Balance to be repaid: cash price $12480; computer bought on hire purchase

[richiesmasher]

Hey, I have a question here and I'm mad at myself I can't figure it out, it's literally on the easy section.

It goes, : ''The cash price of a computer is $12480. It can also be bought on hire purchase by making a 25% deposit and paying simple interest that is equivalent to 20% on the outstanding balance which must be repaid in 18 equal monthly installments.''

Calculate (i) the deposit paid, (ii) the outstanding balance (iii) the balance to be repaid.

There's two extra part to this question but I can figure it out.

My problem is part (iii).

I worked out the deposit to be $3120, and the outstanding balance to be $9360.

So now to get the balance to be repaid, I get 20% of 9360 and multiply it by 18.


This will give me $33696.

This is wrong.

I then convert 18 months to years, 2.5 years.

I multiply my 20% value by that instead, I get $4680.

Still wrong.

What is the secret behind this? I know it's not hard, I just feel like I'm lacking a fundamental understanding on how these interest and hire purchase questions work.

Edit: I've figured it out, I did ($9360 *20%*1/18)/100 then multiplied that answer by 18, then added that to 9360.

But now my question is, why does that work, why does using one month as the time frame in the simple interest formula get me to that answer?

My instinct was to use all 18 months

 

[richiesmasher]

Sorry...that's not correct...it is NOT an easy solution!

You are correct with 9360 being the amount financed at 20%.

You now need to calculate the monthly payment required
to repay above over 18 months.

You need the monthly rate to do so : .20 / 12 = .0166...

If you have a financial calculator, you need these steps:
enter 9360 as AMT
enter .10/12 as INT
enter 18 as N
Answer will be PMT = 606.18

Or you can do so by formula: google "loan payment formula".

I've assumed the 20% means 20% APR compounded monthly.
It is possible that something else is meant: needs clarification.

I believe what you're saying, however I believe that at the level I'm doing that is not required, as the answer I got corresponds to the one that I see the book. I dont doubt that the proper way of doing things is the way you've presented however.

 

[Dr.Peterson]

Hey, I have a question here and I'm mad at myself I can't figure it out, it's literally on the easy section.

It goes, : ''The cash price of a computer is $12480. It can also be bought on hire purchase by making a 25% deposit and paying simple interest that is equivalent to 20% on the outstanding balance which must be repaid in 18 equal monthly installments.''

Calculate (i) the deposit paid, (ii) the outstanding balance (iii) the balance to be repaid.

There's two extra part to this question but I can figure it out.

My problem is part (iii).

I worked out the deposit to be $3120, and the outstanding balance to be $9360.

So now to get the balance to be repaid, I get 20% of 9360 and multiply it by 18.


This will give me $33696.

This is wrong.

I then convert 18 months to years, 2.5 years.

I multiply my 20% value by that instead, I get $4680.

Still wrong.

What is the secret behind this? I know it's not hard, I just feel like I'm lacking a fundamental understanding on how these interest and hire purchase questions work.

Edit: I've figured it out, I did ($9360 *20%*1/18)/100 then multiplied that answer by 18, then added that to 9360.

But now my question is, why does that work, why does using one month as the time frame in the simple interest formula get me to that answer?

My instinct was to use all 18 months

First, 18 months is 1.5 years, not 2.5.

Similarly, in "($9360 *20%*1/18)/100", I think you must have meant 1/12 (monthly) rather than 1/18. And if you do that, you are doing essentially the same thing as multiplying by 18/12 = 1/5.

The first method uses annual interest rate; the second uses the equivalent monthly interest rate. That is why both work.

And I find the problem ambiguous, particularly in the phrase "20% on the outstanding balance". Denis took this as it would usually be taken in practice: 20% (annual interest rate) on the outstanding balance after each payment, which reduces. This amounts to compound interest. Since they explicitly say it is simple interest, and you evidently are not working at that level, it seems that what they mean is to precompute the simple interest on the balance after the deposit, and divide that into equal amounts. Which means that what you did, once you use the right numbers, is correct.

 

[tkhunny]

What is the secret behind this?

The First secret is a realization that the entire amount is not charged interest for all 18 months.

Do one month at a time to see the mechanics:

Your Monthly Payment is "P"

Month 1: $9,360 * (0.20 / 12) = $156 ==> $9,360 + $156 - P = $8,909.82
Month 2: $8,909.82 * (0.20 / 12) = $148.50 ==> $8,909.82 + $148.50 - P = $8,452.13
...
Month 18: $596.25 * (0.20 / 12) = $9.94 ==> $596.25 + $9.94 - P = 0

See how the Simple Interest is different every month? This is the important secret.
You will probably need to round P to the higher penny.

The second secret is listen to Denis.

The third secret is to be very clear. You were clear on the EQUAL monthly payments.

 

[richiesmasher]

First, 18 months is 1.5 years, not 2.5.

Similarly, in "($9360 *20%*1/18)/100", I think you must have meant 1/12 (monthly) rather than 1/18. And if you do that, you are doing essentially the same thing as multiplying by 18/12 = 1/5.

The first method uses annual interest rate; the second uses the equivalent monthly interest rate. That is why both work.

And I find the problem ambiguous, particularly in the phrase "20% on the outstanding balance". Denis took this as it would usually be taken in practice: 20% (annual interest rate) on the outstanding balance after each payment, which reduces. This amounts to compound interest. Since they explicitly say it is simple interest, and you evidently are not working at that level, it seems that what they mean is to precompute the simple interest on the balance after the deposit, and divide that into equal amounts. Which means that what you did, once you use the right numbers, is correct.

Hmm well the answer in the book is $11232, which I get only when using 1/18.

Let me show you the results: ($9360 *20%*1/18)/100 = $104

$104*18 = $1872

$9360 +$1872 = $11232

 

[tkhunny]

Let me show you the results: ($9360 *20%*1/18)/100 = $104

$104*18 = $1872

$9360 +$1872 = $11232

This is where the "more clear" part comes in.

If this is the correct answer, then the interest is pre-computed and divided up in to 18 parts. Kudos to DrP who imagined this might be the prescribed methodology.

Note 1: This may be illegal as it likely falls under usury laws. This is equivalent to an interest rate just under 85%, using a more typical method.

Note 2: Why would you divide by 18 only to multiply by 18 immediately thereafter? $9,360 * 0.20= $1,872

Note 3: Get rid of the "/100". That's just wrong. You don't use it and you shouldn't.

 

[richiesmasher]

Sorry, but you've completely lost me.
What does the book give as the amount of the monthly payment?

Well, the answers in the book are as follows
(I) the deposit paid = $3120
(II)the outstanding balance=$9360
(III)the balance to be repaid =$11232
(IV)the hire purchase price=$14352
(V)the difference between the cash price and hire purchase price = $1872

SO let me be as clear as possible.

I read the question, and initally thought it was compound interest, as every month you'd pay off an interest and then take the interest of how much you have left.


But the question specifically says in equal amounts.

SO I thought, ok 18 months equal amounts.

They state simple interest.

SO I used the simple interest formula, my principal being $9360, my rate being 20%, and my time, my time being one month, or in other words 1/18 representing the interest for one month in the alotted time period.

SO then I calculated, and worked it out, and got $104 for one month, from here it was just do multiply by 18, which gives me $1872.

Now the Balance to be repaid would be the outstanding balance plus the simple interest over the 18 months, which will ultimately give me $9360+$1872 and that gives me $11232.

ALSO I AGREE WITH YOU GUYS, Soon I will be over with exams, I'm currently preparing for them, these are the types of questions on the exam, but after I will pursue higher education.
Thanks for the tip TK HUNNY multiplying by 0.2 is much better.

Thanks for the proper formats Dennis, I will not ignore that and shove your valuable contributions aside
Peace and love <3

 

[Dr.Peterson]

Well, the answers in the book are as follows
(I) the deposit paid = $3120
(II)the outstanding balance=$9360
(III)the balance to be repaid =$11232
(IV)the hire purchase price=$14352
(V)the difference between the cash price and hire purchase price = $1872

SO let me be as clear as possible.

I read the question, and initally thought it was compound interest, as every month you'd pay off an interest and then take the interest of how much you have left.


But the question specifically says in equal amounts.

The fact that it is paid in equal amounts doesn't prevent it from using compound interest; we get that only from the word "simple" that they used! Ordinary installment loans pay equal amounts, too.

SO I thought, ok 18 months equal amounts.

They state simple interest.

SO I used the simple interest formula, my principal being $9360, my rate being 20%, and my time, my time being one month, or in other words 1/18 representing the interest for one month in the alotted time period.

SO then I calculated, and worked it out, and got $104 for one month, from here it was just do multiply by 18, which gives me $1872.

Now the Balance to be repaid would be the outstanding balance plus the simple interest over the 18 months, which will ultimately give me $9360+$1872 and that gives me $11232.

Based on the answer, the 20% is not an annual rate but a total percentage of interest. I've never heard that done, but I suppose it's literally what the words tell you!

So your calculation for the "balance to be repaid" is simply

balance + 20% of balance = 9360 + 9360 * 0.20 = 9360 + 1872 = 11,232
​

It could have been calculated merely as a 20% increase over 9360: 9360 * 1.20 = 11,232.

Taking 1/18 only makes sense if you take 20% as the rate for the entire 18 months, not an annual rate. And since you multiplied by 18 anyway, it was not needed at all.

ALSO I AGREE WITH YOU GUYS, Soon I will be over with exams, I'm currently preparing for them, these are the types of questions on the exam, but after I will pursue higher education.
Thanks for the tip TK HUNNY multiplying by 0.2 is much better.

Thanks for the proper formats Dennis, I will not ignore that and shove your valuable contributions aside
Peace and love <3

The important thing about 20 vs. 0.2 is that 20% MEANS 20/100 (that is, "%" means "/100"), so that when you wrote 20% ... /100, you were being redundant. You can call it multiplication by 20%, or multiplication by 20 and division by 100, but not both. And just rewriting 20% as 0.20 is easiest.

Now, the remaining question in my mind is, did they actually teach you what they are doing? Have they told you anywhere that "paying simple interest that is equivalent to 20% on the outstanding balance" should be read as "paying 20% of the outstanding balance as interest"? And do they claim this is actually done in business? (You'll have to ask the others about that; I know the math far better than finance or the law.)

And what Denis (and tkhunny at first) were talking about is what would really be done for a typical installment loan, unless perhaps 18-month loans are often done without all that bother.

 

[tkhunny]

from Investopedia.com

DEFINITION of 'Usury Rate'

A rate of interest that is usually considerably above current market rates. Usury rates are often charged by unsecured lenders on loans. These rates can be illegal in some countries and situations because they often take advantage of unsuspecting individuals.

More from me: Even the relatively benign "Rule of 78s" can get into usury on some loans. ALWAYS read your terms carefully BEFORE signing anything. This is one of the best answers to the obtuse question(s), "What good is math?" or "When will I ever use this?" The bad guys know enough. You can defend yourself against such if you have enough math in your head. Mathematics will save you from very serious errors. I once dominated a national mortgage lending company in a brief dispute we had about escrow payment calculation. Eventually, the company wrote me a very polite note - after they received a threatening letter from a federal agency who could make their lives miserable. Mathematics WILL save you!

 

[tkhunny]

Good "catch" Dr P.

Calculating the effective annual rate from that gives 1.9939 per month;
rounded to 2%: 1.02^12 = 1.2682 or 26.82% effective.

And that's not likely high enough to be illegal. That's just bad credit card. Nowhere near loan-sharking.