Answer:
Allison: age 67
Leslie: 46 years
Explanation:
We are asked to find at which time the fund equal 1,000,000
both sister fund compose of a 15,000 lump sum and ordinary annuity of 5,000 dollars the difference will be the rate at which each capital works:
Alison:
[tex]15,000\times (1.06)^{n} + 5,000 \frac{1.06^{n}-1}{0.06} = 1,000,000\\[/tex]
We divide by 5,000
[tex]3\times (1.06)^{n} + \frac{1.06^{n}-1}{0.06} = 200\\[/tex]
Then we clear the part:
[tex] 1.06^{n} = (200 - 3 (1.06)^{n})*0.06 +1 \\[/tex]
[tex] 1.06^{n} = 13 - 0.18(1.06)^{n}[/tex]
[tex] 1.18 (1.06)^{n} = 13[/tex]
[tex] (1.06)^{n} = 13/1.18[/tex]
[tex] (1.06)^{n} = 11.0169491[/tex]
we use logarithmic properties to solve for n:
[tex] log 11.0169491 \div log 1.06 [/tex]
41.17864898
Allison will be a millionaire after 41.17 year. She start at age 26 so at age 67 she achieve his goal.
Leslie:
[tex]15,000\times (1.17)^{n} + 5,000 \frac{1.17^{n}-1}{0.17} = 1,000,000\\[/tex]
[tex]3\times (1.17)^{n} + \frac{1.17^{n}-1}{0.17} = 200\\[/tex]
[tex] 1.17^{n} = (200 - 3 (1.17)^{n})*0.17 +1 \\[/tex]
[tex] 1.17^{n} = 35 - 0.51(1.17)^{n}[/tex]
[tex] 1.17^{n} = 35/1.51[/tex]
[tex] log 23.178807947 \div log 1.17 [/tex]
n = 20.02014878
Leslie will achieve it in 20.20 years
thus, at age 46
