Problem 66 - Algebra and Trigonometry

Question

Express the repeating decimal as a fraction. $$ 2.11 \overline{25} $$

Step-by-Step Solution

Answer

The repeating decimal \( 2.11\overline{25} \) as a fraction is \( \frac{10457}{4950} \).

Step 1: Define the repeating decimal

Identify the repeating section of the decimal number. The number given is \( 2.11\overline{25} \), meaning the digits '25' repeat indefinitely.

Step 2: Assign Variables

Let \( x = 2.11252525... \). We will express this repeating decimal as a fraction by using algebra.

Step 3: Multiply to Isolate Repetition

Multiply \( x \) by a power of 10 that shifts the repeating part to the right of the decimal point. Since '25' is a two-digit repeat, multiply by 100.\[ 100x = 211.252525... \]

Step 4: Subtract Equations

Subtract the original \( x \) from \( 100x \):\[100x - x = 211.252525... - 2.11252525...\]This simplifies to:\[99x = 209.14\]

Step 5: Solve for x

Divide both sides by 99 to isolate \( x \):\[ x = \frac{209}{99} + \frac{0.14}{99} \]

Step 6: Convert Remainder to Fraction

To convert \( 0.14 \) as a fraction, note that \( 0.14 = \frac{14}{100} = \frac{7}{50} \):So add this to the integer part:\[ x = \frac{209}{99} + \frac{7}{4950} \]

Step 7: Simplify the Fraction

Combine fractions over a common denominator:\[ x = \frac{209 \times 50 + 7}{4950} \]Calculate:\[ 209 \times 50 = 10450 \ 10450 + 7 = 10457 \ x = \frac{10457}{4950} \]Check the simplest form for divisibility extras.

Step 8: State the Final Fraction

The fraction that represents the repeating decimal \( 2.11\overline{25} \) is \( \frac{10457}{4950} \) in its simplified form.

Key Concepts

Repeating Decimals

Repeating decimals are decimals that have one or more repeating digits or a repeating sequence of digits after the decimal point.
For example, in the decimal number \(2.11\overline{25}\), the digits '25' repeat indefinitely. Recognizing this repeating pattern is crucial as it helps convert the decimal into a fraction.

• A repeating decimal can be expressed in different notations such as \(2.11252525...\).
• The bar notation \(\overline{25}\) indicates that the '25' will repeat endlessly.

Knowing how to identify the repeating section sets the foundation for converting the decimal into a fraction which is a core skill required for more complex mathematical problems.

Algebraic Techniques

Algebraic techniques come into play when you wish to convert a repeating decimal like \(2.11\overline{25}\) into a fraction.
This process mainly involves setting up an equation where the decimal is expressed using a variable (usually \(x\)). Here, we let \(x = 2.11252525...\).

• First, identify the repeating part. For two repeating digits like '25', multiply \(x\) by 100 to move the repetition out of the way.
• This creates an equation: \(100x = 211.252525...\).
• Set up a second equation with the original \(x\) and subtract it from this new equation.

Subtracting these equations helps to eliminate the repeating part, simplifying the equation and allowing for an easier solution to isolate \(x\). This is a fundamental algebraic step to solve challenges involving repeating decimals.

Fraction Simplification

Once you have created a fraction from the repeating decimal, the next step is simplifying it. This involves reducing the fraction to its smallest possible form without changing its value.
After isolating \(x\) and getting the fraction \(\frac{209}{99} + \frac{7}{4950}\), combine the terms over a common denominator.

• Utilize basic arithmetic to calculate the sum: \(209 \times 50 + 7\).
• Combine these calculations to make: \(\frac{10457}{4950}\).
• Check for any common factors in the numerator and denominator that you can divide out to simplify the fraction further.

Fraction simplification is a consistent part of various mathematical calculations and helps ensure that final results are both accurate and easy to interpret.

Decimal Conversion

Decimal conversion entails changing a repeating decimal into a fraction and sometimes converting back, so we understand both formats. It is a crucial step that requires accuracy and understanding.
For \(2.11\overline{25}\), converting it into a fraction involves a series of steps including the use of algebraic equations.

• By initially setting \(x = 2.11252525...\) and using algebraic manipulation, we can come up with \(x = \frac{10457}{4950}\).
• Gaining proficiency in decimal conversion allows one to go between fractions and repeating decimals comfortably, bridging the formats to meet specific scientific and real-world applications.
• Always double-check conversions for accuracy, especially if it plays a role in larger processes or calculations.

This knowledge is indispensable for many fields, from computing to engineering, where precise fractional values are frequently necessary.