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Chapter 4: Problem 2

For exercises 1-4, rewrite the repeating decimal as a fraction. $$ 0 . \overline{7} $$

Short Answer

Expert verified
0.\overline{7} = \frac{7}{9}

Step by step solution

01

- Set up the equation

Let \( x = 0.\overline{7} \). This means that \( x \) is the repeating decimal we want to convert into a fraction.
02

- Multiply by 10

Multiply both sides of the equation by 10 to shift the decimal point one place to the right: \( 10x = 7.\overline{7} \).
03

- Subtract the original equation

Subtract the original equation \( x = 0.\overline{7} \) from the new equation \( 10x = 7.\overline{7} \): \[ 10x - x = 7.\overline{7} - 0.\overline{7} \] This simplifies to \[ 9x = 7 \].
04

- Solve for x

Divide both sides by 9 to isolate \( x \): \[ x = \frac{7}{9} \].
05

- Verify the result

To verify, divide 7 by 9: \[ \frac{7}{9} = 0.\overline{7} \]. This confirms that \( 0.\overline{7} \) can be rewritten as \( \frac{7}{9} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

decimal conversion
Decimal conversion is essential in math. It helps us transform non-whole numbers into other formats, like fractions.
Repeating decimals are a special case. They have a sequence of digits that repeats forever.
For example, in the exercise, we convert the repeating decimal \( 0.\overline{7} \) to a fraction.
The key steps start from setting up an equation, multiplying to shift the decimal point, and then solving for the unknown.
fractions
Fractions represent parts of a whole. They use two numbers, the numerator (top) and the denominator (bottom).
Understanding fractions helps us deal with repeating decimals. By converting a repeating decimal to a fraction, we can express the repeating pattern in a concise way.
For instance, in the exercise, we converted \( 0.\overline{7} \) to the fraction \( \frac{7}{9} \). This means 7 parts of 9 make up the repeating decimal.
algebraic equations
Algebraic equations help us solve problems with unknowns. Here, we used an equation to convert a repeating decimal to a fraction.
Let's break down the steps:
First, we let \( x = 0.\overline{7} \). Next, we multiplied both sides by 10 to shift the decimal point: \( 10x = 7.\overline{7} \). We then subtracted the original equation from this new equation:\( 10x - x = 7.\overline{7} - 0.\overline{7} \). This gave us \( 9x = 7 \). Lastly, we solved for \( x \) by dividing both sides by 9, resulting in \( x = \frac{7}{9} \).
basic math operations
Basic math operations are fundamental skills. They include addition, subtraction, multiplication, and division.
We used all these in converting a repeating decimal to a fraction.
  • We multiplied to shift the decimal.
  • We subtracted equations to eliminate the repeating part.
  • We divided to isolate the variable.
Ensuring a solid grasp of these basics is crucial for tackling more complex math problems, like this one.
Practicing these steps helps build confidence and accuracy in mathematical calculations.

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