Repeating decimals

When studying "Converting a fraction to a repeating decimal," focus on these key points:

1. Understanding Fractions: Recognize that a fraction consists of a numerator (top number) and a denominator (bottom number).

2. Long Division: Use long division to divide the numerator by the denominator. This is the main method for converting fractions to decimals.

3. Identifying Repeats: If the long division process produces a remainder that has already occurred, the decimal will begin to repeat. Document the sequence of remainders to spot this.

4. Repeating Decimal Notation: Use notation such as a bar (e.g., 0.3̅) to indicate the repeating part of the decimal.

5. Terminating vs. Repeating: Understand the difference between fractions that yield terminating decimals (exact decimals that end) and those that yield repeating decimals.

6. Prime Factorization: Know that a fraction will have a terminating decimal if, after simplification, its denominator contains only the prime factors 2 and/or 5.

7. Converting Back: Be able to convert a repeating decimal back into a fraction if needed by setting up an equation and solving for the variable.

By mastering these points, you'll be well-equipped to convert fractions into repeating decimals.

Here are the key points for converting repeating decimals to fractions:

1. Understanding Repeating Decimals: Recognize that a repeating decimal has a digit or group of digits that repeat indefinitely (e.g., 0.333... or 0.142857142857...).

2. Identifying the Decimal: Clearly identify the repeating part of the decimal. For example, in 0.666..., "6" is the repeating digit.

3. Setting Up an Equation: Let the repeating decimal be equal to a variable (e.g., let $x = 0.666...$).

4. Multiplying to Eliminate the Decimal: Multiply the variable by a power of 10 that matches the number of repeating digits. For 0.666..., multiply by 10 to get $10x = 6.666...$.

5. Subtracting the Original Equation: Subtract the original equation from this new equation to eliminate the repeating part. For example:

   $$10x - x = 6.666... - 0.666...$$

   This simplifies to $9x = 6$.

6. Solving for the Variable: Solve for $x$. Continuing from the previous step, divide both sides by 9:

   $$x = \frac{6}{9}$$

7. Simplifying the Fraction: Simplify the fraction to its lowest terms. In this case, $\frac{6}{9}$ simplifies to $\frac{2}{3}$.

8. Result Verification: Optionally, convert the fraction back to a decimal to confirm the result matches the original repeating decimal.

By following these steps, you can convert any repeating decimal into a fraction accurately.

Sure! Here are the key points to learn when studying "Converting repeating decimals to fractions (part 2 of 2)":

1. Setup of the Equation: When converting a repeating decimal to a fraction, start by letting the repeating decimal equal a variable (e.g., $x = 0.666...$).

2. Multiply to Shift Decimal: Multiply both sides of the equation by a power of 10 that matches the length of the repeating segment (e.g., for 0.666..., multiply by 10 to get $10x = 6.666...$).

3. Subtract the Original Equation: Set up a subtraction between the two equations (the original and the multiplied version). This helps eliminate the repeating part (e.g., $10x - x$).

4. Solve for $x$: Simplify the resulting equation to solve for the variable. This will give you a numeric result (e.g., $9x = 6$, thus $x = \frac{6}{9}$).

5. Reduce the Fraction: Reduce the resulting fraction to its simplest form (e.g., $\frac{6}{9} = \frac{2}{3}$).

6. Use of Mixed Numbers: If the repeating decimal is a mixed number (e.g., $2.3\overline{4}$), convert the whole number separately and combine it with the fraction derived from the decimal part.

7. Special Cases: Be aware of any special cases, such as terminating decimals or non-repeating parts, and how they affect the conversion.

By following these steps, you can effectively convert any repeating decimal into a fraction.