Final answer:
The recurring decimal 0.7 with a line over the 7 can be represented as the simplified fraction ⅔. This is done by setting the decimal equal to x, multiplying by 10 to create 10x, and then subtracting x to solve for x and find the fraction.
Step-by-step explanation:
When you see a number like 0.7 with a line over the 7, it indicates that the 7 is repeating indefinitely, making it a recurring decimal. To convert this recurring decimal to a simplified fraction, you can use a simple algebraic method. Let's denote the recurring decimal as x:
x = 0.777...
Multiplying both sides of the equation by 10 shifts the decimal point one position to the right:
10x = 7.777...
Now, subtract the original x from 10x to eliminate the repeating digits:
10x - x = 7.777... - 0.777...
9x = 7
Dividing both sides by 9 gives us the fraction:
x = ⅔
Therefore, 0.7 with a line over the 7 is equal to ⅔ as a simplified fraction.