What is 0.7619047619 as a fraction?
The Correct Answer and Explanation is:
The decimal 0.7619047619 can be expressed as the fraction ( \frac{16}{21} ).
Explanation:
To convert a repeating decimal into a fraction, we can use a method that involves algebraic manipulation. Here, the repeating part of 0.7619047619 is “761904,” which repeats indefinitely. This kind of decimal is known as a “repeating decimal.”
- Set up an equation: Let’s define ( x ) as the repeating decimal: [
x = 0.7619047619047619 \ldots
] - Multiply by a power of 10 to align the repeating section: Since “761904” repeats every six digits, we can multiply both sides of the equation by ( 10^6 = 1,000,000 ) to shift the decimal point six places to the right. [
1,000,000x = 761904.7619047619 \ldots
] - Set up a second equation for elimination: We also have the original equation ( x = 0.7619047619047619 \ldots ). Now, we can subtract this equation from the one above to eliminate the repeating decimal part: [
1,000,000x – x = 761904.7619047619 – 0.7619047619
] Simplifying this gives: [
999,999x = 761904
] - Solve for ( x ): To find the fraction, divide both sides by 999,999: [
x = \frac{761904}{999999}
] - Simplify the fraction: The fraction ( \frac{761904}{999999} ) simplifies to ( \frac{16}{21} ) by dividing both the numerator and denominator by their greatest common divisor, 47,619.
Verification:
To confirm, dividing ( 16 ) by ( 21 ) gives approximately 0.7619047619, matching the original decimal. This confirms that the decimal 0.7619047619 is accurately represented by the fraction ( \frac{16}{21} ).