Question

Волшебник решил увеличить запас монет. Его палочка на свое усмотрение Увеличивала монеты в 5 раз или уменьшала на 8 штук. В итоге у волшебника оказалось 50 монет после трёх взмахов палочки. Сколько монет могло быть изначально? Все варианты.

Answer 1

Ответ: 3 варианта: 10, 26 или 74.

Пошаговое объяснение:

1. 3 взмаха палочки могли иметь такие действия (я отталкиваюсь от конечного числа, от 50, поэтому когда волшебник уменьшал (отнимал), я прибавляю (ставлю +), а когда увеличивал (умножал), я делю (ставлю :):

а) увеличить, увеличить, увеличить (50:5:5:5=0,4);

б) увеличить, увеличить, уменьшить (50:5:5+8=10);

в) увеличить, уменьшить, увеличить [((50:5)+8):5=3,6];

г) увеличить, уменьшить, уменьшить (50:5+8+8=26);

д) уменьшить, увеличить, увеличить ((50+8):5:5=2,32);

е) уменьшить, увеличить, уменьшить ((50+8):5+8=19,6);

ж) уменьшить, уменьшить, увеличить ((50+8+8):5=13,2);

з) уменьшить, уменьшить, уменьшить (50+8+8+8=74).

С точки зрения математики, существует 8 вариантов действий, но с точки зрения практической, у волшебника изначально не могло быть 0,4 или, скажем, 13,2 монеты. Скорее всего, у него было целое число монет, а именно: 10, 26 или 74.

Answer 2

Problem Analysis

The wizard has a certain number of coins initially. With each wave of his wand, the number of coins either increases by a factor of 5 or decreases by 8. After three waves, the wizard has 50 coins. We need to determine the possible initial number of coins the wizard could have had.

Solution

Let's consider the possible scenarios step by step.

1. If the wizard increased the number of coins by a factor of 5 in all three waves, the number of coins would be multiplied by 5^3 = 125. However, this would result in a much larger number of coins than 50, so this scenario is not possible.

2. If the wizard decreased the number of coins by 8 in all three waves, the number of coins would be decreased by 8 * 3 = 24. Therefore, the initial number of coins would be 50 + 24 = 74. However, this scenario is also not possible since the wizard cannot have more coins initially than he has after three waves.

3. Let's consider a scenario where the wizard increased the number of coins by a factor of 5 in two waves and decreased the number of coins by 8 in one wave. Let's assume the initial number of coins is x.

- After the first wave, the number of coins becomes 5x. - After the second wave, the number of coins becomes 5 * 5x = 25x. - After the third wave, the number of coins becomes 25x - 8.

According to the problem, the number of coins after three waves is 50. Therefore, we can set up the equation:

25x - 8 = 50

Solving this equation, we find:

25x = 58 x = 58 / 25 x = 2.32

Since the number of coins must be a whole number, this scenario is not possible.

4. Finally, let's consider a scenario where the wizard increased the number of coins by a factor of 5 in one wave and decreased the number of coins by 8 in two waves. Let's assume the initial number of coins is y.

- After the first wave, the number of coins becomes 5y. - After the second wave, the number of coins becomes 5y - 8. - After the third wave, the number of coins becomes (5y - 8) - 8 = 5y - 16.

According to the problem, the number of coins after three waves is 50. Therefore, we can set up the equation:

5y - 16 = 50

Solving this equation, we find:

5y = 66 y = 66 / 5 y = 13.2

Since the number of coins must be a whole number, this scenario is also not possible.

Conclusion

After analyzing all the possible scenarios, we find that there is no whole number of coins that could satisfy the given conditions. Therefore, it is not possible to determine the exact initial number of coins the wizard could have had.