If $1,000 is deposited in a certain bank account

[zaarathelab]

Statement 1: Equation has only one unknown. Also interest has to converge to a unique value as per the formula - hence SUFFICIENT

Statement 2: I first substituted 8 for r and found out that Interest(I) has to be > 176.4$. If i solve by substituting 1.15, I get Interest> 150$. I can be < or > than 176$. Hence INSUFFICIENT.

[RonPurewal]

that's basically the deal.

[tanyatomar]

Hi Ron,
the way i solved this question i got D.
for the 2. i opened the square on left side and ended up with an quadratic equation for r. r^2 + 2r -15 >0 was the equation i got. After solving it i got that 3<r<5
=> r is definitely !> 8.
isn't this sufficient..

why is there a difference when u square root the right side instead of sqaring left one :(

[RonPurewal]

r^2 + 2r -15 >0 was the equation i got. After solving it i got that 3<r<5

you solved this inequality incorrectly; the correct solution to the inequality is r < -5 or r > 3.
that means there are actually two mistakes. first, you've got the wrong factors (you have positive 3, rather than negative 3, as a boundary value). second, you've mistakenly identified the solution set as the region between those values; it's actually the region outside those values.

you may want to review how to solve these kinds of inequalities. our algebra strategy guide has some pretty thorough coverage of this.
you should also try evaluating your solution by plugging in values. if you think the solution is just 3 < x < 5, then that's problematic -- because it's easy to see that r^2 + 2r - 15 > 0 is true for gigantic values of r, like r = a thousand million billion.

[tanyatomar]

you r right ron, i did not take the second case of both (r+5) and (r-3) <0. and then take union of both cases.... silly mistake
lesson learnt..
thanks a ton.. :)

Tanya

[RonPurewal]

you r right ron, i did not take the second case of both (r+5) and (r-3) <0. and then take union of both cases.... silly mistake
lesson learnt..
thanks a ton.. :)

Tanya

you're welcome.

[Abhinav-]

Given that r is within a square term. Is it safe to assume from stmt 1 that r will have a unique value after solving the equation?

Thanks!

Ron--

the term (1 + r/100) can't be negative in this problem, because "r" is the interest rate on an investment and is therefore a positive number.

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Hi Ron,

Thanks for clarifying that. I know I'm missing something here. 'r' must be positive, but can we be certain that the quadratic in terms of second degree powers of 'r' NEVER solves to give us two different positive solutions?

In other words, in such interest rate DS problems, can we be confident (and if yes, how) that the solution for 'r' will always give us 1 positive and 1 negative root, of which the negative one can be ignored.

Thanks in advance

[RonPurewal]

Algebraically, you can notice that you have (1 + r/100) within a square term, and there is no "r" anywhere else. So, by whatever routine algebraic steps, you can isolate (1 + r/100)^2 = some number, so that (1 + r/100) = ±√some number.
It's clear that only the positive root counts here"”the negative root will give a value of r < -100, which is obviously not a thing"”so there's only going to be one value for r.

On the other hand, you can also realize that statement 1 is sufficient without considering any mathematics, just by using pure common sense (see next post). That's much better from the standpoint of efficiency, though you should be equally comfortable with both.

[RonPurewal]

Pure common sense:
We're talking about solving for the interest rate that will return $210 after a fixed period of time, on a fixed interest scheme.

Just think about it for a moment. The greater the interest rate (= the value of "r"), the more interest the deposit will earn.

In this context, your question is equivalent to "For the same deposit, the same interest scheme, and the same length of time, could two different interest rates return the same amount of interest?"

It should be pretty clear that the answer is no. In fact, even if no formula is given at all, it should still be perfectly clear that this statement is sufficient, since "higher interest rate --> more interest" is universally true.

[Abhinav-]

Thanks so much, Ron.

Two things:

1. Algebraic Approach
Yes, from your algebraic representation, it seems pretty obvious that (1 + r/100) = ±√some number and we should ignore the negative root. I however, made the mistake (took the rather pointless and tedious route) of simplifying the quadratic which turned out as (r^2)+200r-2100=0. At this juncture during the test, I could not tell immediately whether this quadratic would simplify to give "one positive and one negative root" or return "two positive roots" (which is likely to create trouble and not give a conclusive answer). Algebraically, is there a way that by looking at a quadratic equation of this nature (ax^2+bx+c), whether it will return two positive roots or a negative and a positive root, without actually solving the equation. I know that via the discriminant method (D=b^2-4ac), I can identify whether the equation has two distinct roots, a double root or no real root. But that does not tell me anything about the signs of the root prior to actually solving for them!


Intuitive approach:
I much prefer your highly intuitive approach that only ONE positive value of r, could return the same interest, with all other conditions (length of time and investment scheme) staying equal. I somehow missed the obvious because I was focussing way too much on the complex quadratic! Again, this doesn't preclude a negative 'r' (neg root) from existing, just that it doesn't concern us.


With this takeaway, in DS problems concerning interest, if ever there appears a quadratic in terms of r equated with some "interest amount", can I be confident that it will always yield only one unique solution?

[tim]

Algebraically, is there a way that by looking at a quadratic equation of this nature (ax^2+bx+c), whether it will return two positive roots or a negative and a positive root, without actually solving the equation.

If c is negative, there will be one negative and one positive root (assuming a is positive).

I wouldn't advise you to conclude that a quadratic will only have one viable solution; many times there will be two in fact, and this is the source of many traps on the GMAT.

[RonPurewal]

If you have to solve a quadratic on the GMAT, you will always be able to factor it. I.e., you will never have to use the quadratic formula.

With this fact in mind especially, the decision seems easy to me: Do the work.
Get out the proverbial shovel, and dig.
This test specializes in being unpredictable. If you just do the work, then unpredictability is a non-issue; if you try to predict what will happen, then unpredictability is a huge issue.

[NL]

Do the work.

What do you mean here? Following a process that one summarized for oneself?

This test specializes in being unpredictable. If you just do the work, then unpredictability is a non-issue; if you try to predict what will happen, then unpredictability is a huge issue.

A lot of hair here! (insightful)

My problem is: When I got questions incorrect under timing, the main reason was that I unintentionally didn’t follow a process. Even I myself set up timing, my mind started running and unconsciously skipped steps, then ended up at the starting point (went in a circle). The questions become 5 times harder than under normal conditions.
If I try to calm myself down, sometimes I lose the sense of timing, and may do questions too slowly.

How do I "fix" this problem?

[RonPurewal]

"Do the work" means ... do the work.
Don't guess what will happen if you solve equations; just solve them.

This discussion is now unrelated to the problem in the thread, so please post a new thread in the general math folder. Thanks.

[NL]

This discussion is now unrelated to the problem in the thread, so please post a new thread in the general math folder. Thanks.

This is called a relevant point. So, move.