Two places \( P \) and \( Q \) are at a distance 960 km from each other. Ahmed and Rami start walking from \( P \) to \( Q \) at 2 hours difference, where Ahmed leaves from \( P \) before Rami. Ahmed and Rami both have \( 60 \mathrm{~km} / \mathrm{h} \) and \( 80 \mathrm{~km} / \mathrm{h} \) speed resepectively. Rami goes ahead of Ahmed at point \( M \), which falls somewhere between \( P \) and \( Q \), then how long will it take for Rami to get ahead of ahmed?

Let's denote the time it takes for Ahmed to cover the distance from point \( P \) to point \( M \) as \( t \) hours. Since Ahmed and Rami start walking at a 2-hour difference, Ahmed starts walking 2 hours before Rami. The distance covered by Ahmed in \( t \) hours is given by the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \] Substituting the values, we have: \[ \text{Distance} = 60t \] The distance covered by Rami in \( t \) hours is given by the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \] Substituting the values, we have: \[ \text{Distance} = 80(t-2) \] Since both Ahmed and Rami cover the same distance from point \( P \) to point \( M \), we can equate the distances covered by them: \[ 60t = 80(t-2) \] Now, we can solve this equation to find the value of \( t \), which represents the time it takes for Rami to get ahead of Ahmed. Solve the equation by following steps: - step0: Solve for \(t\): \(60t=80\left(t-2\right)\) - step1: Expand the expression: \(60t=80t-160\) - step2: Move the variable to the left side: \(60t-80t=-160\) - step3: Subtract the terms: \(-20t=-160\) - step4: Change the signs: \(20t=160\) - step5: Divide both sides: \(\frac{20t}{20}=\frac{160}{20}\) - step6: Divide the numbers: \(t=8\) The time it takes for Rami to get ahead of Ahmed is 8 hours.