Compound Interest (1 Viewer)

nrumble42 · Feb 3, 2017

Megan is on a fixed salary of $83 000 per year and is offered a performance bonus of 5% of this salary each year. Paul is on a salary of $83 000 per year with an annual increase of 3%p.a.

When will Paul's annual earnings first exceed Megan's annual earnings, if Megan receives her bonus every year?

Mahan1 · Apr 1, 2017

nrumble42 said:
Megan is on a fixed salary of $83 000 per year and is offered a performance bonus of 5% of this salary each year. Paul is on a salary of $83 000 per year with an annual increase of 3%p.a.

When will Paul's annual earnings first exceed Megan's annual earnings, if Megan receives her bonus every year?

$\textrm{Well, if Megan receives the bonus every year then each she gets :}$

$83000 + 83000*0.05 = 830000*(1.05) = 87150 \ \textrm{every year}$

$\textrm{let's see what Paul's Salary is going to be after 1 and 2 years :}$

$\textrm{After one year:} \ 83000+83000*0.03 = 83000*(1.03) = 85490$

$\textrm{After two years:} \ 83000*(1.03)*(1.03) = 83000*(1.03)^{2} =88054.7$

$\textrm{Hence after two years,Paul's salary exceeds Magan's}$

$\textrm{This is a simple question but in general, You can solve these questions like this: }$

$\textrm{Magan's salary after bonus:} \ P*(1+r) \ \textrm{where P = starting salary, r = bonus interest}$

$\tetrm{Paul's salary after n years: } \ P*(1+s)^{n} \ \textrm{where} \ P \ \textrm{= starting salary,} \ s% \ \textrm{= annual salary increase}$

$\textrm{find n when,} \ P*(1+s)^{n} > P*(1+r) \Rightarrow (1+s)^{n} > (1+r) \Rightarrow ln( (1+s)^{n}) = n ln(1+s) > ln(1+r)$

$\Rightarrow n > \frac{ln(1+r)}{ln(1+s)} \ \textrm{note} \ ln(1+s) > 0 \ \textrm{so the direction of the inequality won't change}$

Compound Interest (1 Viewer)

nrumble42

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Mahan1

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