Question

Solitary Savings Bank received an initial deposit of $2000. It kept a percentage of this money in reserve based on the reserve rate and loaned out the rest. The amount it loaned out was eventually all deposited back into the bank. If this cycle continued indefinitely and eventually the $2000 turned into $40,000, what was the reserve rate?

Answer 1

Answer:

5%

Step-by-step explanation:

Given that Solitary savings bank received intially amount  of 2000/-

Let x% percent be kept as reserve then (100-x)% will be loaned out which would be deposited in other bank. The other bank would keep x% of 100-x and again loan the remaining amount.

Thus the total deposits of this cycle would be a geometric series as

2000[1+(100-x)%+(100-x)%^2+(100-x)^3+...]

Here common ratio is

$\frac{100-x}{100} =1-0.01x$ for all positive x

This series is infinite and sum of this would be

sum of geometric series

=$2000(\frac{1}{\frac{x}{100} } )=\frac{200000}{x}$

This is given as 40000

Hence equate and simplify to get

x=5%

So reserve rate is 5%

Answer 2

Let n be the reserve rate. Then:
2000/n=40000
2000=40000n
n=2000/40000=.05=5% reserve rate
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