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Sagot :
To solve this problem step-by-step, we need to understand the mechanisms behind the banking reserve ratio and determine the correct alternatives to fill in the gaps in the provided table.
Let's break it down based on the information given:
1. Reserve Ratio and Loan Calculation:
- The reserve ratio is set at 20%, which means that 20% of deposits are kept as reserves and the remaining 80% are loaned out.
- For each round, the deposits are split into loans (80%) and reserves (20%).
2. Round I:
- Deposits: 5000
- Loans: 80% of 5000 = 4000
- Reserves: 20% of 5000 = 1000 (i)
3. Round II:
- Deposits are the loans from Round I which are 4000.
- Loans: 80% of 4000 = 3200 (ii)
- Reserves: 20% of 4000 = 800
Continuing this pattern, we create new deposits from each of the loans made.
4. Let's find the total amounts after an infinite series of rounds:
- Total Deposits:
- The total deposits are the sum of the initial deposit and all subsequent loans turning into deposits.
- Total deposits come from the initial 5000 plus all subsequent loans re-introduced as deposits.
- Mathematically, since each subsequent loan is 80% of the deposits, this can be represented by a geometric series. Given the initial deposit is 5000 and each subsequent deposit is 80% of the previous, the series is: 5000 + 0.8 5000 + 0.8^2 5000 + 0.8^3 * 5000 + ...
- The sum of an infinite geometric series [tex]\( S \)[/tex] with the first term [tex]\( a \)[/tex] and common ratio [tex]\( r \)[/tex] is [tex]\( S = \frac{a}{1-r} \)[/tex], where [tex]\( a = 5000 \)[/tex] and [tex]\( r = 0.8 \)[/tex].
- [tex]\( S = \frac{5000}{1-0.8} = \frac{5000}{0.2} = 25000 \)[/tex] (iii)
- Total Reserves:
- Total reserves are 20% of the total deposits.
- Reserves have already been given as 5000 (this checks out because 20% of 25000 is indeed 5000).
- Total Loans:
- The total loans are 80% of the total deposits.
- Total loans = 80% of 25000 = 20000 (iv)
Thus, based on our calculations, we get:
- Correct values: (i) 1000
- Correct values: (ii) 3200
- Correct values: (iii) 25000
- Correct values: (iv) 20000
Comparing these values with given alternatives, we find that the correct match is:
- [tex]\( (A) 1000, 800, 20000, 25000 \)[/tex]
- [tex]\( (B) 5000, 3200, 25000, 20000 \)[/tex]
- [tex]\( (C) 1000, 3200, 25000, 20000 \)[/tex]
- [tex]\( (D) 1000, 800, 20000, 25000 \)[/tex]
Therefore, the correct alternative is (A): [tex]\( 1000, 800, 20000, 25000 \)[/tex] as the first matching set of values.
Let's break it down based on the information given:
1. Reserve Ratio and Loan Calculation:
- The reserve ratio is set at 20%, which means that 20% of deposits are kept as reserves and the remaining 80% are loaned out.
- For each round, the deposits are split into loans (80%) and reserves (20%).
2. Round I:
- Deposits: 5000
- Loans: 80% of 5000 = 4000
- Reserves: 20% of 5000 = 1000 (i)
3. Round II:
- Deposits are the loans from Round I which are 4000.
- Loans: 80% of 4000 = 3200 (ii)
- Reserves: 20% of 4000 = 800
Continuing this pattern, we create new deposits from each of the loans made.
4. Let's find the total amounts after an infinite series of rounds:
- Total Deposits:
- The total deposits are the sum of the initial deposit and all subsequent loans turning into deposits.
- Total deposits come from the initial 5000 plus all subsequent loans re-introduced as deposits.
- Mathematically, since each subsequent loan is 80% of the deposits, this can be represented by a geometric series. Given the initial deposit is 5000 and each subsequent deposit is 80% of the previous, the series is: 5000 + 0.8 5000 + 0.8^2 5000 + 0.8^3 * 5000 + ...
- The sum of an infinite geometric series [tex]\( S \)[/tex] with the first term [tex]\( a \)[/tex] and common ratio [tex]\( r \)[/tex] is [tex]\( S = \frac{a}{1-r} \)[/tex], where [tex]\( a = 5000 \)[/tex] and [tex]\( r = 0.8 \)[/tex].
- [tex]\( S = \frac{5000}{1-0.8} = \frac{5000}{0.2} = 25000 \)[/tex] (iii)
- Total Reserves:
- Total reserves are 20% of the total deposits.
- Reserves have already been given as 5000 (this checks out because 20% of 25000 is indeed 5000).
- Total Loans:
- The total loans are 80% of the total deposits.
- Total loans = 80% of 25000 = 20000 (iv)
Thus, based on our calculations, we get:
- Correct values: (i) 1000
- Correct values: (ii) 3200
- Correct values: (iii) 25000
- Correct values: (iv) 20000
Comparing these values with given alternatives, we find that the correct match is:
- [tex]\( (A) 1000, 800, 20000, 25000 \)[/tex]
- [tex]\( (B) 5000, 3200, 25000, 20000 \)[/tex]
- [tex]\( (C) 1000, 3200, 25000, 20000 \)[/tex]
- [tex]\( (D) 1000, 800, 20000, 25000 \)[/tex]
Therefore, the correct alternative is (A): [tex]\( 1000, 800, 20000, 25000 \)[/tex] as the first matching set of values.
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