Express the recurring decimal $0.1 \dot{3} \dot{6}$ as a fraction in its simplest form. Use $x$ in your working.
Answer (2)
Let x = 0.1\[36\] .
Multiply by 10: 10x = 1.\[36\] .
Multiply by 1000: 1000x = 136.\[36\] .
Subtract to eliminate the repeating part: 990 x = 135 , so x = 990 135 = 22 3 .
The fraction in its simplest form is 22 3 .
Explanation
Understanding the Problem We are asked to express the recurring decimal 0.1\[36\] as a fraction in its simplest form, using x in our working. This means that the digits '36' repeat indefinitely after the digit 1.
Multiplying by 10 Let x = 0.1\[36\] = 0.1363636... . To eliminate the repeating part, we'll multiply by powers of 10. First, multiply by 10 to move the decimal point one place to the right: 10 x = 1.363636...
Multiplying by 1000 Next, we want to shift the decimal two places to the right so that the repeating block lines up. Multiply by 1000: 1000 x = 136.363636...
Subtracting to Eliminate the Repeating Part Now, subtract 10 x from 1000 x to eliminate the repeating decimal part: 1000 x − 10 x = 136.363636... − 1.363636... This simplifies to: 990 x = 135
Solving for x Now, solve for x by dividing both sides by 990: x = 990 135
Simplifying the Fraction To simplify the fraction, we need to find the greatest common divisor (GCD) of 135 and 990. The GCD of 135 and 990 is 45. Divide both the numerator and the denominator by 45: x = 990 ÷ 45 135 ÷ 45 = 22 3
Final Answer Therefore, the recurring decimal 0.1\[36\] expressed as a fraction in its simplest form is 22 3 .
Examples
Recurring decimals appear in various real-world scenarios, such as when converting units or dealing with financial calculations involving percentages. For example, when calculating compound interest or converting currencies, recurring decimals may arise. Expressing these decimals as fractions allows for more precise calculations and easier manipulation of the values. Understanding how to convert recurring decimals to fractions is essential for accurate financial analysis and unit conversions.
The recurring decimal 0.1 3 ˙ 6 ˙ can be expressed as the fraction 22 3 after a series of steps involving multiplication and subtraction to eliminate the repeating part. Through this method, we simplify the fraction to its simplest form. This process ensures accurate representation of the recurring decimal as a fraction.
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