Question

Вова идет по прямой дороге. Сначала он делает 10 шагов вперед, потом-два шага назад, потом- опять 10 вперед и затем два шага назад.Сколько шагов он сделает к моменту, когда впервые окажется в 100 шагах от места старта?

Answer 1

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Answer 2

Problem Analysis

Vova takes 10 steps forward, then 2 steps backward, then 10 steps forward again, and finally 2 steps backward. We need to determine how many steps he will have taken by the time he is 100 steps away from the starting point.

Solution

To solve this problem, we can break it down into steps and calculate the total number of steps Vova takes.

1. Vova takes 10 steps forward. 2. Vova takes 2 steps backward. 3. Vova takes 10 steps forward. 4. Vova takes 2 steps backward.

We can see that Vova completes this cycle of 4 steps, which consists of 10 forward steps and 2 backward steps. This cycle repeats until he reaches the desired distance.

Let's calculate the number of cycles Vova needs to complete to reach 100 steps away from the starting point:

1. The first cycle covers 10 forward steps and 2 backward steps, resulting in a net gain of 8 steps. 2. The second cycle also covers 10 forward steps and 2 backward steps, resulting in another net gain of 8 steps. 3. This pattern continues until Vova reaches the desired distance.

To calculate the number of cycles needed, we can divide the desired distance by the net gain per cycle:

Number of cycles = Desired distance / Net gain per cycle

In this case, the desired distance is 100 steps, and the net gain per cycle is 8 steps.

Let's calculate the number of cycles:

Number of cycles = 100 / 8 = 12.5

Since we cannot have a fraction of a cycle, we will consider only the integer part of the result. Therefore, Vova needs to complete 12 cycles to reach 100 steps away from the starting point.

Now, let's calculate the total number of steps Vova takes:

Total number of steps = Number of cycles * Total steps per cycle

The total steps per cycle consist of 10 forward steps and 2 backward steps:

Total steps per cycle = 10 + 2 = 12

Let's calculate the total number of steps:

Total number of steps = 12 * 12 = 144

Therefore, Vova will have taken 144 steps by the time he is 100 steps away from the starting point.

Answer

Vova will have taken 144 steps by the time he is 100 steps away from the starting point.