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Compound Interest - GMAT Prep

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Compound Interest - GMAT Prep

by rahulvsd » Thu May 17, 2012 9:15 am
An investment of $1000 was made in a certain account and earned interest that was compounded annually. The annual interest rate was fixed for the duration of the investment, and after 12 years the $1000 increased to $4000 by earning interest. In how many years after the initial investment was made the $1000 have increased to $8000 by earning interest at that rate?

A. 16
B. 18
C. 20
D. 24
E. 30

[spoiler]OA: B.[/spoiler]
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by Mike@Magoosh » Thu May 17, 2012 3:20 pm
Hi, there. I'm happy to help with this. :)

This problem is enormously simplified by understanding a deep property of exponential functions. (NB: the total amount resulting from compound interest is indeed an exponential function.)

In a linear function (y = mx + b), every horizontal "step" you take results in adding (or subtracting) the same figure to the output. Linear functions are close kin with arithmetic series, series in which you add a fixed difference to produce successive terms.

In an exponential function (y = a*(b^x)), every horizontal "step" you take results in multiplying (or dividing) the output by the same figure. Exponential functions are close kin with geometric series, series in which you multiply a fixed ratio to produce successive terms.

Thus, in this problem, "steps" of the same size in time, in years, will produce the same factors of growth.
In twelve years, the total was multiplied by 4, so in twelve more years (24 years total) it would be multiplied by four again --- after 24 years, there will be $16,000.
We know the factor 4 can be written as 2*2, as the product of two equal factors, and so if we divide that twelve year period into two equal steps, each would produce one of those factors in growth. Thus, in six years, there's $2000. In twelve years, there's $4000. In eighteen years, there's $8000.

Answer = B

Does this analysis make sense? Please let me know if you have any follow up questions.

Mike :)
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by GMATGuruNY » Thu May 17, 2012 5:28 pm
rahulvsd wrote:An investment of $1000 was made in a certain account and earned interest that was compounded annually. The annual interest rate was fixed for the duration of the investment, and after 12 years the $1000 increased to $4000 by earning interest. In how many years after the initial investment was made the $1000 have increased to $8000 by earning interest at that rate?

A. 16
B. 18
C. 20
D. 24
E. 30

[spoiler]OA: B.[/spoiler]
If it takes 12 years for the amount to increase by a factor of 4 (from 1000 to 4000), then it will take half that time -- 6 more years -- for the amount to increase by a factor of 2 (from 4000 to 8000).
Thus, the total amount of time needed = 12+6 = 18 years.

The correct answer is B.
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by russ9 » Sun Dec 07, 2014 1:52 pm
Mike@Magoosh wrote:Hi, there. I'm happy to help with this. :)

This problem is enormously simplified by understanding a deep property of exponential functions. (NB: the total amount resulting from compound interest is indeed an exponential function.)

In a linear function (y = mx + b), every horizontal "step" you take results in adding (or subtracting) the same figure to the output. Linear functions are close kin with arithmetic series, series in which you add a fixed difference to produce successive terms.

In an exponential function (y = a*(b^x)), every horizontal "step" you take results in multiplying (or dividing) the output by the same figure. Exponential functions are close kin with geometric series, series in which you multiply a fixed ratio to produce successive terms.

Thus, in this problem, "steps" of the same size in time, in years, will produce the same factors of growth.
In twelve years, the total was multiplied by 4, so in twelve more years (24 years total) it would be multiplied by four again --- after 24 years, there will be $16,000.
We know the factor 4 can be written as 2*2, as the product of two equal factors, and so if we divide that twelve year period into two equal steps, each would produce one of those factors in growth. Thus, in six years, there's $2000. In twelve years, there's $4000. In eighteen years, there's $8000.

Answer = B

Does this analysis make sense? Please let me know if you have any follow up questions.

Mike :)
Hi Mike

Two questions regarding your approach:

1) Since interest is compounded, isn't the amount that increases after every annual period change? For example, if I invest $100 at a rate of 20% compounded annual -- after year 1, I have 120. After year 2, I have 144 and so on. It rises a lot quicker because they are exponents -- doesn't it?

By the same token, how can we assume that it will take the same amount of time to double from $1000 to $2000 as $4000 to $800. I would assume that the latter would double a lot quicker?

2) If I was to follow the math, I get 4 = (1+r)^12. What exactly can we do to simplify this further. I tried to multiply both sides by (1/12) but that didn't really get me anywhere? If I simplified it further, I ended up getting:

(2^6) = (100 + R)/(100)
R = 6200?
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by [email protected] » Sun Dec 07, 2014 8:20 pm
Hi russ9,

Both Mike and Mitch are talking about a math concept called "the rule of 72" - it's a quick math rule that you can use when dealing with compound interest. Here's how it works:

If you have a fixed interest rate that is compounded annually, then you can make a quick estimation for how long it would take your initial investment to DOUBLE.

72/(interest rate) = number of years for the principle to double

eg. $100 invested at 8% compounded annually will DOUBLE in 9 years.

72/8 = 9

Here, we're told that the $1,000 become $4,000, so it DOUBLES and then DOUBLES AGAIN. The twelve years that it takes for this to happen can be broken down into two 6-year periods in which the amount doubled each time.

We now know that it takes 6 years to DOUBLE an amount, regardless of what the initial investment is.

The question asks how long it takes to go from $1,000 to $8,000

$1,000 to $2,000 = 6 years
$2,000 to $4,000 = another 6 years
$4,000 to $8,000 = another 6 years

Total = 18 years.

Final Answer: B

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by parry » Fri Jun 03, 2016 4:58 pm
[email protected] wrote:Hi russ9,

Both Mike and Mitch are talking about a math concept called "the rule of 72" - it's a quick math rule that you can use when dealing with compound interest. Here's how it works:

If you have a fixed interest rate that is compounded annually, then you can make a quick estimation for how long it would take your initial investment to DOUBLE.

72/(interest rate) = number of years for the principle to double

eg. $100 invested at 8% compounded annually will DOUBLE in 9 years.

72/8 = 9

Here, we're told that the $1,000 become $4,000, so it DOUBLES and then DOUBLES AGAIN. The twelve years that it takes for this to happen can be broken down into two 6-year periods in which the amount doubled each time.

We now know that it takes 6 years to DOUBLE an amount, regardless of what the initial investment is.

The question asks how long it takes to go from $1,000 to $8,000

$1,000 to $2,000 = 6 years
$2,000 to $4,000 = another 6 years
$4,000 to $8,000 = another 6 years

Total = 18 years.

Final Answer: B

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Hi Rich,

Thanks for the great explanation.

I have a query.

What if the interest rate was compounded half yearly, can we still apply the same rule?
If not, please explain how to solve the same problem with interest rate compounded half yearly.

Thanks in advance for your time.

Regards,
PS
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by [email protected] » Sat Jun 04, 2016 9:24 am
Hi parry,

Yes, the concept is the same regardless of how often you calculate the interest. If you change the variables though (re: the interest rate and how often the interest is calculated), then the result of the calculations would be different, but you'd still do the same calculations.

By calculating interest half-yearly, you DOUBLE the number of interest calculations per year and you HALVE the interest rate. Everything else is the same.

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by parry » Sat Jun 04, 2016 10:32 pm
Thanks a lot Mr. Rich,

it is very helpful.
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by NandishSS » Sat Jan 07, 2017 7:42 pm
[email protected] wrote:Hi russ9,

Both Mike and Mitch are talking about a math concept called "the rule of 72" - it's a quick math rule that you can use when dealing with compound interest. Here's how it works:

If you have a fixed interest rate that is compounded annually, then you can make a quick estimation for how long it would take your initial investment to DOUBLE.

72/(interest rate) = number of years for the principle to double

eg. $100 invested at 8% compounded annually will DOUBLE in 9 years.

72/8 = 9

Here, we're told that the $1,000 become $4,000, so it DOUBLES and then DOUBLES AGAIN. The twelve years that it takes for this to happen can be broken down into two 6-year periods in which the amount doubled each time.

We now know that it takes 6 years to DOUBLE an amount, regardless of what the initial investment is.

The question asks how long it takes to go from $1,000 to $8,000

$1,000 to $2,000 = 6 years
$2,000 to $4,000 = another 6 years
$4,000 to $8,000 = another 6 years

Total = 18 years.

Final Answer: B

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Hi Rich,

Could you please explain how did you derive 72/(interest rate) = number of years for the principle to double.Can it be applied to all Compound interest problem?

Please share the links of problem for the practice :-)

Thanks
Nandish
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by crackverbal » Sat Jan 07, 2017 10:30 pm
Hi,

Here is my 2 cents to this question,

Everyone has explained the logical way of solving this question, which is of course short and quick,

But math is not so tough here,

Everyone knows the total amount formulae when compounded annually,

T = P(1+(r/100))^n

Given: After 12 years amount increased from 1000 to 4000, that is

1000(1+(r/100))^12 = 4000,

(1+(r/100))^12 = 4,


Now take the 12th root on both sides,

(1+(r/100))= 4^(1/12),

(1+(r/100))= 2^(1/6), ----(1)

Question:

How many years it will take 1000 to become 8000 at this same constant rate?

1000(1+(r/100))^n = 8000,

(1+(r/100))^n = 8

Just substitute 1 here, we get,

(2^(1/6))^n = 8

2^(n/6) = 2^3,

Same base both sides, so we can equate the power

n/6 = 3

So n = 18.

So there it will take 18 years with same constant rate to become 1000 to 8000.

So the answer is B.

Hope this helps �
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by [email protected] » Sun Jan 08, 2017 11:40 am
Hi NandishSS,

The 'rule of 72' (some people call it the 'rule of 70') is a quick way for bankers (and other investment professions) to estimate how long it would take for a fixed investment to double in value - and it applies to all compound interest questions. You can actually prove that it works with ANY example that you can come up with.

Here's a simple example: How long would it take for $100 to double if you invested it at 10% compound interest? According to the rule of 72, it should be about 72/10 = 7 years....

Start = $100
Yr. 1 = $110
Yr. 2 = $121
Yr. 3 = $133.1
Yr. 4 = $146.4
Yr. 5 = $161
Yr. 6 = $177
Yr. 7 = $195

$195 is almost double $100.

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