Convert each of the following repeating decimals to $\frac{a}{b}$ form, where $a$ and $b$ are integers and $b \neq 0$.
a. $0 . \overline{7}$
Answer (1)
Let x = 0. 7 .
Multiply by 10: 10 x = 7. 7 .
Subtract the equations: 10 x − x = 7. 7 − 0. 7 , which simplifies to 9 x = 7 .
Solve for x : x = 9 7 . The final answer is 9 7 .
Explanation
Define the Repeating Decimal Let x = 0. 7 . This means x = 0.7777 … . We want to express x as a fraction b a , where a and b are integers and b = 0 .
Multiply by 10 Multiply both sides of the equation x = 0. 7 by 10. This gives us 10 x = 7. 7 , which means 10 x = 7.7777 … .
Subtract the Equations Subtract the equation x = 0. 7 from the equation 10 x = 7. 7 . This gives us 10 x − x = 7. 7 − 0. 7 , which simplifies to 9 x = 7 .
Solve for x Solve for x by dividing both sides of the equation 9 x = 7 by 9. This gives us x = 9 7 .
Final Answer Therefore, the repeating decimal 0. 7 is equal to the fraction 9 7 . We can verify that 7 and 9 are integers and 9 = 0 .
Examples
Repeating decimals can be used to represent fractions in a different form. For example, when measuring ingredients for a recipe, you might encounter a repeating decimal in the measurements. Converting it to a fraction helps in accurately measuring the quantity. Understanding how to convert repeating decimals to fractions is also fundamental in understanding number systems and algebra.
