Problem 75 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(0.272727 \ldots, x\) and \(0.727272 \ldots\) are in H.P., then \(x\) must be (A) rational (B) integer (C) irrational (D) None of these

Step-by-Step Solution

Answer

(A) rational.

Step 1: Understanding Harmonic Progression (H.P.)

When three numbers are in Harmonic Progression, their reciprocals are in Arithmetic Progression (A.P.). Therefore, if \(0.272727 \ldots\), \(x\), and \(0.727272 \ldots\) are in H.P., then the reciprocals \(0.363636\ldots\), \(\frac{1}{x}\), and \(1.375\) must be in A.P.

Step 2: Convert Repeating Decimals to Fractions

The repeating decimal \(0.272727\ldots\) is equal to \(\frac{3}{11}\) and \(0.727272\ldots\) is equal to \(\frac{8}{11}\). Now, find their reciprocals: the reciprocal of \(\frac{3}{11}\) is \(\frac{11}{3}\) and the reciprocal of \(\frac{8}{11}\) is \(\frac{11}{8}\).

Step 3: Set Up the Arithmetic Progression Equation

In an A.P., the difference between consecutive terms is constant. Let the reciprocals form the A.P. sequence as such: \(\frac{11}{3}, \frac{1}{x}, \frac{11}{8}\). The A.P. condition is \[\left(\frac{1}{x} - \frac{11}{3}\right) = \left(\frac{11}{8} - \frac{1}{x}\right)\].

Step 4: Solve for \(\frac{1}{x}\)

Simplify and solve the equation: \[\frac{1}{x} - \frac{11}{3} = \frac{11}{8} - \frac{1}{x}\]. Combine terms to get \[2 \cdot \frac{1}{x} = \frac{11}{3} + \frac{11}{8}\]. Take the LCM of \(3\) and \(8\), which is \(24\). Hence, the equation becomes \[2 \cdot \frac{1}{x} = \frac{88 + 33}{24} = \frac{121}{24}\].

Step 5: Find \(x\)

Solving \[2 \cdot \frac{1}{x} = \frac{121}{24}\] gives \[\frac{1}{x} = \frac{121}{48}\]. Therefore, \(x\) is \(\frac{48}{121}\).

Step 6: Determine the Nature of \(x\)

Since \(x = \frac{48}{121}\) and both \(48\) and \(121\) are integers with no common factor besides \(1\), \(x\) is a rational number. Therefore, the correct choice is (A) rational.

Key Concepts

Arithmetic Progression

When we talk about arithmetic progression, often abbreviated as A.P., we mean a sequence of numbers where the difference between consecutive terms is always the same. This difference is known as the 'common difference.'
For example, in the sequence 3, 7, 11, 15, each term after the first is obtained by adding 4. Here, 4 is the common difference. A.P.s are characterized by their simplicity and regularity.

• General Form: If you're given an A.P. starting with an initial term 'a' and having a common difference 'd', it can be expressed generally as: a, a+d, a+2d, and so on.
• Properties: The nth term of an A.P. can be calculated by the formula: \(a_n = a + (n-1)d\).

In the context of the solution, to understand Harmonic Progression (H.P.) in relation to arithmetic progression, we realize that when numbers are said to be in an H.P., their reciprocals follow a pattern described by an arithmetic progression. By this transformation, we're able to solve harmonic sequences with techniques applicable to A.P.s.

Repeating Decimals

Repeating decimals, sometimes known as recurring decimals, are decimal numbers in which a pattern of one or more digits repeatedly continues forever. Such decimals are interesting because they demonstrate a connection between decimal fractions and rational numbers.
For example, the decimal 0.272727... is repeating because the digits '27' go on infinitely. Similarly, 0.727272... repeats the digits '72'.

• Conversion to Fraction: To convert a repeating decimal to a fraction, first identify the repeating sequence. For 0.272727..., '27' is the repeating part. Here is how you can convert 0.272727... into a fraction:
  • Set the repeating decimal as a variable, e.g., let \( x = 0.272727\ldots \).
  • Multiply the equation by 100 (because '27' has two digits), so \( 100x = 27.272727\ldots \).
  • Subtract the original equation from this new equation: \( 100x - x = 27.272727\ldots - 0.272727\ldots \).
  • This simplifies to \( 99x = 27 \), so \( x = \frac{27}{99} \).
  • Simplify the fraction: \( \frac{27}{99} = \frac{3}{11} \).

Thus, repeating decimals are indeed rational numbers as they can convert into fractions.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, with the denominator \( q \) not equal to zero.
Here's how you can think about rational numbers in the context of this exercise:

• Characteristics of Rational Numbers:
  • Any integer is a rational number because it can be written as itself over 1, \( e.g., 5 = \frac{5}{1}\).
  • Decimals that terminate (end) or repeat (have a specific digit or set of digits that repeat) are also rational numbers.
• Connection to the Exercise: In this problem, the number \( x \) was determined as \( \frac{48}{121} \), a simple fraction with integers in the numerator and denominator. Because both integers 48 and 121 are whole numbers and share no common factors, besides 1, \( x \) is confirmed to be a rational number.

Rational numbers are integral to many branches of mathematics because they help form a bridge between integers and real numbers, facilitating calculations and transformations.