Problem 78 - A Complete Resource Book in Mathematics for JEE Main

Question

The consecutive numbers of a three digit number form a G.P. If we subtract 792 from this number, we get a number consisting of the same digits written in the reverse order and if we increase the second digit of the required number by 2, the resulting number forms an A.P. The number is (A) 139 (B) 193 (C) 931 (D) None of these

Step-by-Step Solution

Answer

The number is 931.

Step 1: Define the Problem

Let's denote the three-digit number as \(abc\), where \(a, b, c\) are its digits. According to the problem, these digits form a geometric progression (G.P.). Therefore, we can express this relationship as \(b^2 = ac\).

Step 2: Use Reverse Number Property

According to the problem, if we subtract 792 from the number, we get the reversed number. Mathematically, this can be written as \(100a + 10b + c - 792 = 100c + 10b + a\). Simplifying this equation, we get \(99a + c - 792 = 99c + a\), which further simplifies to \(98a = 98c + 792\). Thus, \(a = c + 8\).

Step 3: Use Arithmetic Progression Condition

When we increase the second digit \(b\) by 2, the digits form an arithmetic progression (A.P.). Therefore, \(b + 2\) is the arithmetic mean of \(a\) and \(c\), or \(b + 2 = \frac{a + c}{2}\). Simplifying this gives \(2b + 4 = a + c\).

Step 4: Represent and Solve Equations

Now we have three equations:1. \(a = c + 8\)2. \(b^2 = ac\)3. \(2b + 4 = a + c\) Substitute \(a = c + 8\) into the other two equations.For the G.P. relation: \(b^2 = (c + 8)c\).For the A.P. relation: \(2b + 4 = (c + 8) + c\). Simplifying the A.P. equation gives \(2b + 4 = 2c + 8\), resulting in \(2b = 2c + 4\), hence \(b = c + 2\).

Step 5: Check for Consistency

Substitute \(b = c + 2\) into the G.P. equation: \((c + 2)^2 = (c + 8)c\).Now expand and simplify the equation to: \(c^2 + 4c + 4 = c^2 + 8c\).Thus, \(4c + 4 = 8c\).From which, \(4c = 4\) leading to \(c = 1\).

Step 6: Identify the Digits

With \(c = 1\), we substitute back to find \(b\) and \(a\).From \(b = c + 2\), \(b = 1 + 2 = 3\).From \(a = c + 8\), \(a = 1 + 8 = 9\).

Step 7: Form the Number

The digits are \(a = 9\), \(b = 3\), and \(c = 1\). Therefore, the number \(abc\) is 931.

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers where each term after the first is obtained by adding a constant difference to the previous term. In simple terms, it's a series of numbers where each number is equally spaced from the next.
For example, in the sequence 2, 4, 6, 8, the difference between consecutive terms is 2, which is the constant difference and can be denoted as \(d\).

• The formula for the nth term \(a_n\) of an arithmetic progression is: \(a_n = a_1 + (n-1) \times d\)
• The sum \(S_n\) of the first n terms of an A.P. is given by: \(S_n = \frac{n}{2} \times (2a_1 + (n-1)d)\)

In the context of our problem, it indicates that when the second digit of the three-digit number is increased by 2, the resulting sequence formed by its digits will exhibit this property, meaning they will be equal-spaced.

Three-digit numbers

Three-digit numbers range from 100 to 999 and consist of three places: the hundreds, tens, and units. Each place represents a specific digit, and the value of the entire number is calculated based on these positions. For instance, the number 931 is a three-digit number where:

• 9 is in the hundreds place
• 3 is in the tens place
• 1 is in the units place
• Thus, \(931 = (9 \times 100) + (3 \times 10) + (1 \times 1)\)

Understanding the concepts of place value and how digits form numbers is crucial in deciphering mathematical problems involving three-digit numbers, like the one in this exercise.

Reversal of numbers

The reversal of a three-digit number involves switching the positions of the digits. For example, reversing 735 gives 537. This concept is integral in solving the current problem as one of the conditions states that subtracting 792 from the number results in its reverse.
Mathematically, given a number \(abc\), its reversal is \(cba\). The place values change as follows:

• The hundreds digit \(a\) becomes the units digit \(c\).
• The tens digit \(b\) remains the same.
• The units digit \(c\) becomes the hundreds digit \(a\).

Using this rule, and knowing the numerical values, you can set up algebraic equations to find relationships between the original and reversed numbers, aiding in problem-solving.

Problem-solving in mathematics

Problem-solving in mathematics involves understanding the problem, devising a plan, executing it, and reviewing the solution. The process can be applied to various mathematical scenarios, such as the exercise we're discussing.
For this specific problem: - **Identify and Define**: Recognize the digits form a geometric progression, reverse order, and form an arithmetic progression under certain conditions. - **Set Up Equations**: Use known relationships to set up equations that illustrate the given conditions. - **Solve Systematically**: Substitute known values into these equations to find unknown digits, which form a solution. - **Verify Solution**: Check the solution meets all conditions given initially. Using these steps systematically improves understanding and efficiency when solving complex mathematical challenges, allowing solutions like determining the three-digit number 931.