Solving a Real-life Distance Problem Using Algebra: A Journey Spread Across Different Modes of Transport

Suppose a person travels a combined distance of 61 km over a span of 9 hours. This journey is divided between walking and cycling, with speeds of 4 km/hr and 9 km/hr, respectively. The challenge is to determine the distance covered on foot.

Setting Up the Problem with Algebra

Let us denote the distance covered on foot as d_f (kilometers) and the distance covered by bicycle as d_b (kilometers). The conditions given are:

The total distance covered is 61 km. Therefore, d_f d_b 61. The time spent on foot is d_f / 4 and on bicycle is d_b / 9. The total time is 9 hours, so (d_f / 4) (d_b / 9) 9.

Solving the System of Equations

Let's solve this system step by step:

Step 1: Express d_b in terms of d_f

From the first equation: d_b 61 - d_f

Step 2: Substitute d_b into the second equation

Substitute d_b into the second equation: (d_f / 4) (61 - d_f) / 9 9

Step 3: Solve for d_f

To eliminate the fractions, multiply the entire equation by 36 (the least common multiple of 4 and 9): 36(d_f / 4) 36(61 - d_f) / 9 36 * 9

This simplifies to: 9d_f 461 - d_f 324

Further simplifying, we get: 8d_f 244 324

Therefore, 8d_f 324 - 244 8d_f 80 d_f 80 / 8 16 km

Conclusion

The distance traveled on foot is 16 km. Let's verify this solution. The distance covered by bicycle is: d_b 61 - 16 45 km

Now we check the time spent:

Time by foot: 16 km / 4 km/hr 4 hours Time by bicycle: 45 km / 9 km/hr 5 hours Total time: 4 5 9 hours, which is correct

Thus, the solution is confirmed: 16 km was traveled on foot.

Alternative Approach

Alternatively, let's consider the following:

Let t1 be the time of travel by bicycle and t2 be the time of travel on foot. Thus, t1 t2 9. The distances can also be expressed as x/9 and 61-x/4, where x is the distance covered by bicycle. This gives the equation x/9 (61-x)/4 9.

Solving this:

Multiply through by 36: 4x 549 - 9x 324 -5x -225 x 45 km Distance on foot: 61 - 45 16 km

Hence, the distance traveled by bicycle is 45 km, and by walking is 16 km.

Final Verification

To confirm, the times are:

Bicycle: 45 km / 9 km/hr 5 hours Walking: 16 km / 4 km/hr 4 hours Total time: 5 4 9 hours (correct)

The solution is verified: 16 km was the distance traveled on foot.