, 1000x - x = integer
Fraction simplification

Theorems

Conversion of repeating decimals to fractions

Suitable Grade Level

Grades 8-10