You plan to accumulate 100,000 at the end of 42 years by making the follow-

ing deposits:

X at the beginning of years 1-14

No deposits at the beginning of years 15-32; and

Y at the beginning of years 33-42.

The annual effective interest rate is 7%.

Suppose X - Y = 100. Calculate Y.

Y = 479.17

Step-by-step explanation:

At the end of year 14, the balance from the deposits of X can be found using the annuity due formula:

A = P (1+r/n) ((1 + r/n) ^ (nt) - 1) / (r/n)

where P is the periodic payment, n is the number of payments and compoundings per year, t is the number of years, and r is the annual interest rate.

A = X (1.07) (1.07^14 - 1) / 0.07 ≈ 24.129022X

This accumulated amount continues to earn interest for the next 28 years, so will further be multiplied by 1.07^28. Then the final balance due to deposits of X will be ...

Ax = (24.129022X) (1.07^28) = 160.429967X

__

The same annuity due formula can be used for the deposits of Y for the last 10 years of the interval:

Ay = Y (1.07) (1.07^10 - 1) /.07 = 14.783599Y

__

Now we can write the two equations in the two unknowns:

Ax + Ay = 100,000

X - Y = 100

From the latter, we have ...

X = Y + 100

So the first equation becomes ...

160.429967 (Y + 100) + 14.783599Y = 100000

175.213566Y + 16,043.00 = 100,000

Y = (100,000 - 16,043) / 175.213566 ≈ 479.17

Y is 479.17