The sum of the accumulated value of 1 at the end of four years at a certain effective rate of interest, and the present value of 1 to be paid at the end of four years at an effective rate of discount numerically equal to I is 2.0098
Find the rate
To find the rate, we need to solve the equation provided. Let's break down the equation and solve it step by step.
Let:
- A be the accumulated value of 1 at the end of four years
- P be the present value of 1 to be paid at the end of four years
According to the given information, we have:
A + P = 2.0098 --- (Equation 1)
Now, let's calculate the values of A and P by using the formula for accumulated value and present value.
The formula for accumulated value is:
A = 1 * (1 + i)^n
where:
- i is the rate of interest
- n is the number of years
Similarly, the formula for present value is:
P = 1 / (1 + d)^n
where:
- d is the rate of discount
Since the effective rate of interest and discount are numerically equal (i = d = I), we can denote them as I in our calculations.
So, at the end of four years, we have:
A = 1 * (1 + I)^4 --- (Equation 2)
P = 1 / (1 + I)^4 --- (Equation 3)
Substituting Equations 2 and 3 into Equation 1, we get:
1 * (1 + I)^4 + 1 / (1 + I)^4 = 2.0098
To solve this equation, we can simplify it by using a common denominator:
[(1 + I)^4 + 1] / (1 + I)^4 = 2.0098
Multiplying both sides of the equation by (1 + I)^4:
1 + (1 + I)^4 = 2.0098 * (1 + I)^4
Expanding (1 + I)^4 using the binomial theorem, we get:
1 + 4I + 6I^2 + 4I^3 + I^4 = 2.0098 + 2.0098I + 6.0394I^2 + 8.0528I^3 + 4.0804I^4
Now, let's simplify it further by grouping the terms with the same powers of I:
1 + 4I + 6I^2 + 4I^3 + I^4 - 2.0098 - 2.0098I - 6.0394I^2 - 8.0528I^3 - 4.0804I^4 = 0
Combining the like terms:
1 - 2.0098 - 4I^3 - 2.0098I - 4I^4 + 6I^2 + 4I^3 - 6.0394I^2 - 4.0804I^4 = 0
Simplifying further:
-1.0098 - 0.0098I - I - 0.0804I^2 - 4.0804I^4 = 0
Now, we have a polynomial equation in terms of I. To find the value of I, we can either solve it manually by factoring or by using a numerical method such as Newton's method or a calculator that can solve polynomial equations.
Once you find the value of I, it will represent both the rate of interest (i) and the rate of discount (d) that satisfy the given equation.