Convert the decimal $0 . \overline{423}$ to a fraction.
Answer (2)
Let x = 0. 423 .
Multiply by 1000: 1000 x = 423. 423 .
Subtract the equations: 999 x = 423 .
Solve for x and simplify: x = 999 423 โ = 111 47 โ .
The fraction is 111 47 โ โ .
Explanation
Understanding the Problem We are asked to convert the repeating decimal 0. 423 to a fraction. This means the decimal representation is 0.423423423... where the block '423' repeats indefinitely.
Setting up the Equation Let x = 0. 423 . Our goal is to express x as a fraction q p โ , where p and q are integers and q ๎ = 0 .
Multiplying by 1000 To eliminate the repeating part, we multiply x by 1000, since the repeating block has a length of 3 digits. This gives us 1000 x = 423. 423 = 423.423423423...
Subtracting the Equations Now, we subtract the original equation x = 0. 423 from the new equation 1000 x = 423. 423 :
1000 x โ x = 423. 423 โ 0. 423 which simplifies to 999 x = 423 .
Solving for x Next, we solve for x by dividing both sides of the equation 999 x = 423 by 999: x = 999 423 โ .
Simplifying the Fraction Finally, we simplify the fraction 999 423 โ by finding the greatest common divisor (GCD) of 423 and 999. The GCD of 423 and 999 is 9. Dividing both the numerator and the denominator by 9, we get: 999 รท 9 423 รท 9 โ = 111 47 โ . Therefore, the simplified fraction is 111 47 โ .
Final Answer Thus, the decimal 0. 423 is equal to the fraction 111 47 โ .
Examples
Repeating decimals appear in various real-world scenarios, such as when dividing quantities that don't result in whole numbers. For example, if you divide 47 by 111 using a calculator, you'll get approximately 0.423423423. Converting repeating decimals to fractions allows for more precise calculations and is essential in fields like engineering, finance, and physics, where accuracy is crucial. Understanding how to convert repeating decimals to fractions helps in accurately representing and manipulating these values in mathematical and practical contexts.
The repeating decimal 0.ar{423} is converted into the fraction 111 47 โ through a series of algebraic steps. By representing the decimal as x , multiplying to eliminate the repeating part, and simplifying, we arrive at the final fraction. The GCD method helps in ensuring the fraction is in its simplest form.
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