$\begin{array}{l}x=0 . \overline{7} \\ x \cdot 10^1=0 . \overline{7} \cdot 10^1 \\ 10 x=5 \\ 10 x-x=\overline{7}-0.7\end{array}$
Answer (1)
The question is about converting a repeating decimal to a fraction.
Let's break down the process of converting the repeating decimal x = 0. 7 into a fraction.
Define the repeating decimal:
Start by setting the repeating decimal equal to a variable. Let's say x = 0. 7 .
Express the repeating part using multiplication:
Since 0. 7 repeats every one decimal place, multiply both sides of the equation by 10:
10 x = 7. 7
Set up the subtraction to eliminate the repeating part:
Now, subtract the original equation from this new equation:
10 x − x = 7. 7 − 0. 7
Simplifying, we get:
9 x = 7
Solve for x :
Divide both sides by 9 to get:
x = 9 7
So, the decimal 0. 7 is equal to the fraction 9 7 . This process effectively converts the given repeating decimal into a fraction by eliminating the repeating part through balancing equations and operations. This concept is commonly dealt with in high school algebra.
