Sophie borrowed $2000 and took 3 years to repay the loan and $900 interest.
a)What was the per annum simple interest rate?
b) If Sophie�s interest was calculated at the same rate on the balance owing, how much would she have owed after 6 months if she repaid $40 per month? Give your answer to the nearest dollar.
I did a) which is 15% but I can't do b. Thanks for any help
Well, it seems like Sophie borrowed money and had a fun time with interest rates. Let's see if we can help her out.
a) To find the per annum simple interest rate, we divide the interest by the principal amount and the number of years. So, in this case, the interest rate would be:
$900 / $2000 / 3 years = $900 / $6000 = 0.15 or 15%.
So, you got that right!
Now, let's go to part b:
If Sophie's interest was calculated at the same rate on the balance owing, we need to figure out how much she would owe after 6 months.
First, let's find out how much she repays in total in 6 months. She repays $40 per month, so in 6 months, she repays:
$40/month * 6 months = $240.
Now, we need to calculate how much interest she will accrue on the remaining balance. To do this, we can consider the remaining balance as a new principal amount, and calculate the interest for 6 months. We will use the same interest rate of 15%.
The remaining balance would be:
$2000 - $240 = $1760.
To calculate the interest, we use the formula: Interest = Principal * Rate * Time.
Interest = $1760 * 0.15 * (6/12) years.
Simplifying this gives us:
Interest = $1760 * 0.15 * 0.5 = $132.
So, after 6 months, if Sophie repays $40 per month and the interest is calculated at the same rate on the remaining balance, she would owe approximately $1760 + $132 = $1892.
Remember, this is an estimation to the nearest dollar.
To calculate the per annum simple interest rate, we can use the formula:
Simple Interest = Principal × Rate × Time
a) Given that Sophie borrowed $2000, took 3 years to repay the loan, and paid $900 in interest, we can substitute these values into the formula:
$900 = $2000 × Rate × 3
To find the rate, divide both sides of the equation by ($2000 × 3):
Rate = $900 / ($2000 × 3) = 0.15
The rate is 0.15, which translates to 15% per annum.
b) If Sophie's interest is calculated at the same rate on the balance owing, we can calculate the interest for 6 months. Given that Sophie is repaying $40 per month:
Total repayments in 6 months = $40 × 6 = $240
To calculate the interest, subtract the total repayments from the original balance:
Interest = Balance – Total Repayments
Initially, the balance is $2000. Therefore:
Interest = $2000 – $240 = $1760
After 6 months, the balance owing would be $1760.
To solve part b), we need to first find out the monthly interest rate based on the per annum interest rate from part a).
a) To find the per annum simple interest rate:
Given:
Principal amount (P) = $2000
Interest (I) = $900
Time period in years (t) = 3
Simple interest formula:
I = P * r * t
Substituting the given values:
900 = 2000 * r * 3
Now, solve for r (the per annum interest rate):
r = 900 / (2000 * 3)
r = 0.15
Therefore, the per annum simple interest rate is 0.15 or 15%.
b) To find out how much Sophie would owe after 6 months:
Given:
Monthly repayment (M) = $40
We first need to find out the monthly interest rate:
Per annum interest rate (r) = 15%
Monthly interest rate (i) = r / 12
Substituting the values:
i = 0.15 / 12
i = 0.0125
To calculate the balance owing after 6 months, we'll use the compound interest formula:
Balance after n months = Principal * (1 + i)^n - (M/r) * [(1 + i)^n - 1]
Substituting the values:
Principal (P) = $2000
Interest rate per month (i) = 0.0125
Number of months (n) = 6
Monthly repayment (M) = $40
Balance after 6 months = 2000 * (1 + 0.0125)^6 - (40 / 0.0125) * [(1 + 0.0125)^6 - 1]
Calculating:
Balance after 6 months = 1961 - (3200 * 0.0125) * 0.207134
Balance after 6 months ≈ $1917 (rounded to the nearest dollar)
Therefore, if Sophie repaid $40 per month and the interest was calculated at the same rate, she would owe approximately $1917 after 6 months.