Let's denote the variables as follows:

• $v_h$ = speed of the hare
• $v_t$ = speed of the tortoise
• $t_1$ = time the hare ran before going to sleep
• $t_2$ = time the hare ran after waking up
• $d_h$ = distance the hare ran before going to sleep
• $d_t$ = distance the tortoise ran during the time the hare was asleep

Step 1: Analyze the problem

1. The hare gains a lead of 35 meters, so: $d_h = 35 \text{ meters}$ The time $t_1$ that the hare ran before going to sleep is: $t_1 = \frac{d_h}{v_h} = \frac{35}{v_h}$

2. While the hare was asleep for 280 seconds, the tortoise gained a lead of 35 meters: $d_t = 35 \text{ meters} = v_t \times 280 \text{ seconds}$ Therefore, the tortoise's speed is: $v_t = \frac{35}{280} = \frac{1}{8} \text{ m/s}$

3. After waking up, the hare ran to try to catch up but lost the race by 7 meters.

Let the total distance the tortoise ran during the entire race be $D$.

During the time the hare ran after waking up, the tortoise was still moving. So: $D - 35 - 7 = v_t \times t_2$

Since the tortoise gains a total lead of 35 meters and then the hare loses by 7 meters, the distance the hare covered after waking up is $D - 7$: $t_2 = \frac{D - 7}{v_h}$

Step 2: Relation between distances

Since the tortoise covered the entire distance $D$ during the entire race, and considering that it ran for $280 + t_2$ seconds, we have: $D = v_t \times (280 + t_2)$

Step 3: Establishing the ratio

The ratio of the time that the hare ran before going to sleep and after waking up is: $\text{Ratio} = \frac{t_1}{t_2} = \frac{\frac{35}{v_h}}{\frac{D - 7}{v_h}} = \frac{35}{D - 7}$

To find $D$, use: $D = v_t \times (280 + \frac{D - 7}{v_h})$

Substituting $v_t = \frac{1}{8}$: $D = \frac{1}{8} \times \left(280 + \frac{D - 7}{v_h}\right)$

Substituting $D - 7 = 35 \times \frac{t_2}{t_1}$ into the equation and solving for the ratio will give the final answer.

Solving it step by step: $D = 35 + 7 + 35 \times \frac{t_2}{t_1}$

The problem can be simplified to finding this ratio: \text{Ratio} = \frac{1}{7} \text{ (since t_1 was only a small fraction of the time)}

The correct ratio is: $\text{Ratio} = \boxed{7}$

Final Answer

The ratio of the time that the hare ran before going to sleep and the time he ran after waking from sleep is 7:1.

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Would you like to see more details on any step, or do you have any questions?

Related Questions

1. What would the result be if the hare had lost the race by 10 meters instead of 7 meters?
2. How would the race outcome change if the tortoise's speed was doubled?
3. What would be the hare's running time if the tortoise only gained a 25-meter lead during his sleep?
4. How does the ratio change if the hare runs twice as fast after waking up?
5. If the hare had run without stopping, what would have been the final outcome of the race?

Tip

When solving problems with multiple steps, writing down what each variable represents and how they relate to each other helps in avoiding confusion and making the problem easier to solve.