Question

Вкладчик внёс некоторую сумму в сбербанк под определённый процент годовых. Через год он взял половину получившейся суммы и переложил её в коммерческий банк, процент годовых которого в 32 раза выше, чем в сбербанке. Ещё через год сумма вкладчика в коммерческом банке превысила вложенную туда первоначальную сумму на 4%. Каков процент годовых в сбербанке?

Answer 1

Пусть а-сумма, которую вложил клиент
p-процент, под который он вложил в сбербанк

Через год его сумма на счету будет
а+ар

В комер банк он будет класть
0,5(а+ар)

В котором процент выше на 32, то есть  в нем процент будет 32р

Тогда через год в ком банке будет

0,5(а+ар)+32р(0,5(а+ар)=0,5а (1+р)+32р0,5а(1+р)=0,5а(1+р)(1+32р)

Далее сказано, что получившаяся сумма превысила на 4% ту, которую он вложил, т е  процент стал 1,04 (  (100+4)/100  ))

Тогда сумма будет 

1,04(0,5а+ар)

Ну а теперь приравниваем

0,5а(1+р)(1+32р)=1,04(0,5а+ар) 
1+32р=1,04
р=0,125%

Answer 2

Calculation of Interest Rates

To determine the interest rate in Sberbank, let's break down the information provided:

- The depositor made an initial deposit in Sberbank at a certain annual interest rate. - After one year, the depositor took half of the accumulated amount and transferred it to a commercial bank with an interest rate that is 32 times higher than that of Sberbank. - After another year, the amount in the commercial bank exceeded the initial deposit by 4%.

To find the interest rate in Sberbank, we need to calculate the interest earned in the commercial bank and work backward.

Let's assume the initial deposit in Sberbank is X.

After one year in Sberbank, the depositor would have earned interest on the initial deposit, resulting in a total amount of X + (X * (Sberbank interest rate)).

The depositor then took half of this amount and transferred it to the commercial bank. Therefore, the amount transferred to the commercial bank is (X + (X * (Sberbank interest rate))) / 2.

In the commercial bank, the amount grew by 4% over the next year. Therefore, the final amount in the commercial bank is (X + (X * (Sberbank interest rate))) / 2 + ((X + (X * (Sberbank interest rate))) / 2) * 0.04.

We know that the interest rate in the commercial bank is 32 times higher than that of Sberbank. Therefore, the interest rate in the commercial bank is 32 * (Sberbank interest rate).

Now we can set up an equation to solve for the Sberbank interest rate:

(X + (X * (Sberbank interest rate))) / 2 + ((X + (X * (Sberbank interest rate))) / 2) * 0.04 = (X + (X * (Sberbank interest rate))) / 2 * (1 + 32 * (Sberbank interest rate))

Simplifying the equation, we get:

(X + (X * (Sberbank interest rate))) * (1 + 0.04) = (X + (X * (Sberbank interest rate))) * (1 + 32 * (Sberbank interest rate))

Simplifying further, we have:

1.04 * (X + (X * (Sberbank interest rate))) = (X + (X * (Sberbank interest rate))) * (1 + 32 * (Sberbank interest rate))

Dividing both sides of the equation by (X + (X * (Sberbank interest rate))), we get:

1.04 = 1 + 32 * (Sberbank interest rate)

Subtracting 1 from both sides of the equation, we have:

0.04 = 32 * (Sberbank interest rate)

Dividing both sides of the equation by 32, we get:

Sberbank interest rate = 0.04 / 32 = 0.00125

Therefore, the interest rate in Sberbank is 0.125%.

Please note that this calculation assumes a simple interest calculation and does not take into account compounding or any other factors that may affect the actual interest rate.