Series & Sequences (Loans) (1 Viewer)

[nightweaver066]

I keep getting the answer wrong whilst doing superannuation and loan type questions..

Could someone please read over my working out and help me out? Thanks

Qn: Jan borrows $8000 and agrees to repay it in equal instalments each year for 10 years. If interest is charged at 7 percent per annum on any part of the unpaid debt, calculate the amount of each yearly instalment.

Working out:
























However the answers at the back of the book say: $1139

 

[Dubble-u25]

A₁= $8000x1.07 - M (where M is the yearly repayment)
A₂= ($8000x1.07 - M)x1.07 - M
= $8000x1.07²- M(1.07+1)
Therefore:
A_n= $8000(1.07)ⁿ - M(1.07^n-1+......+1)
A₁₀=0
$8000(1.07)¹⁰ - M(1.07⁹+1.07⁸+.....+1)= 0

M= $8000(1.07)¹⁰/1.07⁹+1.07⁸+.....+1
M= $8000(1.07)¹⁰/ [1(1.07¹⁰ - 1)/ 1.07 - 1]
M= $8000(1.07)¹⁰/13.816.......
M= $1139

Sorry not sure if I'm right plus have no idea how to use latex

In your working the $x in the first instalment shouldn't be multiplied by the 7% only the $8000.
Also, when working out the sum of the gp, a=1 not 1.07

 

[nightweaver066]

A₁= $8000x1.07 - M (where M is the yearly repayment)
A₂= ($8000x1.07 - M)x1.07 - M
= $8000x1.07²- M(1.07+1)
Therefore:
A_n= $8000(1.07)ⁿ - M(1.07^n-1+......+1)
A₁₀=0
$8000(1.07)¹⁰ - M(1.07⁹+1.07⁸+.....+1)= 0

M= $8000(1.07)¹⁰/1.07⁹+1.07⁸+.....+1
M= $8000(1.07)¹⁰/ [1(1.07¹⁰ - 1)/ 1.07 - 1]
M= $8000(1.07)¹⁰/13.816.......
M= $1139

Sorry not sure if I'm right plus have no idea how to use latex

In your working the $x in the first instalment shouldn't be multiplied by the 7% only the $8000.
Also, when working out the sum of the gp, a=1 not 1.07

Thanks for that, your answer is correct.

I assumed that they paid the first instalment in the same year they borrowed the money..

 

[Dubble-u25]

Your welcome, also yes they can/do pay the first instalment in the same year they borrowed the money hence why the $8000 is multiplied by 1.07. Which means that they are paying back the 8000 they borrowed plus the interest which is calculated before the first instalment. So the first instalment is unaffected by the interest. This also applies to the subsequent yearly installments.