Question

The recurring decimal \(0.555\) is equal to the rational number (a) \(\frac{9}{5}\) (b) \(\frac{5}{55}\) (c) \(\frac{5}{9}\) (d) \(\frac{8}{11}\)

Short answer

Answer: (c) \(\frac{5}{9}\)

Step-by-step solution

Express the recurring decimal as a fraction

Let \(x\) be the given recurring decimal; in this case, \(x = 0.555\). To express \(x\) as a fraction, we will multiply it by 10 and subtract the original equation from the new one to eliminate the recurring part. So, let \(10x = 5.555\). Now, subtract the original equation from the new equation: \(10x - x = 5.555 - 0.555\)

Solve for x

From the previous step, simplify the equation: \(9x = 5\) Now, divide both sides by 9 to find the value of \(x\): \(x = \frac{5}{9}\)

Compare the fraction to the given options

Now that we have expressed the recurring decimal as a fraction, we will compare it with the given options. We see that the fraction \(\frac{5}{9}\) matches option (c). So, the correct answer is: (c) \(\frac{5}{9}\)

Key concepts

Expressing Decimals as Fractions

When you encounter a recurring decimal like 0.555..., the goal is to convert it into a fraction. This process involves eliminating the repeating part of the decimal. Here's how you can do it.

• First, represent the decimal as a variable: let \( x = 0.555... \).
• Next, multiply the variable by a power of 10 to move the decimal point past the repeating part. Since 0.555... has one digit repeating, multiply by 10: \( 10x = 5.555... \).
• Now, subtract the original decimal equation from this new equation to get rid of the repeating parts: \( 10x - x = 5.555... - 0.555... \)

After performing these steps, you get \( 9x = 5 \), which simplifies down to \( x = \frac{5}{9} \). By doing this, you have expressed the recurring decimal as a simple fraction.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). Every recurring decimal can be transformed into a rational number.

This is because recursion indicates a repeated pattern that can be represented with a fixed ratio of integers. For example, the decimal 0.555... is recurring and it was successfully converted to the fraction \( \frac{5}{9} \).

• The numerator (top number of the fraction) reflects the part of the decimal beyond its repeating sequence.
• The denominator (bottom number of the fraction) comes from the subtraction and balancing done during conversion, representing how many parts the number is divided into in fractional format.

Understanding rational numbers is essential because they provide a clear way of expressing repeating decimals and terminating decimals in a compact number form.

Solving Equations

In this task, solving an equation was crucial to convert the decimal into a fraction. The primary equation derived from expressing the recurring decimal was \( 10x - x = 5.555... - 0.555... \).

• Firstly, simplification: By simplifying the left side to \( 9x \), we isolate the variable \( x \) on one side.
• Balancing the equation: Ensure the equation remains balanced by matching both sides. Here, this means correctly aligning \( 9x \) with 5 on the right side of the equation.
• Solution: Finally, solve for \( x \) by dividing both sides of the simplified equation \( 9x = 5 \) by 9, giving us \( x = \frac{5}{9} \).

This method demonstrates the fundamental approach in solving equations — manipulate the equation to isolate a single variable, maintaining equality throughout to derive a solution. Recognizing these steps is beneficial for handling any decimal-to-fraction conversion and beyond.