Convert 2.3 repeating into a fraction.
Understand the Problem
The question is asking how to convert the repeating decimal 2.3 (where the 3 continues indefinitely) into a fraction. This requires understanding the concept of converting repeating decimals into fractional form.
Answer
The fraction form of the repeating decimal \( 2.3\overline{3} \) is \( \frac{7}{3} \).
Answer for screen readers
The repeating decimal ( 2.3\overline{3} ) can be expressed as the fraction ( \frac{7}{3} ).
Steps to Solve
- Set up the repeating decimal as a variable
Let ( x ) represent the repeating decimal.
$$ x = 2.3333... $$
- Multiply by a power of 10 to move the decimal point
Since the digits after the decimal point are repeating every one digit, multiply both sides by 10.
$$ 10x = 23.3333... $$
- Set up the equation to eliminate the repeating part
Now, subtract the first equation from the second equation to eliminate the repeating decimals.
$$ 10x - x = 23.3333... - 2.3333... $$
This simplifies to:
$$ 9x = 21 $$
- Solve for ( x )
Now, divide both sides by 9 to isolate ( x ).
$$ x = \frac{21}{9} $$
- Simplify the fraction
Next, simplify the fraction. Both the numerator and the denominator can be divided by 3.
$$ x = \frac{21 \div 3}{9 \div 3} = \frac{7}{3} $$
The repeating decimal ( 2.3\overline{3} ) can be expressed as the fraction ( \frac{7}{3} ).
More Information
The conversion of repeating decimals to fractions is a common technique in mathematics. Understanding this process can help with other similar problems involving decimals.
Tips
- Failing to align the equations correctly during subtraction.
- Forgetting to simplify the final fraction.
- Misidentifying the repeating part of the decimal.
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