MAGMAT - A certain investment grows

[Ruben]

A certain investment grows at an annual interest rate of 8%, compounded quarterly. Which of the following equations can be solved to find the number of years, x, that it would take for the investment to increase by a factor of 16?


16 = (1.02)x/4
2 = (1.02)x
16 = (1.08)4x
2 = (1.02)x/4
1/16 = (1.02)4x


I can't understand what the question is really asking. The explanations says:

At the end of the x years, the final value, F, will be equal to 16 times the principal (the money is growing by a factor of 16).
Therefore, F = 16P.
r = .08 (8% annual interest rate)
n = 4 (compounded quarterly)
t = x (the question is asking us to express the time in terms of x number of years)
We can write the equation
16P = P (1 + .08/4)4x
16 = (1.02)4x

Could you please help me rephrasing the question? Is it asking to find a formula that allows us to obtain a a value of 16 after x years of compunding interest?

Thanks,

Ruben

[RonPurewal]

Could you please help me rephrasing the question? Is it asking to find a formula that allows us to obtain a a value of 16 after x years of compunding interest?

not really, although the '16' is what you end up with after the appropriate cancellations.

make sure you know how to use the compound interest formula, P*(1 + r/n)^(nt), where P is the principal at the beginning, r is the interest rate PER YEAR (as a decimal, e.g., 0.07 for 7%), n is the number of compounding periods per year (e.g., n = 4 for quarterly compounding), and t is time IN YEARS.
that's what gives the formula in this problem.

note that the 16 is originally 16P: you start with principal 'P', and you want to find the time that the investment takes to reach '16P'. you can then cancel the 'P's that appear on both sides. (alternatively, since the phrasing of the problem makes it clear that the original principal is irrelevant - because it's never mentioned - you can just pick a number for P.)

make sense?

[Ruben]

Ron,

I think I got it. I was thrown off by the wording a factor of 16. When y increasey by a factor of x means I multiply y by x right?

Thanks,

Ruben

[RonPurewal]

When y increasey by a factor of x means I multiply y by x right?

correct.

[lanzhou1]

My solution on the CAT is 2 = (1.02)^x.

There is a paragraph in addition to the explanation copied above. After the solution calculated 16 = (1.02)^(4x) (which is what I got) it also says:

_______________________________________________
"Now we can take the fourth root of both sides of the equation. (i.e.the equivalent of taking the square root twice) We will only consider the positive root because a negative 2 doesn't make sense here.
161/4 = [(1.02)^(4x)]^(1/4)
2 = (1.02)^x

The correct answer is B."
________________________________________________

My question is... why do you take the 4th root? I'm not sure of the logic here

[lanzhou1]

Nevermind, I got it. Just needed to find an equivalent equation.

[tim]

Glad to hear it!

[marc.gagnon]

Can you help clarify "increase by a factor of 16".

If I have $5 and it increases by a factor of 1, do I now I have $5 or $10 (original $5 + increase $5)?

[jnelson0612]

Marc, check this post out:

When y increases by a factor of x means I multiply y by x right?

correct.

[marc.gagnon]

Thanks

Can you help confirm/clarify my understanding for similiar phrasings:

Three times as many as X: Means X * 3
Four times greater than X: Means X * 4 + X, which simplifies to X * 5

Am I missing any other ones?

[jnelson0612]

Thanks

Can you help confirm/clarify my understanding for similiar phrasings:

Three times as many as X: Means X * 3
Four times greater than X: Means X * 4 + X, which simplifies to X * 5

Am I missing any other ones?

Those look correct to me.
Marc has three times as many apples as I do. If you have 12 apples, I have 4.

Marc earns a salary that is 4 times greater than mine. If mine is $20, and Marc's is 4 times greater than that, his must be $80 greater, or $100.

I don't think you're necessarily missing anything; if you ever are in doubt, plug real numbers to help you figure these out.

[asharma8080]

I was trying to do this problem without using the formula; however, run into a road-block. what's going on here with my approach?

Let starting investment = y
Wanted investment = 16y

if we were to do quarterly increases:
1st q-- y * 1.02
2nd q--y * 1.02 * 1.02
3rd q--y * (1.02)^3

so there is a pattern... the exponent on 1.02 gives the quarters it has passed.

Thus- 16y = y * (1.02)^x
divide by y
16 = (1.02)^ x
Or,
2= (1.02)^(x/4)
x = Number of Quarters needed to make the money 16 times.

If I want to convert quarters into years, straightforward way is to divide by 4 (e.g., 4 quarters / 4 = 1 year). In this case, it does not work.

2 = (1.02)^(x/16)--- NOT AN ANSWER CHOICE

Now...how do I convert this into YEARS??? or, what conceptual understanding I am missing here?

[RonPurewal]

yeah ok, I see what you're doing there.

Here's what you're actually doing:
You're converting (x/4) quarters into years. I.e., you're thinking that x/4 is the actual number of quarters, in which case x/16 would be the number of years.

There are actually 2 incongruities with what you're trying to do here.

1/
(x/4) is not a number of quarters. In that equation, x (by itself) is the number of quarters.

2/
You are trying to switch from a variable representing quarters to a variable representing years. This is NOT the same as doing a unit conversion on a fixed quantity (= what your actual work represents, as described above).

To make it clear what's happening, let's say that "Q" is the number of quarters in your equation:
2= (1.02)^(Q/4)

Now let's say "Y" is the corresponding number of years. In this case, there are 4 times as many quarters as there are years, so 4Y = Q.
Substituting that gives
2= (1.02)^(4Y/4)
2 = 1.02^Y
which is the answer to the problem.

[RonPurewal]

yeah ok, I see what you're doing there.

Here's what you're actually doing:
You're converting (x/4) quarters into years. I.e., you're thinking that x/4 is the actual number of quarters, in which case x/16 would be the number of years.

There are actually 2 incongruities with what you're trying to do here.

1/
(x/4) is not a number of quarters. In that equation, x (by itself) is the number of quarters.

2/
You are trying to switch from a variable representing quarters to a variable representing years. This is NOT the same as doing a unit conversion on a fixed quantity (= what your actual work represents, as described above).

To make it clear what's happening, let's say that "Q" is the number of quarters in your equation:
2= (1.02)^(Q/4)

Now let's say "Y" is the corresponding number of years. In this case, there are 4 times as many quarters as there are years, so 4Y = Q.
Substituting that gives
2= (1.02)^(4Y/4)
2 = 1.02^Y
which is the answer to the problem.

[michaelbrettgonzalez]

Bottom line this for me Ron...

I got it wrong on my CAT
The minute I saw the formula at the top of the explanation I tried it again and it was EASY... so just memorize the formula? or should i try to solve with conceptual understanding of this topic IOT help me on future problems?

Thanks

Mike