Understanding Time, Speed, and Distance Problems

This problem involves the concepts of time, speed, and distance, specifically focusing on scenarios where two individuals move towards each other and the times taken after their meeting point are given. Such problems often require understanding the relationship between speed and time, and a specific formula can be applied when the travel times after meeting are known.

Analyzing the Given Information

• Person A starts from place X and person B starts from place Y at the same time.
• They move towards each other and meet somewhere between X and Y.
• Let the meeting point be M.
• Let \(v_A\) be the speed of A and \(v_B\) be the speed of B.
• It is given that the speed of A is 20% more than that of B.
  • \(v_A = v_B + 0.20 v_B = 1.20 v_B\)
  • Therefore, the ratio of their speeds is \(\frac{v_A}{v_B} = 1.2\) or \(\frac{v_A}{v_B} = \frac{12}{10} = \frac{6}{5}\).
• After meeting at M, A takes \(t_A = p\) hours to reach Y (the remaining distance MY).
• After meeting at M, B takes \(t_B = 7\frac{1}{5}\) hours to reach X (the remaining distance MX).
  • \(t_B = 7 + \frac{1}{5} = 7 + 0.2 = 7.2\) hours.

Key Formula for Meeting Point Problems

When two people start at the same time from two points and move towards each other, and after meeting, they take \(t_A\) and \(t_B\) time respectively to reach the opposite starting points, the ratio of their speeds is related to these times by the formula:

\( \frac{\text{Speed of A}}{\text{Speed of B}} = \sqrt{\frac{\text{Time B takes after meeting}}{\text{Time A takes after meeting}}} \)

\( \frac{v_A}{v_B} = \sqrt{\frac{t_B}{t_A}} \)

Solving for the Unknown Time (p)

We have the ratio of speeds \(\frac{v_A}{v_B} = 1.2\) and the times \(t_A = p\) hours and \(t_B = 7.2\) hours. We can substitute these values into the formula:

\( 1.2 = \sqrt{\frac{7.2}{p}} \)

To solve for \(p\), we first square both sides of the equation:

\( (1.2)^2 = \left(\sqrt{\frac{7.2}{p}}\right)^2 \)

\( 1.44 = \frac{7.2}{p} \)

Now, we can rearrange the equation to find \(p\):

\( p \times 1.44 = 7.2 \)

\( p = \frac{7.2}{1.44} \)

To simplify the division, we can multiply both the numerator and denominator by 100 to remove decimals:

\( p = \frac{7.2 \times 100}{1.44 \times 100} = \frac{720}{144} \)

Now, perform the division:

\( \frac{720}{144} = 5 \)

So, the value of \(p\) is 5 hours.

Summary of Calculation

The value of \(p\) is 5.

Revision Table: Key Concepts in Time and Distance

Additional Information: Understanding the Meeting Point Formula Derivation

Let the distance from X to M be \(d_A\) and the distance from M to Y be \(d_B\). Since A and B start at the same time and meet at M, the time taken by A to reach M is the same as the time taken by B to reach M. Let this time be \(T\).

• Distance \(d_A = v_A \times T\) (Distance covered by A before meeting)
• Distance \(d_B = v_B \times T\) (Distance covered by B before meeting)

After meeting at M:

• A travels distance \(d_B\) (from M to Y) in time \(t_A = p\). So, \(d_B = v_A \times t_A = v_A \times p\).
• B travels distance \(d_A\) (from M to X) in time \(t_B = 7.2\). So, \(d_A = v_B \times t_B = v_B \times 7.2\).

We have two expressions for \(d_A\) and \(d_B\):

• \(d_A = v_A T\) and \(d_A = v_B \times 7.2\)
• \(d_B = v_B T\) and \(d_B = v_A \times p\)

From the first pair: \(v_A T = v_B \times 7.2 \implies \frac{v_A}{v_B} = \frac{7.2}{T}\) (Equation 1)

From the second pair: \(v_B T = v_A \times p \implies \frac{v_B}{v_A} = \frac{p}{T} \implies \frac{v_A}{v_B} = \frac{T}{p}\) (Equation 2)

Equating the expressions for \(\frac{v_A}{v_B}\) from Equation 1 and Equation 2:

\( \frac{7.2}{T} = \frac{T}{p} \)

\( T^2 = 7.2 \times p \)

Now, substitute \(T\) from Equation 2 into \(T^2 = 7.2 \times p\):

\( \left(p \times \frac{v_A}{v_B}\right)^2 = 7.2 \times p \)

\( p^2 \times \left(\frac{v_A}{v_B}\right)^2 = 7.2 \times p \)

Since \(p \ne 0\) (time taken is non-zero), we can divide by \(p\):

\( p \times \left(\frac{v_A}{v_B}\right)^2 = 7.2 \)

\( p = \frac{7.2}{\left(\frac{v_A}{v_B}\right)^2} \)

This gives \(p\) in terms of the speed ratio. However, the standard formula is expressed as:

\( \frac{v_A}{v_B} = \sqrt{\frac{t_B}{t_A}} \)

Let's use the relationship \(T^2 = 7.2 \times p\). From Equation 1, \(T = \frac{7.2}{v_A/v_B}\). From Equation 2, \(T = p \times \frac{v_A}{v_B}\).

\( \frac{7.2}{v_A/v_B} = p \times \frac{v_A}{v_B} \)

\( 7.2 = p \times \left(\frac{v_A}{v_B}\right)^2 \)

\( \frac{7.2}{p} = \left(\frac{v_A}{v_B}\right)^2 \)

\( \sqrt{\frac{7.2}{p}} = \frac{v_A}{v_B} \)

This confirms the formula used, where \(t_B = 7.2\) and \(t_A = p\).