Step-by-Step Solution

Step-by-Step Solution

Step 1: Define variables and formulate time equation

Let $v$ be the uniform speed in km/h ($v > 2$).

• Actual time taken: $t_1 = \dfrac{48}{v}\text{ hours}$
• Slower time taken: $t_2 = \dfrac{48}{v - 2}\text{ hours}$

Given that the slower journey takes 2 hours longer: $\frac{48}{v - 2} - \frac{48}{v} = 2$

Step 2: Simplify and form the quadratic equation

Divide the entire equation by $2$: $\frac{24}{v - 2} - \frac{24}{v} = 1$

Combine fractions over $v(v - 2)$: $\frac{24v - 24(v - 2)}{v(v - 2)} = 1$

$\frac{48}{v^2 - 2v} = 1 \implies v^2 - 2v - 48 = 0$

Step 3: Factorize for speed $v$

$(v - 8)(v + 6) = 0$

$v = 8 \quad \text{or} \quad v = -6$

Since speed must be positive ($v > 0$), we reject $v = -6$, so $v = 8\text{ km/h}$.

Step 4: Compute the time taken

$\text{Time taken} = \frac{48}{v} = \frac{48}{8} = 6\text{ hours}$

Step 5: Check solution

• Slower speed $= 8 - 2 = 6\text{ km/h}$.
• Slower time $= \dfrac{48}{6} = 8\text{ hours}$.
• Difference $= 8 - 6 = 2\text{ hours}$ $\checkmark$

$\boxed{\text{The cyclist took } 6 \text{ hours to cover 48 km (at } 8\text{ km/h)}}$