Problem 146 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the value of \(0.4 \overline{23}\).

Step-by-Step Solution

Answer

The value of 0.4\( \overline{23} \) is \( \frac{42}{99} \).

Step 1: Assign the repeating decimal to a variable

Assigning 0.4\( \overline{23} \) to a variable, let's say \( x \), so we have \( x = 0.4 \overline{23} \). It is good to represent a large number in terms of variables, as it makes calculation much easier.

Step 2: Create an equation to get rid of decimal

To remove the decimals, multiply \( x \) by 100. We get \( 100x = 42.\overline{23} \).

Step 3: Subtract the initial equation from this new equation

Subtracting the first equation from the second to eliminate the decimals, gives us 99x = 42. This simplification effectively eliminates the repeating decimal.

Step 4: Solve for x

After getting rid of decimals by making a second equation and subtraction, solve for \( x \) to get the value. Dividing both sides by 99 gives \( x = \frac{42}{99} \).

Key Concepts

Rational Numbers

Rational numbers are fundamental in understanding different types of numbers we encounter in mathematics. These numbers can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. For example, the fraction \( \frac{42}{99} \) is a rational number because it consists of two integers. Rational numbers are contrasted with irrational numbers, which cannot be expressed as such fractions.
To illustrate, consider the decimal form of certain rational numbers. They might appear as either terminating decimals, like \( 0.5 = \frac{1}{2} \), or repeating decimals, such as \( 0.333\ldots = \frac{1}{3} \). This concept is important when converting decimals to fractions, as it reveals the underlying rational number behind a seemingly endless decimal cycle.

Decimal Representation

Decimal representation refers to expressing numbers in a base-ten numeral system. It's the most common way we write numbers today. This system allows us to easily handle both whole numbers and decimals, which is particularly useful for calculations and measurements. Each digit represents a power of ten, helping us understand the size and scale of the number.
Non-integer numbers can be either terminating or repeating decimals. While terminating decimals have a fixed number of digits (e.g., \( 0.75 \)), repeating decimals have one or more digits that repeat indefinitely, such as \( 0.4\overline{23} \). This repeating pattern provides a clue that a decimal can be converted into a rational number. To deal with repeating decimals, techniques like assigning variables and solving algebraic equations are employed. This allows us to find the equivalent fraction form.

Algebraic Manipulation

Algebraic manipulation involves rearranging and modifying mathematical expressions to solve equations or simplify expressions. It is a handy tool in converting repeating decimals into fractions, as demonstrated in the original exercise.
By assigning the repeating decimal \( x = 0.4\overline{23} \), and multiplying by a power of ten, we create an algebraic equation that helps isolate the repeating part. In this case, multiplying by 100 results in \( 100x = 42.\overline{23} \). Through subtraction of the initial equation \( x = 0.4\overline{23} \) from the multiplied equation, the repeating decimal is effectively removed, leaving a simple linear equation. Solving for \( x \) involves straightforward algebraic steps, concluding with \( x = \frac{42}{99} \). This process showcases the power of algebra to deal with repeating decimal patterns.