Question
Suppose Tim borrows $200,000 for three years from his rich grandparent, who agrees to charge Tim simple interest at 5% annually. How much would Tim have to pay his grandparent in interest altogether for three years?
Suppose Tim takes a loan from the bank instead at an interest rate of 5% per year compounded annually, with the full loan amount and interest payable after three years. What would be the total interest paid to the bank by Tim?
Responses
A Simple interest=$230,000; Compound interest=$231,525Simple interest=$230,000; Compound interest=$231,525
B Simple interest = $31,525; Compound interest = $30,000Simple interest = $31,525; Compound interest = $30,000
C Simple interest = $30,000; Compound interest = $31,525Simple interest = $30,000; Compound interest = $31,525
D Simple interest = $30,000; Compound interest=$231,525
Suppose Tim takes a loan from the bank instead at an interest rate of 5% per year compounded annually, with the full loan amount and interest payable after three years. What would be the total interest paid to the bank by Tim?
Responses
A Simple interest=$230,000; Compound interest=$231,525Simple interest=$230,000; Compound interest=$231,525
B Simple interest = $31,525; Compound interest = $30,000Simple interest = $31,525; Compound interest = $30,000
C Simple interest = $30,000; Compound interest = $31,525Simple interest = $30,000; Compound interest = $31,525
D Simple interest = $30,000; Compound interest=$231,525
Answers
GPT-4o mini
To calculate the total interest paid in each scenario, let's break it down step by step.
**1. Simple Interest Calculation:**
The formula for simple interest is:
\[ \text{Interest} = P \times r \times t \]
Where:
- \( P \) = principal amount (the initial loan amount)
- \( r \) = annual interest rate (as a decimal)
- \( t \) = time (in years)
For Tim's loan to his grandparent:
- \( P = 200,000 \)
- \( r = 0.05 \)
- \( t = 3 \)
Calculating the interest:
\[ \text{Interest} = 200,000 \times 0.05 \times 3 = 200,000 \times 0.15 = 30,000 \]
So, the total simple interest paid to his grandparent is **$30,000**.
---
**2. Compound Interest Calculation:**
The formula for compound interest when interest is compounded annually is:
\[ A = P \times (1 + r)^t \]
Where:
- \( A \) = the total amount after time \( t \)
- \( P \) = principal amount (the initial loan amount)
- \( r \) = annual interest rate (as a decimal)
- \( t \) = time (in years)
We need to calculate the total amount \( A \):
- \( P = 200,000 \)
- \( r = 0.05 \)
- \( t = 3 \)
Now calculating \( A \):
\[ A = 200,000 \times (1 + 0.05)^3 \]
\[ A = 200,000 \times (1.05)^3 \]
\[ A = 200,000 \times 1.157625 \]
\[ A = 231,525 \]
The total interest paid to the bank is the total amount minus the principal:
\[ \text{Interest} = A - P = 231,525 - 200,000 = 31,525 \]
---
**Final Results:**
- Total simple interest paid to the grandparent: **$30,000**
- Total compound interest paid to the bank: **$31,525**
Comparing the results with the provided options:
**C. Simple interest = $30,000; Compound interest = $31,525**
**1. Simple Interest Calculation:**
The formula for simple interest is:
\[ \text{Interest} = P \times r \times t \]
Where:
- \( P \) = principal amount (the initial loan amount)
- \( r \) = annual interest rate (as a decimal)
- \( t \) = time (in years)
For Tim's loan to his grandparent:
- \( P = 200,000 \)
- \( r = 0.05 \)
- \( t = 3 \)
Calculating the interest:
\[ \text{Interest} = 200,000 \times 0.05 \times 3 = 200,000 \times 0.15 = 30,000 \]
So, the total simple interest paid to his grandparent is **$30,000**.
---
**2. Compound Interest Calculation:**
The formula for compound interest when interest is compounded annually is:
\[ A = P \times (1 + r)^t \]
Where:
- \( A \) = the total amount after time \( t \)
- \( P \) = principal amount (the initial loan amount)
- \( r \) = annual interest rate (as a decimal)
- \( t \) = time (in years)
We need to calculate the total amount \( A \):
- \( P = 200,000 \)
- \( r = 0.05 \)
- \( t = 3 \)
Now calculating \( A \):
\[ A = 200,000 \times (1 + 0.05)^3 \]
\[ A = 200,000 \times (1.05)^3 \]
\[ A = 200,000 \times 1.157625 \]
\[ A = 231,525 \]
The total interest paid to the bank is the total amount minus the principal:
\[ \text{Interest} = A - P = 231,525 - 200,000 = 31,525 \]
---
**Final Results:**
- Total simple interest paid to the grandparent: **$30,000**
- Total compound interest paid to the bank: **$31,525**
Comparing the results with the provided options:
**C. Simple interest = $30,000; Compound interest = $31,525**