Robin split a total of $24,000 between 2 investments, X and

[BTGmoderatorDC]

Robin split a total of $24,000 between 2 investments, X and Y. If investment Y earns 7% simple annual interest, how much of that total did Robin put into investment Y?

(1) Each investment earns the same dollar amount of the interest annually.
(2) Investment X earns 5 percent simple annual interest.

OA C

Source: GMAT Prep

[fskilnik@GMATH]

Robin split a total of $24,000 between 2 investments, X and Y. If investment Y earns 7% simple annual interest, how much of that total did Robin put into investment Y?

(1) Each investment earns the same dollar amount of the interest annually.
(2) Investment X earns 5 percent simple annual interest.

Source: GMAT Prep

\[\left. \begin{gathered}
X:\,\,\,24 \cdot k\,\,\,\,\,\,\,;\,\,\,\,\,\,\frac{i}{{100}}\left( {24k} \right)\,\,\,\,\,{\text{annual}}\,\,{\text{simple}}\,\,{\text{interest}}\,\,\left( * \right) \hfill \\
Y:\,\,\,24 \cdot \left( {1 - k} \right)\,\,\,\,\,;\,\,\,\,\,\,\frac{7}{{100}}\left[ {24\left( {1 - k} \right)} \right]\,\,\,\,{\text{annual}}\,\,{\text{simple}}\,\,{\text{interest}}\,\,\, \hfill \\
\end{gathered} \right\}\,\,\,\,\,\,\,\,\left[ {{\text{unit}}:\,\,\,\$ \,\,{{10}^3}} \right]\]
\[\left( * \right)\,\,{\text{just}}\,\,{\text{a}}\,\,{\text{initial}}\,\,{\text{assumption}}\,\]

\[? = 24\left( {1 - k} \right)\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,\,\boxed{? = k}\,\,\,\,\,\,\left( {0 < k < 1} \right)\,\]

\[\left( 1 \right)\,\,\,\frac{i}{{100}}\left( {24k} \right) = \frac{7}{{100}}\left[ {24\left( {1 - k} \right)} \right]\,\,\,\,\,\, \Rightarrow \,\,\,\,\,ik = 7\left( {1 - k} \right)\,\,\,\,\,\left( {**} \right)\]
\[\left\{ \begin{gathered}
\,{\text{Take}}\,\,i = 7\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,? = k = 0.5\,\,\,\, \hfill \\
\,{\text{Take}}\,\,i = 0.0001\,\,\,\left( { \cong 0} \right)\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,? = k \cong 1\,\,\,\, \hfill \\
\end{gathered} \right.\]

$$\left( 2 \right)\,\,i = 5\,\,\,\left( {{\rm{trivial}}\,\,{\rm{bifurcation}}} \right)\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,k = 0.5\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,k = 0.6\,\,\,\, \hfill \cr} \right.\,\,$$

$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{
\,\,\left( {**} \right)\,\,\,ik = 7\left( {1 - k} \right)\,\,\,\, \hfill \cr
\,\,\,i = 5 \hfill \cr} \right.\,\,\, \Rightarrow \,\,\,\,k\,\,{\rm{unique}}\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.\,\,\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[Jay@ManhattanReview]

Robin split a total of $24,000 between 2 investments, X and Y. If investment Y earns 7% simple annual interest, how much of that total did Robin put into investment Y?

(1) Each investment earns the same dollar amount of the interest annually.
(2) Investment X earns 5 percent simple annual interest.

OA C

Source: GMAT Prep

Given:

* X + Y = 24,000;
*Investment Y earns 7% simple annual interest;

Question: Value of Y?

Let's take each statement one by one.

(1) Each investment earns the same dollar amount of the interest annually.

Say Investment Y earns n% simple annual interest

=> X*n/100 = Y*7/100

n(24000 - Y) = 7Y

Since the value of n is not known, we can't get the value of Y. insufficient.

(2) Investment X earns 5 percent simple annual interest.

Certainly insufficient. Insufficient.

(1) and (2) together

Plugging-in the value of n = 5 in n(24000 - Y) = 7Y, we can get the unique value of Y. Sufficient.

Since we are sure that 5(24000 - Y) = 7Y is a linear equation, not a quadratic one, we are sure that we get a unique value.

The correct answer: C

Hope this helps!

-Jay
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[GMATGuruNY]

Robin split a total of $24,000 between 2 investments, X and Y. If investment Y earns 7% simple annual interest, how much of that total did Robin put into investment Y?

(1) Each investment earns the same dollar amount of the interest annually.
(2) Investment X earns 5 percent simple annual interest.

Statement 1:
Test one case that also satisfies Statement 2.
Case 1: X = 5% and Y = 7%
For 5% interest and 7% percent interest to yield the same dollar amount,
$7 must be invested at 5% interest for every $5 invested at 7% interest:
X --> $7(0.05) = $0.35.
Y --> $5(0.07) = $0.35.
Thus:
X:Y = $7 : $5, implying that 5/12 of the $24,000 must be invested in Y:
(5/12)(24,000) = 10,000.

Test one case that does NOT also satisfy Statement 2.
Case 2: X = 3% and Y = 7%
For 3% interest and 7% percent interest to yield the same dollar amount,
$7 must be invested at 3% interest for every $3 invested at 7% interest:
X --> $7(0.03) = $0.21.
Y --> $3(0.07) = $0.21.
Thus:
X:Y = $7 : $3, implying that 3/10 of the $24,000 must be invested in Y:
(3/10)(24,000) = 7200.

Since the amount invested in Y can be different values, INSUFFICIENT.

Statement 2:
Here, the amount invested in Y can be any monetary value between $0 and $24,000.
INSUFFICIENT.

Statements combined:
Only Case 1 satisfies both statements.
In Case 1, the amount invested in Y = 10,000.
SUFFICIENT.

The correct answer is C.