0.87 repeating as a fraction
Understand the Problem
The question is asking how to express the decimal number 0.87 repeating as a fraction. This involves converting a repeating decimal into a fraction form, using algebraic methods to represent the repeating part.
Answer
The decimal $0.87\overline{7}$ is expressed as the fraction $\frac{8}{9}$.
Answer for screen readers
The decimal number $0.87\overline{7}$ can be expressed as the fraction $\frac{8}{9}$.
Steps to Solve
- Define the repeating decimal as a variable
Let ( x = 0.877777...). We can see that the digit '7' is repeating.
- Align the repeating part
To eliminate the repeating part, multiply ( x ) by 10 to shift the decimal point one place to the right: $$ 10x = 8.77777... $$
- Set up the equation
Now, set up the equation using the initial definition of ( x ): $$ 10x = 8.77777... $$ $$ x = 0.87777... $$
- Subtract the two equations
Now, subtract the second equation from the first: $$ 10x - x = 8.77777... - 0.87777... $$ This simplifies to: $$ 9x = 8.0 $$
- Solve for ( x )
Now, divide both sides by 9: $$ x = \frac{8.0}{9} $$
Thus, ( x ) can be expressed as: $$ x = \frac{8}{9} $$
The decimal number $0.87\overline{7}$ can be expressed as the fraction $\frac{8}{9}$.
More Information
This fraction $\frac{8}{9}$ indicates that the repeating decimal is slightly less than 1, specifically 0.88. Understanding how to convert repeating decimals into fractions helps with simplifying and figuring out exact values that can be used in further mathematical operations.
Tips
- A common mistake is failing to properly align and subtract the equations. Always ensure you subtract the correct repeating decimals to achieve the right equation.
- Another mistake can be forgetting to reduce the final fraction to its simplest form. Make sure all fractions are in simplest form whenever applicable.
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