Question

Из пункта А в направлении пункта В выехал первый велосипедист со скоростью 12 2/3 км/ч. Одновременно из пункта В в том же направлении выехал второй велосипедист, скорость которого в 1 16/41 раза меньше скорости первого. Через сколько часов после начала движения первый велосипедист догонет второго, если расстояния между пунктами А и В равно 8 км.

Answer 1

1) 12 2/3 : 1 16/41 = 38/3 : 57/41 = 38/3 * 41/57 = (2*41)/(3*3) = 82/9 = 9 1/9 (км/ч) - скорость второго велосипедиста;
2) 12 2/3 - 9 1/9 = 12 6/9 - 9 1/9 = 3 5/9 (км/ч) - скорость сближения при движении вдогонку;
3) 8 : 3 5/9 = 8 : 32/9 = 8/1 * 9/32 = (1*9)/(1*4) = 9/4 = 2 1/4 (ч) - через 2 1/4 часа после начала движения первый велосипедист догонит второго.
1 час = 60 мин
1/4 ч = 60 : 4 = 15 мин
Ответ: через 2 часа 15 минут первый велосипедист догонит второго. 

Answer 2

Problem Analysis

We are given that two cyclists start from point A and point B, respectively, and travel towards each other. The first cyclist has a speed of 12 2/3 km/h, while the second cyclist's speed is 1 16/41 times smaller than the first cyclist's speed. The distance between points A and B is 8 km. We need to determine how many hours it will take for the first cyclist to catch up to the second cyclist.

Solution

To solve this problem, we can set up an equation based on the given information. Let's denote the time it takes for the first cyclist to catch up to the second cyclist as t.

The distance traveled by the first cyclist is given by distance = speed × time. Therefore, the distance traveled by the first cyclist is (12 2/3) × t.

The distance traveled by the second cyclist is given by distance = speed × time. The speed of the second cyclist is 1 16/41 times smaller than the speed of the first cyclist, so the speed of the second cyclist is (12 2/3) × (1 16/41).

Since the two cyclists are traveling towards each other, the sum of their distances traveled should be equal to the total distance between points A and B, which is 8 km. Therefore, we can set up the equation:

(12 2/3) × t + (12 2/3) × (1 16/41) × t = 8

Now, let's solve this equation to find the value of t.

Calculation

To solve the equation, we need to convert the mixed numbers to improper fractions.

The first cyclist's speed is 12 2/3 km/h, which can be written as (38/3) km/h.

The second cyclist's speed is (1 16/41) times smaller than the first cyclist's speed. We can calculate this as follows:

1 16/41 = (41 + 16)/41 = 57/41

Therefore, the second cyclist's speed is (57/41) × (38/3) km/h.

Now, let's substitute these values into the equation and solve for t:

(38/3) × t + (38/3) × (57/41) × t = 8

To simplify the equation, we can multiply both sides by 3 to eliminate the fractions:

38t + 38 × (57/41) × t = 24

Now, let's calculate the value of t.

Calculation Continued

To solve the equation, we can simplify the expression 38 × (57/41):

38 × (57/41) = (38 × 57) / 41 = 5316 / 41

Now, let's substitute this value back into the equation and solve for t:

38t + (5316 / 41) × t = 24

To eliminate the fraction, we can multiply both sides by 41:

41 × 38t + 5316t = 24 × 41

Simplifying further:

1558t + 5316t = 984

Combining like terms:

6874t = 984

Now, let's solve for t:

t = 984 / 6874

Calculating the value of t:

t ≈ 0.143 hours

Answer

After approximately 0.143 hours (or 8.6 minutes) from the start of the first cyclist's journey, the first cyclist will catch up to the second cyclist.

Please note that the above solution is based on the given information and calculations.