Question

On January 1, 1980, Jack deposited 1,000 into Bank X to earn interest at the rate of j per annum compounded semi-annually. On January 1, 1985, he transferred his account to Bank Y to earn interest at the rate of k per annum compounded quarterly. On January 1, 1988, the balance at Bank Y is 1,990.76. If Jack could have earned interest at the rate of k per annum compounded quarterly from January 1, 1980 through January 1, 1988, his balance would have been 2,203.76. Calculate the ratio kjkj.

Answer 1

Answer:

Ratio k/j is 1.25.

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