Write 0.78 repeating as a fraction is simplest form,
The Correct Answer and Explanation is :
To express the repeating decimal (0.\overline{78}) (which means 0.787878… repeating) as a fraction in its simplest form, follow these steps:
Step 1: Set up the equation
Let (x = 0.\overline{78}). This means that:
[
x = 0.787878\ldots
]
Step 2: Eliminate the repeating decimal
Multiply both sides of the equation by 100 to shift the decimal point two places to the right, since the repeating block “78” is two digits long:
[
100x = 78.787878\ldots
]
Now you have two equations:
[
x = 0.787878\ldots \quad \text{(Equation 1)}
]
[
100x = 78.787878\ldots \quad \text{(Equation 2)}
]
Step 3: Subtract the two equations
Subtract Equation 1 from Equation 2 to eliminate the repeating decimal part:
[
100x – x = 78.787878\ldots – 0.787878\ldots
]
Simplifying this:
[
99x = 78
]
Step 4: Solve for (x)
Now, solve for (x) by dividing both sides of the equation by 99:
[
x = \frac{78}{99}
]
Step 5: Simplify the fraction
To simplify the fraction (\frac{78}{99}), find the greatest common divisor (GCD) of 78 and 99. The GCD of 78 and 99 is 3. Now, divide both the numerator and the denominator by 3:
[
\frac{78}{99} = \frac{78 \div 3}{99 \div 3} = \frac{26}{33}
]
Thus, the fraction representing (0.\overline{78}) in its simplest form is:
[
\boxed{\frac{26}{33}}
]
Explanation
The process of converting a repeating decimal to a fraction involves creating an equation for the repeating decimal, manipulating it algebraically by shifting the decimal point, and then eliminating the repeating part through subtraction. After this, the resulting fraction is simplified by dividing both the numerator and denominator by their greatest common divisor. This ensures the fraction is in its simplest form, as (\frac{26}{33}) cannot be reduced further.