GMAT Prep
At a certain bookstore, each notepad costs x dollars and each marker costs y dollars. If $10 is enough to buy 5 notepads and 3 markers, is $10 enough to buy 4 notepads and 4 markers instead?
1) Each notepad cost less than $1.
2) $10 is enough to buy 11 notepads.
OA E.
A certain bookstore, each notepad costs x dollars and each
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- ceilidh.erickson
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Translate the given information:
$10 is enough to buy 5 notepads and 3 markers -->
$$5x+3y\le10$$
Translate the question:
is $10 enough to buy 4 notepads and 4 markers instead? -->
$$4x+4y\le10\ ?$$
Simplify:
$$2x+2y\le5\ ?$$
1) Each notepad cost less than $1.
If x < 1, then 5x < 5. Let's test cases to see whether this is sufficient:
Case 1:
x = $0.90
y = $0.30
5x + 3y = $4.50 + $0.90 = $5.40 --> less than $10, holds statement true
4x + 4y = $3.60 + $1.20 = $4.80 --> less than $10, answer to the question is YES.
Case 1:
x = $0.10
y = $3.00
5x + 3y = $0.50 + $9.00 = $9.50 --> less than $10, holds statement true
4x + 4y = $0.40 + $12.00 = $12.40 --> NOT less than $10, so the answer to the question is NO.
Insufficient
2) $10 is enough to buy 11 notepads.
$$11x\ \le10$$
$$x\ \le0.90$$
Both of the cases we tested for statement 1 are consistent with this information, so this must also be insufficient. And putting the statements together won't help, since both cases will still be relevant (i.e. we can find a "yes" answer or a "no" answer to the question using both statements).
The answer is E.
$10 is enough to buy 5 notepads and 3 markers -->
$$5x+3y\le10$$
Translate the question:
is $10 enough to buy 4 notepads and 4 markers instead? -->
$$4x+4y\le10\ ?$$
Simplify:
$$2x+2y\le5\ ?$$
1) Each notepad cost less than $1.
If x < 1, then 5x < 5. Let's test cases to see whether this is sufficient:
Case 1:
x = $0.90
y = $0.30
5x + 3y = $4.50 + $0.90 = $5.40 --> less than $10, holds statement true
4x + 4y = $3.60 + $1.20 = $4.80 --> less than $10, answer to the question is YES.
Case 1:
x = $0.10
y = $3.00
5x + 3y = $0.50 + $9.00 = $9.50 --> less than $10, holds statement true
4x + 4y = $0.40 + $12.00 = $12.40 --> NOT less than $10, so the answer to the question is NO.
Insufficient
2) $10 is enough to buy 11 notepads.
$$11x\ \le10$$
$$x\ \le0.90$$
Both of the cases we tested for statement 1 are consistent with this information, so this must also be insufficient. And putting the statements together won't help, since both cases will still be relevant (i.e. we can find a "yes" answer or a "no" answer to the question using both statements).
The answer is E.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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- Jay@ManhattanReview
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Global Stats
We have 5x + 3y ≤ 10.AAPL wrote:GMAT Prep
At a certain bookstore, each notepad costs x dollars and each marker costs y dollars. If $10 is enough to buy 5 notepads and 3 markers, is $10 enough to buy 4 notepads and 4 markers instead?
1) Each notepad cost less than $1.
2) $10 is enough to buy 11 notepads.
OA E.
We have to determine whether 4x + 4y ≤ 10 or x + y ≤ 2.5
Question: Is x + y ≤ 2.5?
When we sacrifice x for y in the inequality 5x + 3y ≤ 10, we get 4x + 4y ≤ 10. Thus, for the inequality 4x + 4y ≤ 10 to be true, we must have x ≥ y; otherwise, we cannot determine conclusively whether 4x + 4y ≤ 10 or x + y ≤ 2.5.
Looking at the given inequality 5x + 3y ≤ 10, we see that if x = y, then 8x ≤ 10 => x ≤ 1.25 and y ≤ 1.25. So, here $1.25 is the average price of the items.
Let's take each statement one by one.
1) Each notepad cost less than $1.
=> x < 1. Since the average price per item = 1.25 (considering the prices of both the items equal), and x < 1, we must have y > 1.25; this means that y > x. We already concluded that we can have a unique answer only if x ≥ y. Since x < y, we do not get a unique answer. Insufficient.
2) $10 is enough to buy 11 notepads.
> x = 10/11 < 1. It calls for the same analysis as we did in Statement 1. Insufficient.
(1) and (2) together
Both the statements lead to the same conclusion. Insufficient.
The correct answer: E
Hope this helps!
-Jay
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- fskilnik@GMATH
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Global Stats
Do not forget that there are noninteger dollar values, but all values in cents are integers!AAPL wrote:GMAT Prep
At a certain bookstore, each notepad costs x dollars and each marker costs y dollars. If $10 is enough to buy 5 notepads and 3 markers, is $10 enough to buy 4 notepads and 4 markers instead?
1) Each notepad cost less than $1.
2) $10 is enough to buy 11 notepads.
$$\left\{ \matrix{
\,{\rm{notepads}}\,,\,\,\$ \,n\,\,{\rm{cents}}\,\,{\rm{each}}\,\,\,\left( {n = 100x} \right) \hfill \cr
\,{\rm{markers}}\,,\,\,\$ \,m\,\,{\rm{cents}}\,\,{\rm{each}}\,\,\,\left( {m = 100y} \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,5n + 3m \le 1000\,\,\,\left[ {{\rm{cents}}} \right]\,\,\,\,\left( * \right)$$
$$\left( 1 \right)\,\,n < 100\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {n,m} \right) = \left( {50\,,\,200} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\left[ {\,250 + 600 < 1000\,\,\left( * \right)\,} \right]\,\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {n,m} \right)\, = \left( {50,210} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\,\left[ {\,250 + 630 < 1000\,\,\left( * \right)\,} \right]\,\,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,11n \le 1000\,\,\,\,\mathop \Leftrightarrow \limits^{n\,\,{\mathop{\rm int}} } \,\,\,\,n \le 90\,\,\,\left\{ \matrix{
\,\left( {{\rm{Re}}} \right){\rm{Take}}\,\,\left( {n,m} \right) = \left( {50,200} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {n,m} \right)\, = \left( {50,210} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.\,\,\,\,\,\,\,\,$$
$$ \Rightarrow \,\,\,\,\left( {\rm{E}} \right)$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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