1. To find the rate of interest, we can use the formula:
Interest = Principal * Rate * Time
where Principal is the original amount borrowed, Rate is the interest rate per year (expressed as a decimal), and Time is the length of time the loan is taken out for in years.
Let P be the principal amount (the amount borrowed or ρáíd in instalments), r be the interest rate, and t be the time in years.
For the cash payment of P25,000, there is no interest involved, so we have:
Interest = 0
Principal = P25,000
Time = 1 year (since it is ρáíd in cash)
Substituting these values into the formula yields:
0 = P25,000 * r * 1
Solving for r gives:
r = 0 (or 0%)
This means that there is no interest charged for the cash payment option.
For the instalment payment of P25,950, we have:
Interest = P25,950 - P
Principal = P
Time = 4/12 = 1/3 year (since it is ρáíd over 4 months)
Substituting these values into the formula yields:
P850.00 = P * r * (1/3)
Solving for r gives:
r = (P850.00) / (P * (1/3))
Multiplying both sides by 3P gives:
3P * r = P850.00
Dividing both sides by P gives:
3r = (P850.00) / P
Solving for P gives:
P = (P850.00) / (3r)
Now we can substitute this value of P back into the formula to find the interest rate:
P850.00 = [(P850.00) / (3r)] * r * (1/3)
Multiplying both sides by 3r gives:
P850.00 * 3r = P850.00
Dividing both sides by P850.00 gives:
3r = 1
Solving for r gives:
r = 1/3 (or 33.33%)
Therefore, the rate of interest for the instalment payment option is 33.33%.
2. Let P be the amount invested. We can use the formula for simple interest:
Interest = Principal * Rate * Time
where Principal is the original amount invested, Rate is the interest rate per year (expressed as a decimal), and Time is the length of time the investment is held for in years.
We are given that the interest earned is P850.00, the rate is 10%, and the time is 5/12 year (since it is invested for 5 months). Substituting these values into the formula yields:
P850.00 = P * 0.10 * (5/12)
Solving for P gives:
P = (P850.00) / (0.10 * (5/12))
Simplifying gives:
P = P140,833.33
Therefore, the amount invested is P140,833.33.