Maths puzzle

Tednol

This question was in my Advanced Maths module exam today. Bear in mind it is advanced maths with economics in mind so it's hardly P6 stuff. This question left me confused and angry...

"An individual is entering a mortage contract which specifies the amount borrowed, A = £150,000, and the total length of time over which the loan is repayed, K = 25 years. In addition, the annual interest rate is fixed at 5% for the first 5 years (that is, I = 0.05). Determine the constant annual repayment, b, for the first 5 years, such that the sum that remains to be repaid in 5 years from now is £125,000"

Any ideas? I'm sure I've got an equation that will do it for me somewhere, but I certainly didn't learn it for my exam.

Ed

Probably this:
The amount the loan is down by is 25,000 so you will have paid that plus interest. Multiply by 1.05^5 and divide by 5 and you should probably get the yearly repayment for the first five years.

Reply 2

Tednol

This question was in my Advanced Maths module exam today. Bear in mind it is advanced maths with economics in mind so it's hardly P6 stuff. This question left me confused and angry...

"An individual is entering a mortage contract which specifies the amount borrowed, A = £150,000, and the total length of time over which the loan is repayed, K = 25 years. In addition, the annual interest rate is fixed at 5% for the first 5 years (that is, I = 0.05). Determine the constant annual repayment, b, for the first 5 years, such that the sum that remains to be repaid in 5 years from now is £125,000"

Any ideas? I'm sure I've got an equation that will do it for me somewhere, but I certainly didn't learn it for my exam.

Ed

Can it wait until tomorrow when I've had some sleep? Kind of tired at the minute, but I've done something rather similar before. Just follow the instructions and you should be fine. Thats what I seem to remember about it.

Reply 3

Tednol

OP

ZJuwelH

Probably this:
The amount the loan is down by is 25,000 so you will have paid that plus interest. Multiply by 1.05^5 and divide by 5 and you should probably get the yearly repayment for the first five years.

25,000 x 1.05^5 = 31907 over the 5 years => 6381 each year.

Take 150000 as your start. After one year will have paid 6381 towards it, and have had to multiply it by 1.05. You end up with more than 150000.

Nice try tho man.

Reply 4

Tednol

OP

Sire

Can it wait until tomorrow when I've had some sleep? Kind of tired at the minute, but I've done something rather similar before. Just follow the instructions and you should be fine. Thats what I seem to remember about it.

Course!

Reply 5

Tednol

25,000 x 1.05^5 = 31907 over the 5 years => 6381 each year.

Take 150000 as your start. After one year will have paid 6381 towards it, and have had to multiply it by 1.05. You end up with more than 150000.

Nice try tho man.

If you are paying towards the loan you should be taking away from the initial sum of the loan, so how do you get more than 150,000?

Reply 6

theone

Tednol

This question was in my Advanced Maths module exam today. Bear in mind it is advanced maths with economics in mind so it's hardly P6 stuff. This question left me confused and angry...

"An individual is entering a mortage contract which specifies the amount borrowed, A = £150,000, and the total length of time over which the loan is repayed, K = 25 years. In addition, the annual interest rate is fixed at 5% for the first 5 years (that is, I = 0.05). Determine the constant annual repayment, b, for the first 5 years, such that the sum that remains to be repaid in 5 years from now is £125,000"

Any ideas? I'm sure I've got an equation that will do it for me somewhere, but I certainly didn't learn it for my exam.

Ed

After one year, our amount left if (150000-b).(1.05).

After two years, it's ((150000-b)(1.05) - b).(1.05) = 150000.(1.05)^2 - (2.05)(1.05)b.

After three years it's ((150000.(1.05)^2 - (2.05)(1.05)b))-b)(1.05) = (150000.(1.05)^3 - ((2.05)(1.05)^2 + 1.05)b)

After four years it's (150000.(1.05)^4 - ((2.05)(1.05)^3 + (1.05)^2 + 1.05)b)

After five years it's (15000.(1.05)^5 - ((2.05)(1.05)^4 + (1.05)^3 + (1.05)^2 + 1.05)b = 125000.

Therefore we get ((1.05)^5 + (1.05)^4 ... + 1.05)b = 150000.(1.05)^5 - 125000.

Now solve for b.

Hope this is right :biggrin:

Reply 7

elpaw

15

i think i saw a similar question in a past STEP maths paper, so don't be embarrased that its easy (which it isn't)

Reply 8

Tednol

This question was in my Advanced Maths module exam today. Bear in mind it is advanced maths with economics in mind so it's hardly P6 stuff. This question left me confused and angry...

"An individual is entering a mortage contract which specifies the amount borrowed, A = £150,000, and the total length of time over which the loan is repayed, K = 25 years. In addition, the annual interest rate is fixed at 5% for the first 5 years (that is, I = 0.05). Determine the constant annual repayment, b, for the first 5 years, such that the sum that remains to be repaid in 5 years from now is £125,000"

Any ideas? I'm sure I've got an equation that will do it for me somewhere, but I certainly didn't learn it for my exam.

Ed

I'm doing this in my head so don't trust me. (can't find the calculator, too tired/lazy to find it) but wouldn't the repayment be 5,000 per year?

Reply 9

byb3

8

elpaw

i think i saw a similar question in a past STEP maths paper, so don't be embarrased that its easy (which it isn't)

looks like a wrapped up geometric series to me.

Adam

Reply 10

Sire

I'm doing this in my head so don't trust me. (can't find the calculator, too tired/lazy to find it) but wouldn't the repayment be 5,000 per year?

I'll bet that was just (150000-125000)/5...

Reply 11

have they asked this question properly, or just thrown in a lot of stuff to throw you off? As it stands you have a 150,000 loan. In 5 years it will be down to 125,000. How much have you paid off each year?
Answer to that question without anything about interest etc is simple.
150,000 minus 125,000 = 25,000
25,000 divided by 5 = 5,000
ANS = 5,000.

Are you sure you don't need to calculate the interest rate and find a monthly repayment?

Reply 12

theone

As you can tell, it's not difficult to prove (by induction) the formula for the amount remaining after n years, so once you've done this, all problems of this type are just a case of plugging the numbers in.

Reply 13

ZJuwelH

I'll bet that was just (150000-125000)/5...

It was. I've read the question over, and I can't see anything other than that. It doesn't mention anything about calculating the interest rate, if so, please specify.

Reply 14

theone

Sire

It was. I've read the question over, and I can't see anything other than that. It doesn't mention anything about calculating the interest rate, if so, please specify.

"In addition, the annual interest rate is 1.05%"

Reply 15

theone

"In addition, the annual interest rate is 1.05%"

Truth be told, the question says: "A person takes out a mortgage of 150,000... how much does the annual repayment need to be if the person is to owe 125,000 five years from now?"

All the rest of it seems to be chatcrap if you read it like that so Sire is probably right.

Reply 16

Tednol

OP