0.7777 as a fraction
Understand the Problem
The question is asking for the conversion of the decimal 0.7777 into a fraction. To solve this, we can express 0.7777 as a fraction by recognizing it as a repeating decimal, which can be represented as 7777/10000.
Answer
The repeating decimal $0.7777$ can be expressed as the fraction $\frac{7}{9}$.
Answer for screen readers
The fraction representation of the decimal $0.7777$ is $\frac{7}{9}$.
Steps to Solve
-
Identify the repeating decimal Recognize that the decimal 0.7777 can be expressed as $0.\overline{7}$, where the overline indicates that 7 repeats indefinitely.
-
Set up the equation Let $x = 0.\overline{7}$. This means we are trying to solve for $x$.
-
Multiply to eliminate the repeating part Multiply both sides of the equation by 10 to move the decimal point one place to the right: $$ 10x = 7.7777... $$ Notice that the right-hand side still contains $0.\overline{7}$.
-
Set up the subtraction Now, subtract the original equation ($x = 0.7777...$) from this new equation: $$ 10x - x = 7.7777... - 0.7777... $$ This simplifies to: $$ 9x = 7 $$
-
Solve for x Now, divide both sides by 9: $$ x = \frac{7}{9} $$
-
Final expression Thus, the decimal 0.7777 can be expressed as the fraction: $$ 0.\overline{7} = \frac{7}{9} $$
The fraction representation of the decimal $0.7777$ is $\frac{7}{9}$.
More Information
The fraction $\frac{7}{9}$ is an irreducible fraction, meaning it cannot be simplified further. This fraction represents the repeating decimal $0.7777...$ effectively.
Tips
- Mistaking the decimal for a non-repeating fraction: It’s important to recognize that 0.7777 is a repeating decimal.
- Failing to use proper multiplication to eliminate the decimal: Ensure to multiply by the right power of 10 to shift the decimal.
