(b) Write the recurring decimal 0.18 as a fraction in its lowest terms. [0.18 means 0.181818...]
Answer (2)
Let x = 0.181818...
Multiply by 100: 100 x = 18.181818...
Subtract the equations: 99 x = 18
Solve and simplify: x = 99 18 = 11 2 11 2
Explanation
Understanding the Problem We are asked to express the recurring decimal 0.181818... as a fraction in its simplest form. Let's denote the recurring decimal as x .
Multiplying by 100 Let x = 0.181818... . To convert this recurring decimal to a fraction, we multiply x by 100, since the repeating block has a length of 2. This gives us 100 x = 18.181818... .
Subtracting the Equations Now, we subtract the original equation from the multiplied equation: 100 x − x = 18.181818... − 0.181818... . This simplifies to 99 x = 18 .
Solving for x Next, we solve for x by dividing both sides of the equation by 99: x = 99 18 .
Finding the GCD To express the fraction in its lowest terms, we need to find the greatest common divisor (GCD) of 18 and 99. The GCD of 18 and 99 is 9.
Simplifying the Fraction Finally, we divide both the numerator and the denominator by the GCD: x = 99 ÷ 9 18 ÷ 9 = 11 2 . Therefore, the recurring decimal 0.181818... can be expressed as the fraction 11 2 in its lowest terms.
Examples
Recurring decimals appear in various real-world scenarios, such as when dividing quantities that don't result in whole numbers. For example, if you divide a pizza into 11 slices and want to give 2 slices to each person, the fraction of the pizza each person receives is 11 2 , which is approximately 0.181818... This understanding helps in fair distribution and accurate measurements.
The recurring decimal 0.181818... can be converted into the fraction 11 2 by following a mathematical process involving multiplication, subtraction, and simplification. By defining the decimal as x , multiplying by 100, and solving for x , we find the fraction's simplest form. Therefore, 0.181818... equals 11 2 .
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