A boat covers 24 km upstream and 36 km downstream in 6 hours while it covers 36 km upstream and 24 km downstream in $6 \frac{1}{2}$ hours. The velocity of the current is

### Concept & Simultaneous Linear Equations Problems featuring two distinct mixed-direction journeys can be modeled as a system of linear equations by substituting $\frac{1}{U}$ and $\frac{1}{D}$ with variables. $$ \frac{D_{u1}}{U} + \frac{D_{d1}}{D} = T_1 $$ $$ \frac{D_{u2}}{U} + \frac{D_{d2}}{D} = T_2 $$ ### Step-by-Step Solution * Let upstream speed be $U$ and downstream speed be $D$. Define $u = \frac{1}{U}$ and $v = \frac{1}{D}$. * Journey 1: $24$ km upstream, $36$ km downstream in $6$ hours. $$ 24u + 36v = 6 \quad \text{--- (Equation 1)} $$ * Journey 2: $36$ km upstream, $24$ km downstream in $6.5$ hours. $$ 36u + 24v = 6.5 \quad \text{--- (Equation 2)} $$ * Add Equation 1 and Equation 2: $$ 60u + 60v = 12.5 \implies u + v = \frac{12.5}{60} = \frac{25}{120} = \frac{5}{24} \quad \text{--- (Equation 3)} $$ * Subtract Equation 1 from Equation 2: $$ 12u - 12v = 0.5 \implies u - v = \frac{0.5}{12} = \frac{1}{24} \quad \text{--- (Equation 4)} $$ * Add Equation 3 and Equation 4 to find $u$: $$ 2u = \frac{6}{24} \implies 2u = \frac{1}{4} \implies u = \frac{1}{8} $$ * Since $u = \frac{1}{U}$, $U = 8$ km/hr. * Substitute $u$ into Equation 4 to find $v$: $$ \frac{1}{8} - v = \frac{1}{24} \implies v = \frac{3}{24} - \frac{1}{24} = \frac{2}{24} = \frac{1}{12} $$ * Since $v = \frac{1}{D}$, $D = 12$ km/hr. * Velocity of the current $= \frac{D - U}{2} = \frac{12 - 8}{2} = 2$ km/hr. ### Exam Strategy & Shortcut Adding and subtracting symmetric equations ($Ax + By = C$ and $Bx + Ay = D$) is much faster than traditional substitution or elimination. Once you have $x+y$ and $x-y$, the individual variables are found instantly. ### Common Pitfall Trying to isolate $U$ or $D$ directly without substituting $\frac{1}{U}$ and $\frac{1}{D}$ first. This leads to complex algebraic fractions that are highly prone to calculation errors. ### Final Answer Therefore, the correct answer is **2 km/hr**.