Problem 65 - Precalculus Mathematics for Calculus

Question

Express the repeating decimal as a fraction. $$0.030303 \dots$$

Step-by-Step Solution

Answer

The fraction is \(\frac{1}{33}\).

Step 1: Assign a variable to the repeating decimal

Let's set the repeating decimal to a variable. We have \( x = 0.030303\dots \).

Step 2: Set up an equation to eliminate the repeating part

Multiply \( x \) by 100 (since the repeating part is two digits long) to shift the decimal point two places to the right: \[ 100x = 3.030303\dots \]

Step 3: Subtract original equation from the multiplied equation

Subtract the original equation \( x = 0.030303\dots \) from \( 100x = 3.030303\dots \). This gives us: \[ 100x - x = 3.030303\dots - 0.030303\dots \] Simplifying, \[ 99x = 3 \]

Step 4: Solve for x

Divide both sides of the equation \( 99x = 3 \) by 99 to solve for \( x \): \[ x = \frac{3}{99} \]

Step 5: Simplify the fraction

Simplify \( \frac{3}{99} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3: \[ \frac{3}{99} = \frac{3 \div 3}{99 \div 3} = \frac{1}{33} \]

Step 6: Conclusion

The repeating decimal \( 0.030303\dots \) as a fraction in simplest form is \( \frac{1}{33} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is crucial when dealing with expressions that can be broken down into smaller parts. When we simplify a fraction, we aim to find its form with the smallest possible numbers in the numerator and denominator while maintaining the same value. In the solution provided, the fraction \( \frac{3}{99} \) was simplified to \( \frac{1}{33} \). This process involves identifying common factors between the numerator (3) and the denominator (99).

• First, we need to determine the greatest common divisor (GCD) of the two numbers, which is a key factor in simplifying fractions.
• For \( \frac{3}{99} \), we divided both the numerator and the denominator by 3, which is their GCD, resulting in the simplified form \( \frac{1}{33} \).

Simplification not only makes the fraction easier to work with but also helps in comparing, adding, or subtracting fractions in mathematical problems.
Understanding this concept allows us to ensure that our final fraction results are presented in their simplest and most interpretable form.

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental mathematical concept used to simplify fractions and solve various problems. It is the largest number that divides two or more integers without leaving a remainder. In the context of simplifying fractions, finding the GCD is a straightforward process.

• For integers 3 and 99, we need to find their prime factors. The number 3 is already a prime number, and for 99, the prime factorization is 3 and 33 (since 99 = 3 x 33). Their common factor is 3.
• Using the GCD of 3, the fraction \( \frac{3}{99} \) can be simplified by dividing both the numerator and the denominator by 3, yielding \( \frac{1}{33} \).

Finding the GCD is not only essential in simplifying fractions but also plays a pivotal role in other areas such as least common multiples, algebra and even in number theory. This lays a foundation for deeper mathematical understanding and application.

Precalculus Mathematics

Precalculus Mathematics serves as a bridge between algebra and calculus, introducing essential concepts that pave the way for more complex analysis. One of its primary components involves understanding numbers and expanding or simplifying them.

• Understanding the conversion of repeating decimals into fractions is an integral part of this. Repeating decimals, like 0.030303\( \ldots \), offer an opportunity to apply algebraic techniques to express them as exact fractions.
• Assigning a variable, shifting decimal places, and setting equations to eliminate the repeating component are steps that utilize precalculus techniques to solve problems seamlessly.

Grasping these concepts equips students better for calculus, where precision and accuracy in number representation become increasingly important. It encourages critical thinking and problem-solving skills that are vital across all advanced mathematics disciplines.