1. The distance from the home of Vintik and Shpuntik to school is 6 kilometers. Vintik and Shpuntik left for school at the same time, with Vintik spending half of the time riding a scooter at a speed of 10 km/h and then walking, while Shpuntik traveled half the distance by bicycle and then walked. They arrived at school at the same time. Both walk at a speed of 5 km/h. At what speed does Shpuntik ride his bicycle?
Answer: 15 km/h.

## Solution:

First method.

Since Shtyubik rides twice as fast as he walks, he also covers twice the distance he walks (since he spends the same amount of time on both), which is 4 km. Since Shtyubik and Shpuntyk walk at the same speed, the last 2 km they walked together. Therefore, while Shtyubik was riding 4 km on the scooter, Shpuntyk rode 3 km on the bicycle and then walked another 1 km. Since Shtyubik rides the scooter twice as fast as he walks, while Shpuntyk walked 1 km, Shtyubik rode 2 km. Consequently, while Shtyubik was riding the first 2 km on the scooter, Shpuntyk covered 3 km on his bicycle, so he rides the bicycle one and a half times faster than Shtyubik on the scooter, which is 15 km/h.

Second method.

As in the first method, we get that Shtyubik rode 4 km on the scooter and walked 2 km - for which he spent $4 / 10 + 2 / 5 = 4 / 5$ (hours). Let the speed of Shpuntyk on the bicycle be $x$ km/h. He rode 3 km on the bicycle, spending $3 / x$ (hours) on it, and then walked 3 km, spending $3 / 5$ (hours) on it. Shtyubik and Shpuntyk spent the same amount of time on the road, so $3 / x + 3 / 5 = 4 / 5$, from which $3 / x = 1 / 5$, and then $x = 15$.

## Criteria:

Correct answer without correct justification - 2 points.

Correctly formulated equation (for the time spent by Shtyubik and Shpuntyk) - another 2 points.