Question

Петя идет до школы на 15 минут меньше, чем Коля. Если Петя уменьшит скорость на 25%, то Коля обгонит его на 5 минут. На сколько процентов скорость Пети превосходила скорость Коли изначально?

Answer 1

Путь = 1
х - скорость Пети
у - скорость Коли
15 минут 1/4 часа
5 минут=1/12 часа

путь/скорость=время
1/х+1/4=1/у
1/0,75х-1/12=1/у

1/х + 1/4= 1/0,75х -1/12
1/0,75х-1/х=1/4+1/12
4/3х-3/3х=3/12+1/12
1/3х=1/3
х=1 км/ч

1/1+1/4=1/у
5/4=1/у
у=4/5=0,8 км/ч

1 это х
0,8  это 100

х=100/0,8=125%
125-100=25%

Answer 2

Problem Analysis

We are given that Petya takes 15 minutes less than Kolya to go to school. If Petya reduces his speed by 25%, Kolya will overtake him by 5 minutes. We need to find the percentage by which Petya's initial speed exceeded Kolya's initial speed.

Solution

Let's assume that Kolya's initial speed is x units per minute. Therefore, Petya's initial speed is x + 15 units per minute.

When Petya reduces his speed by 25%, his new speed becomes 0.75 * (x + 15) units per minute.

We are given that if Petya reduces his speed, Kolya will overtake him by 5 minutes. This means that the time taken by Kolya to reach the school is 5 minutes less than the time taken by Petya.

Using the formula time = distance / speed, we can set up the following equation:

(distance / (0.75 * (x + 15))) - (distance / x) = 5

Simplifying the equation, we get:

(x + 15) / (0.75 * (x + 15)) - 1 / x = 5 / distance

Since the distance is not given, we can assume any value for it. Let's assume the distance is 60 units.

Substituting the value of distance, we get:

(x + 15) / (0.75 * (x + 15)) - 1 / x = 5 / 60

Simplifying further, we get:

(x + 15) / (0.75 * (x + 15)) - 1 / x = 1 / 12

To solve this equation, we can multiply both sides by 12 * 0.75 * (x + 15) * x to eliminate the denominators:

12 * x * (x + 15) - 12 * (0.75 * (x + 15)) - (12 * (x + 15) * 0.75 * x) = (x + 15) * x

Simplifying further, we get:

12 * x^2 + 180 * x - 12 * (0.75 * x + 11.25) - 12 * (0.75 * x^2 + 11.25 * x) = x^2 + 15 * x

Simplifying and rearranging the equation, we get:

0.75 * x^2 - 45 * x - 135 = 0

Solving this quadratic equation, we find that x ≈ 60. Therefore, Kolya's initial speed is approximately 60 units per minute.

Now, we can calculate Petya's initial speed:

Petya's initial speed = Kolya's initial speed + 15 = 60 + 15 = 75 units per minute.

To find the percentage by which Petya's initial speed exceeded Kolya's initial speed, we can use the formula:

percentage = ((Petya's initial speed - Kolya's initial speed) / Kolya's initial speed) * 100

Substituting the values, we get:

percentage = ((75 - 60) / 60) * 100 = 25%

Therefore, Petya's initial speed exceeded Kolya's initial speed by 25%.

Answer

The initial speed of Petya exceeded the initial speed of Kolya by 25%.