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# S99-05

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Math Expert
Joined: 02 Sep 2009
Posts: 42249

Kudos [?]: 132619 [0], given: 12326

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16 Sep 2014, 01:53
Expert's post
5
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Difficulty:

55% (hard)

Question Stats:

59% (01:28) correct 41% (01:34) wrong based on 56 sessions

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Samantha invests $$i_1$$ dollars in bond $$X$$, which pays $$r_1$$ percent simple interest annually, and she invests $$i_2$$ dollars in bond $$Y$$, which pays $$r_2$$ percent simple interest annually. After one year, will she have earned more interest, in dollars, from bond $$X$$ than from bond $$Y$$?

(1) $$r_1^2 \gt r_2^2$$

(2) The ratio of $$i_1$$ to $$i_2$$ is larger than the ratio of $$r_1$$ to $$r_2$$.
[Reveal] Spoiler: OA

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Kudos [?]: 132619 [0], given: 12326

Math Expert
Joined: 02 Sep 2009
Posts: 42249

Kudos [?]: 132619 [2], given: 12326

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16 Sep 2014, 01:53
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Official Solution:

The amount of interest, in dollars, that Samantha will receive in one year is equal to the interest rate multiplied by the principal. For bond $$X$$, this product is equal to $$\frac{r_1}{100} \times i_1$$. Likewise, for bond $$Y$$, this product is equal to $$\frac{r_2}{100} \times i_2$$.

The question can be rephrased thus: "Is $$\frac{r_1}{100} \times i_1 \gt \frac{r_2}{100} \times i_2$$?" or, after multiplying through by 100, "Is $$r_1i_1 \gt r_2i_2$$?"

Statement 1: INSUFFICIENT. There is no information about $$i_1$$ or $$i_2$$.

Statement 2: INSUFFICIENT. We can translate this statement to an inequality:
$$\frac{i_1}{i_2} \gt \frac{r_1}{r_2}$$

Since all of the quantities are positive, we can multiply through without worrying about flipping the inequality symbol, and we get the following:
$$i_1r_2 \gt i_2r_1$$

However, we cannot conclude that $$r_1i_1$$ is always larger (or always smaller) than $$r_2i_2$$. You can choose numbers to see why this is so.

Statements 1 &amp; 2 together: SUFFICIENT. We want to combine the inequalities in such a way as to get $$r_1i_1$$ on one side of the inequality symbol and $$r_2i_2$$ on the other side -- if possible. In fact, this combination is possible, and the right way to execute it is first to rearrange the second statement in order to put all the same subscripts on one side. We can start from the product we obtained by cross-multiplying:
$$i_1r_2 \gt i_2r_1$$

Now divide each side by both $$r$$'s. Again, since the interest rates are necessarily positive in this scenario (you cannot be paid "negative interest"), we do not have to worry about flipping the sign. We get the following inequality:
$$\frac{i_1}{r_1} \gt \frac{i_2}{r_2}$$

Finally, we multiply this inequality by the inequality from statement 1. (Normally, this is a dangerous move, but once again, since all the quantities are positive, we are allowed to multiply.) Just make sure that the inequality symbols are in the same direction.
$$\frac{i_1}{r_1} \gt \frac{i_2}{r_2}$$
$$r_1^2 \gt r_2^2$$

We wind up with the inequality we're looking for:
$$r_1i_1 \gt r_2i_2$$

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Joined: 27 May 2016
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23 Aug 2016, 04:41
I think the correct option should be B. Only statement 2 alone is sufficient to answer the question.

This is how I approached the problem:

Question asks IS r1*i1>r2*i2, which can also be written as (i1/i2)>(r2/r1) (assuming none of the numbers are 0 or negative).

Now Statement 1 is definitely insufficient,
but for Statement 2, it says (i1/i2) > (r1/r2), now this expression also means that (i1/i2) < (r2/r1), which is our desired result. Therefore,sufficient.

Here the assumption is that all the numbers are positive, which is valid as we are comparing investments and interest rates.

Please let me know if I have missed anything in the explanation above.

Thanks!

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Math Expert
Joined: 02 Sep 2009
Posts: 42249

Kudos [?]: 132619 [0], given: 12326

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24 Aug 2016, 04:48
jainshubhams92@gmail.com wrote:
I think the correct option should be B. Only statement 2 alone is sufficient to answer the question.

This is how I approached the problem:

Question asks IS r1*i1>r2*i2, which can also be written as (i1/i2)>(r2/r1) (assuming none of the numbers are 0 or negative).

Now Statement 1 is definitely insufficient,
but for Statement 2, it says (i1/i2) > (r1/r2), now this expression also means that (i1/i2) < (r2/r1), which is our desired result. Therefore,sufficient.

Here the assumption is that all the numbers are positive, which is valid as we are comparing investments and interest rates.

Please let me know if I have missed anything in the explanation above.

Thanks!

Please check here: samantha-invests-i1-dollars-in-bond-x-which-pays-r1-102084.html
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06 Sep 2017, 16:17
I have never heard of multiplying an inequality by another inequality ; I didn't think it was even possible.

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Re: S99-05   [#permalink] 06 Sep 2017, 16:17
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# S99-05

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