Book Reading Problem: Ananth and Bharath's Pace

This problem involves understanding the concept of reading pace and calculating the number of pages remaining for two individuals, Ananth and Bharath, who read a book at different constant rates.

Reading Rates Defined

First, let's define the reading rates for both Ananth and Bharath. Let the total number of pages in the book be denoted by \(P\).

• Ananth's time to read the entire book = 6 hours.
• Bharath's time to read the entire book = 4 hours.

From this, we can determine their respective reading rates (pages per hour):

Pages Remaining Formula

Let \(t\) be the number of hours after which we want to find the relationship between the pages remaining for Ananth and Bharath.

After \(t\) hours:

• Pages read by Ananth = Ananth's reading rate \(\times\) time = \(\left(\frac{P}{6}\right) \times t = \frac{Pt}{6}\).
• Pages remaining for Ananth (\(R_A\)) = Total pages - Pages read by Ananth = \(P - \frac{Pt}{6}\).
• Pages read by Bharath = Bharath's reading rate \(\times\) time = \(\left(\frac{P}{4}\right) \times t = \frac{Pt}{4}\).
• Pages remaining for Bharath (\(R_B\)) = Total pages - Pages read by Bharath = \(P - \frac{Pt}{4}\).

Equation Setup for Remaining Pages

The problem states that the number of pages remaining to be read by Ananth is twice that the number of pages remaining to read by Bharath. This can be written as:

\(R_A = 2 \times R_B\)

Substitute the expressions for \(R_A\) and \(R_B\):

\(P - \frac{Pt}{6} = 2 \times \left(P - \frac{Pt}{4}\right)\)

Time Calculation for Remaining Pages

Now, let's solve this equation for \(t\). We can divide both sides by \(P\) (since \(P\) is the total number of pages, it must be greater than 0):

\(1 - \frac{t}{6} = 2 \times \left(1 - \frac{t}{4}\right)\)

Distribute the 2 on the right side:

\(1 - \frac{t}{6} = 2 - \frac{2t}{4}\)

Simplify the term \(\frac{2t}{4}\):

\(1 - \frac{t}{6} = 2 - \frac{t}{2}\)

To solve for \(t\), gather all terms involving \(t\) on one side and constants on the other:

\(\frac{t}{2} - \frac{t}{6} = 2 - 1\)

\(\frac{t}{2} - \frac{t}{6} = 1\)

Find a common denominator for the fractions involving \(t\). The least common multiple of 2 and 6 is 6:

\(\frac{3t}{6} - \frac{t}{6} = 1\)

Combine the fractions:

\(\frac{3t - t}{6} = 1\)

\(\frac{2t}{6} = 1\)

Simplify the fraction:

\(\frac{t}{3} = 1\)

Multiply both sides by 3 to find \(t\):

\(t = 3\)

Answer Verification for Reading Time

Let's check if our calculated time of 3 hours satisfies the condition:

• Pages remaining for Ananth after 3 hours: \(P - \frac{P \times 3}{6} = P - \frac{P}{2} = \frac{P}{2}\).
• Pages remaining for Bharath after 3 hours: \(P - \frac{P \times 3}{4} = P - \frac{3P}{4} = \frac{P}{4}\).

Is the number of pages remaining for Ananth twice that of Bharath?

\(\frac{P}{2} = 2 \times \frac{P}{4}\)

\(\frac{P}{2} = \frac{2P}{4}\)

\(\frac{P}{2} = \frac{P}{2}\)

The condition holds true. Therefore, after 3 hours, the number of pages remaining for Ananth is twice the number of pages remaining for Bharath.

Final Resulting Hours

The number of hours after which the pages remaining to be read by Ananth is twice that of Bharath is 3 hours.