Sabrina travels part of a journey on foot at 8 km/h and the rest by bicycle at 13 km/h. She covers a total distance of 84 km in 8 hours. For how many hours does she travel by bicycle?

Introduction / Context:
This is a typical mixture of speeds problem where a person travels different parts of a journey at different constant speeds but the total time and total distance are known. The question checks your ability to form simultaneous equations from distance and time relationships and then solve for the time spent at each speed.

Given Data / Assumptions:

• Speed on foot = 8 km/h.
• Speed on bicycle = 13 km/h.
• Total distance traveled = 84 km.
• Total time taken = 8 hours.
• Speeds are constant in each mode of travel.

Concept / Approach:
Let t1 be the time in hours spent walking, and t2 be the time spent cycling. Then t1 + t2 = 8 hours. Distance while walking is 8 * t1 and distance while cycling is 13 * t2. The total distance is 84 km, so 8 * t1 + 13 * t2 = 84. Solving these two equations gives the values of t1 and t2, and the question specifically asks for t2, the cycling time.

Step-by-Step Solution:
Let t1 = time on foot (hours), t2 = time on bicycle (hours). Equation 1: t1 + t2 = 8. Equation 2: 8 * t1 + 13 * t2 = 84. From equation 1, t1 = 8 - t2. Substitute into equation 2: 8 * (8 - t2) + 13 * t2 = 84. Compute: 64 - 8 * t2 + 13 * t2 = 84. Simplify: 64 + 5 * t2 = 84. So 5 * t2 = 20, giving t2 = 4 hours.

Verification / Alternative check:
Then t1 = 8 - 4 = 4 hours. Distance walking = 8 * 4 = 32 km. Distance cycling = 13 * 4 = 52 km. Total distance = 32 + 52 = 84 km and total time = 4 + 4 = 8 hours, so all given conditions are satisfied.

Why Other Options Are Wrong:

• 5, 3 or 2 hours do not satisfy both the total time and total distance conditions when substituted into the equations.

Common Pitfalls:
Some students try to average speeds without considering the different times spent at each speed. Others may incorrectly assume equal distances at each speed and use a simpler formula, which does not fit the data here. Keeping track of both distance and time equations is essential.

Final Answer:
Sabrina travels by bicycle for 4 hours.