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Problem in solving question on Installments


magicman_vj

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Q- What annual installment will discharge a debt of 1092 due in 3 years at 12% Simple Interest?

 

Solution says:

{x + x*12*1/100} + {x + x*12*2/100} + x = 1092

Find the value of x which is the annual installment.

 

Could someone please help me understand the concept of above problem.

I checked with many people but cudnt get the exact explanation.:rolleyes:

 

Thanks in advance..[clap]

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OK, I will try.

You have to realize that the problem is asking to find the amount of annual installments at the SIMPLE INTEREST RATE of 12%. This means that CONSTANT amount of interest accrues each time period (here in the problem time period is given to be three years with installments made at the end of each year ) on the principal amount. So, you should be able to get the following equation: 3X*1.12=1092. Be aware of the trap though. Note that ALL INSTALLMENTS should be of the same value. For instance, let us assume that X is our installment after the first year. The interest that accrues on this installment is 0.12x. But the period 2 principal is not (x + 0.12 x). Because, (x + 0.12x) as the principal for period 2 calculation will lead us to calculate the compound interest rate which is not the same as the simple interest rate. So the period 1 installment = period 2 installment = period 3 installment = x. If you add all these installments you get 3x. Now 12% of simple interest on 3x is 3x*(112/100)=3x*1.12 . Now, according to the question: 3X*1,12=1092. Solving for X you get: X=325.:)

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You are always welcome!

Well, I learned this concept in high school. I often use high school math textbooks for reviewing these concepts.

I am sure there must be well written specialized books regarding installments, interest rates, etc. For GMAT purposes however, one of the best quantitative prep materials is, probably,Prep Book with a CD by KAPLAN. My advise to you is to practice, practice and practice various problems by utilizing concepts that you just learned. I am quite sure that you will find a lot of useful material on this forum as well. Let me know if you still need any help.

Cheers!

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OK, guys!

I am sorry for this delay. Well, the first and most important thing to remember about solving problems which involve SIMPLE interest rate calculations is to realize that the amount to which this interest rate is being applied DOESN'T change from period 1 to period 2, say. So, in other words, we are applying the same interest rate to the same 'chunk' of the 'whole' in each period, where the sum of all these "chunks" ad up to a 'whole'. Alternatively, you could just take a 'whole' and apply the same interest rate JUST ONCE. Then, to arrive at your final answer you add up this 'whole' and the resultant amount that you got by applying the interest in the previous step.

This morning I tried to think of a visual explanation, so that all of you could understand this concept. Here's what I came up with. Assume that you have flat Lego pieces of the same length where lenght represents the resultant amount after the interest application. Now, also assume that you have an initial base (in real world this base could be initial amount of money, etc.) that you will apply your interest to. Let's say that this initial base is in the form of a 4 level (storey) identical Lego house. Further, assume that each level of this house represents a separate period of applying interest. Now, all in all you have 4 periods or let's say 4 quarters of a year. Let's say that the interest rate that we will apply to a base for the whole period of one year in 4 separate installments per quarter is X percent. So, there are two ways you can calculate the SIMPLE interest on this given initial base:

1. First, you can take the interest on each individual layer (level) and add up this amount to your initial base. Now, let's assume that the calculation of interest of X percent on level 1 is the lenght of the first Lego piece. Similarly, for each separate level you have the level itself and the same interest rate that you are applying to that level. So, you get the second Lego piece, which is of the same length as the first Lego piece. All in all, you get 4 identical Lego pieces of the same lenght at the end. By adding these four Lego piece lenghts you get a total SIMPLE interest on your initial base (LEGO house in our case).

2. You can take the initial base and apply the SAME interest to the whole base (to all 4 levels at the same time). Here, you just get the whole distance of these 4 identical LEGO pieces as a one measure.

Hope this helps to understand better this concept.

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In the given example: Initial Base (Lego house) is 975, the lenght of each Lego piece is 39, the number of storeys/levels of this house is 3 and the interest rate is 12%.

So, utilizing the first approach you should get:

(1) 39 + 39 + 39 + 975 = 1092

The second methos will give us:

(2) 975 + 117 = 1092

 

Cheers!

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Lock,

 

thanks for the explanation, so you are saying that 975 was the loan amount and we end up paying 1092 due to 12% interest rate

AND

each year or period since the interest amount remains the same, you end up paying more principal in year 3, than year 1, so that all three year the installments are same

AND

12% rate is over 3 years, not year over year

 

did i get it right?

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hardikrs:

In this problem we are given that the total sum (initial base + SIMPLE interest on this base) equals 1092. Now, the problem is asking what is the yearly installment value (the manitude of each level of the LEGO house). So, in our case it will be: 975/3 = 325.

So, as you see 975 is our initial base (total "interest free" loan amount at the end of the whole period).

With regard to the second part of your question, you still seem to have a confusion over interest rate disbursements at the end of each period. I am quoting you here: "each year or period since the interest amount remains the same, you end up paying more principal in year 3, than year 1, so that all three year the installments are same." Here, you should differentiate the interest on a third year installment (interest on a third level of the LEGO house ALONE) vis a vis the sum of interest payments for ALL THREE YEARS taken together (interest on a LEGO house with ALL three levels taken together).

Now, coming to the final part of your query, in SIMPLE interest rate calculations you can utilize BOTH methods that I outlined above. Just remember: if you are separating the initial base into its sub-parts each sub-part should be equal to each other and you apply the same interest on each sub-part separately. So, yes, at the end the result is 12 percent for the whole base of 975 + the base itself, or you could say that it's 12 percent for all sub-parts taken individually + the sum of all sub-parts.

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  • 5 years later...
OK, I will try.

You have to realize that the problem is asking to find the amount of annual installments at the SIMPLE INTEREST RATE of 12%. This means that CONSTANT amount of interest accrues each time period (here in the problem time period is given to be three years with installments made at the end of each year ) on the principal amount. So, you should be able to get the following equation: 3X*1.12=1092. Be aware of the trap though. Note that ALL INSTALLMENTS should be of the same value. For instance, let us assume that X is our installment after the first year. The interest that accrues on this installment is 0.12x. But the period 2 principal is not (x + 0.12 x). Because, (x + 0.12x) as the principal for period 2 calculation will lead us to calculate the compound interest rate which is not the same as the simple interest rate. So the period 1 installment = period 2 installment = period 3 installment = x. If you add all these installments you get 3x. Now 12% of simple interest on 3x is 3x*(112/100)=3x*1.12 . Now, according to the question: 3X*1,12=1092. Solving for X you get: X=325.:)

 

 

 

cool xplanation lock..!!

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