7.5 A businessman was driving to a business meeting. He calculated that if he traveled at a speed of 90 km/h, he would arrive an hour earlier, and if he traveled at 60 km/h, he would be an hour late. What is the minimum speed he should travel to arrive on time? Justify your answer.
Solution. Method 1. Suppose that three businessmen, each in their own car, were heading to a meeting. The first (Speedster) at a speed of 90 km/h, the second (Turtle) at a speed of 60 km/h, and the third - Punctual, whose speed we need to find. Consider the moment when Speedster arrives at the meeting place. The meeting is still one hour away, and Turtle's arrival is still two hours away. Thus, Turtle is currently $60 \cdot 2=120$ km away from the meeting place. Speedster was ahead of Turtle by $90-60=30$ km each hour, so he was on the road for $120: 30=4$ hours. The meeting place, therefore, is $90 \cdot 4=360$ km from the starting point, and the meeting should take place $4+1=5$ hours after the start of the businessmen's journey. Thus, Punctual's speed is $360: 5=72$ km/h.

Method 2. Let the distance the businessman needs to travel be $S$ km, the time until the meeting be $t$ hours, and the speed at which he should travel to arrive on time (which is the required speed) be $v$ km/h. Then we have the system

$\left\{\begin{array}{rl}v t & =S \\ 90(t-1) & =S \\ 60(t+1) & =S\end{array}\right.$. From the last two equations of the system, we find $t=5, S=360$. Then from the first equation $S=72$.

Answer: 72 km/h.

| THERE IS IN THE SOLUTION | SCORE |
| :--- | :---: |
| Correct and justified answer | 7 points |
| There are arithmetic errors in the correct solution process, possibly leading to an incorrect answer | 6 points |
| The system of equations is correctly set up but not solved (or solved incorrectly), fully describing the problem | 4 points |
| At least one of the two quantities is correctly and justifiedly found:  - the distance to be traveled;  - the time the businessman should be on the road | 3 points |
| Answer without justification or with incorrect justification | 0 points |