Question

Вася и Петя накопили жетонов для игрового автомата. Для одной игры необходим 1 жетон. В первый день Вася потратил седьмую часть своих жетонов, а Петя — только пятую часть своих. На второй день Вася потратил пятую часть остатка, а Петя — седьмую часть остатка. Мальчикам всё ещё хватает монет более чем на 70 игр. Сколько жетонов было у мальчиков в начале, если они не могли сыграть и 120 раз.

Answer 1

Ответ:

Вася и Петя накопили жетонов для игрового автомата. Для одной игры необходим 1 жетон. В первый день Вася потратил седьмую часть своих жетонов, а Петя — только пятую часть своих. На второй день Вася потратил пятую часть остатка, а Петя — седьмую часть остатка. Мальчикам всё ещё хватает монет более чем на 70 игр. Сколько жетонов было у мальчиков в начале, если они не могли сыграть и 120 раз.

Объяснение:

Answer 2

Problem Analysis

Vasya and Petya have collected tokens for a game machine. Each game requires 1 token. On the first day, Vasya spent 1/7 of his tokens, and Petya spent 1/5 of his tokens. On the second day, Vasya spent 1/5 of the remaining tokens, and Petya spent 1/7 of the remaining tokens. The boys still have more than enough tokens to play 70 games, and they couldn't play 120 times. We need to determine how many tokens the boys had at the beginning.

Solution

Let's assume that Vasya had x tokens and Petya had y tokens at the beginning.

On the first day, Vasya spent 1/7 of his tokens, so he had (x - x/7) = (6/7)x tokens remaining. Petya spent 1/5 of his tokens, so he had (y - y/5) = (4/5)y tokens remaining.

On the second day, Vasya spent 1/5 of the remaining tokens, so he had (6/7)x - (6/7)x/5 = (24/35)x tokens remaining. Petya spent 1/7 of the remaining tokens, so he had (4/5)y - (4/5)y/7 = (24/35)y tokens remaining.

We know that the boys still have more than enough tokens to play 70 games, which means they have at least 70 tokens left. Additionally, they couldn't play 120 times, which means they have less than 120 tokens left.

To find the number of tokens they had at the beginning, we need to solve the following system of inequalities:

1. (24/35)x + (24/35)y ≥ 70 (They have at least 70 tokens left) 2. (24/35)x + (24/35)y < 120 (They have less than 120 tokens left)

Let's solve this system of inequalities to find the possible values of x and y.

Solution Steps

1. Set up the system of inequalities: - (24/35)x + (24/35)y ≥ 70 - (24/35)x + (24/35)y < 120 2. Solve the system of inequalities to find the possible values of x and y. 3. Calculate the number of tokens the boys had at the beginning by substituting the values of x and y into the equation x + y.

Solution Details

To solve the system of inequalities, we can use algebraic methods or graphical methods. Let's use algebraic methods for simplicity.

1. (24/35)x + (24/35)y ≥ 70 - Multiply both sides of the inequality by 35 to eliminate the denominator: 24x + 24y ≥ 2450

2. (24/35)x + (24/35)y < 120 - Multiply both sides of the inequality by 35 to eliminate the denominator: 24x + 24y < 4200

Now we have the following system of inequalities: - 24x + 24y ≥ 2450 - 24x + 24y < 4200

To solve this system, we can subtract the second inequality from the first inequality: - 24x + 24y - (24x + 24y) ≥ 2450 - 4200 - 0 ≥ -1750

Since the inequality 0 ≥ -1750 is always true, it means that any value of x and y that satisfies the first inequality will also satisfy the second inequality.

Therefore, there are infinitely many possible values of x and y that satisfy the given conditions. We cannot determine the exact number of tokens the boys had at the beginning without additional information.

However, we can calculate the minimum number of tokens they had at the beginning by assuming the minimum values for x and y that satisfy the first inequality.

Let's assume x = 70 and y = 0. Substituting these values into the equation x + y, we get 70 + 0 = 70.

Therefore, the minimum number of tokens the boys had at the beginning is 70.

Note: The maximum number of tokens they had at the beginning cannot be determined without additional information.

Answer

Based on the given information, we cannot determine the exact number of tokens Vasya and Petya had at the beginning. However, the minimum number of tokens they had is 70.