Problem 144 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Express the recurring decimal \(0.125125125 \ldots \ldots\) as a rational number.

Step-by-Step Solution

Answer

The recurring decimal \(0.125125125\ldots\) can be expressed as a rational number \(\frac{125}{999}\).

Step 1: Set the recurring decimal as a variable x

Let x be the given recurring decimal, i.e. \(x = 0.125125125\ldots\)

Step 2: Formulate the equation to isolate the repeating part

We need to isolate the repeating part, which is '125'. Multiply x by 1000 to shift the repeating digits to the left: \(1000x = 125.125125\ldots\)

Step 3: Subtract x from the new equation

Subtract the original equation from the equation obtained in Step 2: \((1000x - x) = (125.125125\ldots - 0.125125\ldots)\), which simplifies to: \(999x = 125\)

Step 4: Solve for x

Divide both sides of the equation by 999: \(x = \frac{125}{999}\)

Step 5: Simplify the result

Since 125 and 999 have no common factors, the fraction \(\frac{125}{999}\) cannot be simplified further. Therefore, the recurring decimal \(0.125125125\ldots\) can be expressed as a rational number \(\frac{125}{999}\).

Key Concepts

Rational Numbers

Rational numbers are a special type of number that can be expressed as the quotient or fraction of two integers. This means any rational number is in the form of \( \frac{a}{b} \), where both \( a \) and \( b \) are integers, and \( b \) is not zero. Here’s why this is exciting: rational numbers include fractions, whole numbers, and repeated decimals.

In the case of recurring decimals, each decimal has a pattern of digits that repeat infinitely. This is why when you see a number like \( 0.333... \) or in our example \( 0.125125125... \), it actually represents a rational number. Even though they might look a bit complex with their repeating nature, they can always find a simpler representation as a fraction of two whole numbers. This characteristic makes dealing with recurring decimals much simpler.

Decimal to Fraction Conversion

The conversion of a recurring decimal to a fraction is a systematic process. Let’s dive into the example of \( 0.125125125... \). First, you assume this repeating decimal is equal to a variable. We’ll call it \( x \), so \( x = 0.125125125... \).

Next, identify the repeating block of digits; in our case, it's '125'. To convert the number, multiply \( x \) by a power of 10 that aligns the decimal point with the start of the repeating sequence. For '125', we use 1000, as '125' contains three digits. This leads us to: \( 1000x = 125.125125... \).

The magic trick here is subtracting the original sequence from this new sequence: \( 1000x - x = 125.125125... - 0.125125... \), simplifying to \( 999x = 125 \). To find \( x \), divide both sides by 999 and you get \( x = \frac{125}{999} \), conveying that this long recurring decimal is actually a neat fraction!

Mathematical Equations

Mathematical equations are tools that express relationships between varying quantities. They come in all shapes and forms, and with recurring decimals, they're especially handy to convert them into fractions. An equation essentially shows that two expressions are equivalent, signified by an equal sign.

In the solution process, the step of creating equations is crucial. Initially, we form the equation \( x = 0.125125125... \) to represent our recurring decimal. By multiplying both sides, we generate the second equation \( 1000x = 125.125125... \).

Subtracting these two equations eliminates the repeating part, leaving a straightforward equation: \( 999x = 125 \), ready to solve. This highlights the power of mathematical equations: they simplify and connect complex ideas into manageable expressions. Solving such an equation involves basic arithmetic operations, one of the most powerful mathematical tools for dealing with recurring decimals.