Answer:
Ratio k/j is 1.25.
Step-by-step explanation:
Note: There is a slight typographical error in the requirement. The requirement is therefore correctly stated as follows:
Calculate the ratio k/j.
The explanation of the answer is given as follows:
This can be calculated using the following formula:
A = P (1 + (r / m))^(tm) ……………………….. (1)
Where;
A = Balance
P = Amount invested
r = annual interest rate
t = number of years
m = number of compounding
From the question, there are 2 alternatives as follows:
Alternative A
For Bank X
P1 = $1,000
r1 = j
t1 = number of years from 1980 to 1985 = 5
m1 = semiannual = 2
A1 = Amount transferred to Bank Y = P2 = ?
For Bank Y
P2 = ?
r2 = k
t2 = number of years from 1985 to 1988 = 3
m2 = quarterly = 4
A2 = $1,990.76
Alternative B
For Bank Y only
P3 = $1,000
r3 = k
t3 = number of years from 1980 to 1988 = 8
m3 = quarterly = 4
A3 = $2,203.76
For Bank Y in alternative B, we can use equation (1) and solve for k as follows:
2,203.76 = 1000 * (1 + (k / 4))^(8 * 4)
2,203.76 / 1000 = (1 + (k / 4))^32
2.20376 = (1 + (k / 4))^32
2.20376^(1/32) = 1 + (k / 4)
1.02500004450895 = 1 + (k / 4)
1.02500004450895 - 1 = k / 4
0.02500004450895 = k / 4
k = 0.02500004450895 * 4
k = 0.10
For Bank Y in alternative A, we can use r2 = k = 0.10. Therefore, using equation (1) for Bank Y in alternative A, we can solve for P2 as follows:
1,990.76 = P2 * (1 + (0.10 / 4))^(3 * 4)
1,990.76 = P2 * 1.3448888242463
P2 = 1,990.76 / 1.3448888242463
P2 = $1,480.24
From Bank X in alternative A, we now have A1 = P2 = $1,480.24. For Bank X in alternative A, we can now use equation (1) and solve for j as follows:
1,480.24 = 1,000 (1 + (j / 2))^(5 * 2)
1,480.24 / 1,000 =(1 + (j / 2))^10
1.480.24 =(1 + (j / 2))^10
1.480.24^(1 / 10) = 1 + (j / 2)
1.03999969894693 - 1 = j / 2
0.03999969894693 = j / 2
j = 0.03999969894693 * 2
j = 0.08
Since k = 0.10 and j = 0.08, ratio k/j can be calculated as follows:
Ratio k/j = 0.10 / 0.08
Ratio k/j = 1.25