$\begin{array}{l}100 n=54 . \overline{4} \\ \frac{-10 n=5.4}{10 n=\frac{49}{49}} \\ \frac{90 n}{90 n}=\frac{49}{90} \\ \left.n=\frac{49}{90} \right\rvert\, 50.05 \overline{4}=\end{array}$
Answer (1)
Define n = 0.05 4 .
Multiply by 100 and 1000 to get 100 n = 5. 4 and 1000 n = 54. 4 .
Subtract 100 n from 1000 n to eliminate the repeating part: 900 n = 49 .
Solve for n : n = 900 49 . The final answer is 900 49 .
Explanation
Understanding the Problem The problem is to convert the repeating decimal 0.05 4 into a fraction. The provided solution attempts to do this but makes an error in the initial setup and subtraction. Let's correct this and find the right fraction.
Setting up the Equations Let n = 0.05 4 . Our goal is to express n as a fraction. To eliminate the repeating part, we'll multiply n by powers of 10.
Multiplying by 100 First, multiply n by 100 to move the decimal point two places to the right: 100 n = 5. 4 = 5.4444...
Multiplying by 1000 Next, multiply n by 1000 to move the decimal point three places to the right: 1000 n = 54. 4 = 54.4444...
Subtracting the Equations Now, subtract 100 n from 1000 n to eliminate the repeating decimal part: 1000 n − 100 n = 54. 4 − 5. 4 900 n = 49
Solving for n Solve for n by dividing both sides by 900: n = 900 49
Final Answer Therefore, the repeating decimal 0.05 4 is equal to the fraction 900 49 .
Examples
Repeating decimals can be converted into fractions, which is useful in various real-world scenarios. For instance, when dealing with precise measurements or financial calculations, converting a repeating decimal to a fraction ensures accuracy. Imagine you're calculating the exact amount of ingredients for a recipe, and one measurement is given as a repeating decimal. Converting it to a fraction allows for precise measurements, ensuring the recipe turns out perfectly. Similarly, in finance, converting repeating decimals to fractions can help in accurate calculations of interest rates or currency conversions.
