) A family wishes to accumulate $50,000 in a college education fund at the end of 20 years. If they deposit $1000 in the fund at the end of each of the first 10 years and $1000 + in the fund at the end of the second 10 years, find X if the fund earns 7% effective.
Answers
Given :- A family wishes to accumulate $50,000 in a college education fund at the end of 20 years. If they deposit $1000 in the fund at the end of each of the first 10 years and ($1000 + X) in the fund at the end of the second 10 years, find X if the fund earns 7% effective.
Solution :-
given that,
- payment amount (PMT) = $1000 at the end of each year .
- T = 10 years .
- r = 7% per annum = (7/100) = 0.07
so,
→ Future value (FV) = PMT[{1 + (r/100)]^T - 1}/r]
→ FV = 1000[{1 + 0.07}¹⁰ - 1]/(0.07)
→ FV = 1000[(1.07)¹⁰ - 1]/(0.07)
→ FV = 1000 * (1.9671 - 1)/(0.07)
→ FV = $13816.45 .
now,
→ Amount after next 10 years of FV = (13816.45) * (1.07)¹⁰ = $27179.04
then,
→ FV for next 10 years = $50000 - $27179.04 = $22820.96
now,
- payment amount (PMT) = $(1000+x) at the end of each year .
- T = 10 years .
- r = 7% per annum = (7/100) = 0.07
- FV = $22820.96.
therefore,
→ 22820.96 = (1000 + x)[{1 + 0.07}¹⁰ - 1]/(0.07)
→ 22820.96 = (1000 + x)[1.07)¹⁰ - 1]/(0.07)
→ 22820.96 = (1000 + x)(1.9671 - 1)/(0.07)
→ 22820.96/13.81645 = (1000 + x)
→ 1651.72 = 1000 + x
→ x = 1651.72 - 1000
→ x = $651.72 (Ans.)
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