0.75 repeating as a fraction in simplest form

0.75 repeating as a fraction in simplest form

The Correct Answer and Explanation is :

To express ( 0.\overline{75} ) (which is 0.757575…) as a fraction in simplest form, we can use a method that involves algebraic manipulation.

Step 1: Let ( x = 0.\overline{75} ).

Let ( x ) represent the repeating decimal, so:
[
x = 0.75757575\ldots
]

Step 2: Eliminate the repeating part.

Multiply both sides of the equation by 100 to shift the decimal point two places to the right:
[
100x = 75.757575\ldots
]

Now, we have two equations:

1. ( x = 0.757575\ldots )
2. ( 100x = 75.757575\ldots )

Step 3: Subtract the two equations.

Subtract the first equation from the second:
[
100x – x = 75.757575\ldots – 0.757575\ldots
]
This simplifies to:
[
99x = 75
]

Step 4: Solve for ( x ).

Now, solve for ( x ) by dividing both sides of the equation by 99:
[
x = \frac{75}{99}
]

Step 5: Simplify the fraction.

Next, simplify the fraction ( \frac{75}{99} ). The greatest common divisor (GCD) of 75 and 99 is 3. To simplify, divide both the numerator and denominator by 3:
[
\frac{75}{99} = \frac{75 \div 3}{99 \div 3} = \frac{25}{33}
]

Thus, ( 0.\overline{75} ) as a fraction is ( \frac{25}{33} ).

Conclusion:

The repeating decimal ( 0.\overline{75} ) is equivalent to the fraction ( \frac{25}{33} ) in its simplest form.

This method of converting repeating decimals to fractions is based on algebraic manipulation and shows how to deal with repeating patterns by eliminating them step by step.