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Chapter 4: Problem 4

For exercises 1-4, rewrite the repeating decimal as a fraction. $$ 0 . \overline{51} $$

Short Answer

Expert verified
\( 0 . \bar{51} = \frac{17}{33} \).

Step by step solution

01

Let the repeating decimal be x

Set the repeating decimal equal to a variable, say \( x \). So, let \( x = 0 . \, \bar{51} \).
02

Multiply both sides by a power of 10

Since the repeating block is two digits long (51), multiply both sides of the equation \( x = 0 . \, \bar{51} \) by 100 to shift the decimal point two places to the right: \( 100x = 51 . \, \bar{51} \).
03

Set up the subtraction equation

Write the two equations one above the other: \( 100x = 51 . \, \bar{51} \) \( - \) \( x = 0 . \, \bar{51} \) Subtract the second equation from the first to eliminate the repeating decimal: \( 100x - x = 51 . \, \bar{51} - 0 . \, \bar{51} \).
04

Simplify the subtraction

Simplify the left-hand side: \( 99x = 51 \). Now solve for \( x \) by dividing both sides of the equation by 99: \( x = \frac{51}{99} \).
05

Simplify the fraction

Simplify the fraction \( \frac{51}{99} \) by finding the greatest common divisor (GCD) of 51 and 99. The GCD is 3, so: \( \frac{51 \div 3}{99 \div 3} = \frac{17}{33} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Repeating Decimals
Repeating decimals are numbers that have one or more digits that repeat infinitely. For example, the repeating decimal in our exercise is \(0.\overline{51}\), where '51' repeats forever. These decimals are also called recurring decimals. Recognizing a repeating decimal is the first step in converting it to a fraction.
To deal with repeating decimals, we need to set the number equal to a variable, like x. This lets us handle it more easily with algebra.
Next, we look at the repeating digits to decide how to proceed. If one digit repeats, we multiply by 10. If two digits repeat, we multiply by 100, and so on.
Fractions
Fractions represent parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). For example, \(\frac{1}{2}\) means 1 part out of 2 equal parts.
In our exercise, we are converting a repeating decimal to a fraction. This means expressing the repeating decimal in a form like \(\frac{a}{b}\), where both a and b are integers. This fraction represents the same value as the repeating decimal but in a different form.
Working with fractions sometimes requires simplification. Simplifying fractions means making the numerator and denominator as small as possible while maintaining the same value.
Algebraic Manipulation
Algebraic manipulation involves using algebraic techniques to rearrange and solve equations. In the given solution, we begin by setting \(x = 0.\overline{51}\) and then multiplying both sides by 100 because the repeating block is two digits long.
This shifts the decimal point two places, converting it to another equation: \(100x = 51.\overline{51}\).
By subtracting the original equation from this new one, we eliminate the repeating part, simplifying our work.
Simplifying the result, like in \(99x = 51\), helps us isolate x and eventually convert the repeating decimal into a fraction.
Greatest Common Divisor
The Greatest Common Divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. It helps in simplifying fractions. In our exercise, the fraction we get is \(\frac{51}{99}\).
To simplify this fraction, we need to find the GCD of 51 and 99. Using techniques like the Euclidean algorithm, we find that the GCD is 3. Therefore, we divide both the numerator and denominator by 3, resulting in the simplified fraction \(\frac{17}{33}\).
Understanding the GCD helps with making fractions simpler and more manageable.

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