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At a certain bookstore, each notepad costs x dollars and

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At a certain bookstore, each notepad costs x dollars and  [#permalink]

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New post 01 Nov 2010, 21:19
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At a certain bookstore, each notepad costs x dollars and each markers 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

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Re: Notepads and markers  [#permalink]

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New post 01 Nov 2010, 21:36
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metallicafan wrote:
At a certain bookstore, each notepad costs x dollars and each markers 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


Given: \(5x+3y\leq{10}\). Question: is \(4x+4y\leq{10}\)?

Note that both statements are telling us basically the same thing:
(1) \(x<1\);
(2) \(11x<10\) --> \(x<\frac{10}{11}\approx{0.9}\);

Now, if both notepad and marker cost very cheap then the answer would be YES but if notepad costs $0.1 and marker costs $3 then the answer would be NO. Not sufficient.

Answer: E.
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Re: Notepads and markers  [#permalink]

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New post 02 Nov 2010, 17:34
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metallicafan wrote:
At a certain bookstore, each notepad costs x dollars and each markers 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


Given: 5x + 3y <= 10 .... (I)
Question: Is 4x + 4y <=10 ?....(II)

Statement (1) x < 1.
If x is a little less than 1 but almost 1, then 5x is little less than 5. Putting in (I) above, we get that y is less than \(\frac{5}{3}\).
Now we need to put these values in equation (II) and check.
4x will be a little less than 4 and 4y will be less than 4x\(\frac{5}{3}\) i.e. 6.66. Together, 4x + 4y will be less than 10.66. It may be less than 10 or a little more than 10, hence this is not sufficient.

Statement (2) x < \(\frac{10}{11}\)
If x is a little less than 10/11 but almost 10/11, then 5x is little less than 50/11. Putting in (I) above, we get that y is less than \(\frac{20}{11}\).
Now we need to put these values in equation (II) and check.
4x will be a little less than 40/11 and 4y will be less than 4x\(\frac{20}{11}\) i.e. \(\frac{80}{11}\). Together, 4x + 4y will be less than 120/11 i.e. less than 10.9. It may be less than 10 or more, we do not know, hence this is not sufficient.

Note: You didn't actually need to calculate the data in the second statement. This is because, if we compare 5x + 3y with 4x + 4y, the x term has reduced but the y term has increased. If x is as great as possible, it will keep y small and the extra y in second statement may not matter. We tried with x a little less than 1 in first statement. In second statement, x is smaller than 10/11. Hence, if first statement was not sufficient, no way could the second statement be sufficient.
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Re: Notepads and markers  [#permalink]

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New post 01 Feb 2011, 22:35
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metallicafan wrote:
At a certain bookstore, each notepad costs x dollars and each markers 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


Let me put it in this way:
given :
\(5x+3y <= 10\)
asking:
is \(4x+4y <= 10\)

NOTE: in both the above cases we buy total of \(8\) items. All it is asking is, with the same amount i.e. \($10\) can i buy \(1\) more\(y\), making \(y\) to \(4\) items from \(3\)items, instead of buying \(1\) \(x\), making \(x\) to \(4\) items from \(5\) items.

Hence, to answer this question, all we need to know is is the price of \(1 y\) less than that of one \(1 x\). ==> \(is Y < X\) If this is the case, then i can safely replace 1 x with 1 y for that/leass than the amount ($10). Can't I?

stmnt1: \(x < 1\). who cares. i need to know \(is Y < X\)..Not suff.
stmnt2: 11x<10. ==> x<1. Same as above...Not suff.
stmnts 1 & 2. Both are bread slices. Can not make the sandwich with out the extra stuff....Not suff.

Answer "E".

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Murali.

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Re: At a certain bookstore, each notepad costs x dollars and  [#permalink]

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New post 06 Oct 2013, 02:08
can someone explain why "If $10 is enough to buy 5 notepads and 3 markers" => the translation can't be 5x+ 3y = 10 ( what's wrong with this)

and why is it => 5x+3y < = 10
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Re: At a certain bookstore, each notepad costs x dollars and  [#permalink]

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New post 06 Oct 2013, 02:20
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fozzzy wrote:
can someone explain why "If $10 is enough to buy 5 notepads and 3 markers" => the translation can't be 5x+ 3y = 10 ( what's wrong with this)

and why is it => 5x+3y < = 10


LET SAY X= 1 Y =1
THEN 5X + 3Y = 8
STILL 10 DOLLAR IS enough TO BUY 5 NOTEPADS AND 3 MARKERS.

so question wants to say that price of 5 notepads and 3 markers is less than or equal to 10 $

hope it is clear.
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Re: At a certain bookstore, each notepad costs x dollars and  [#permalink]

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New post 06 Oct 2013, 19:19
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fozzzy wrote:
can someone explain why "If $10 is enough to buy 5 notepads and 3 markers" => the translation can't be 5x+ 3y = 10 ( what's wrong with this)

and why is it => 5x+3y < = 10


'is enough' means that it suffices. Whether it suffices perfectly or something is leftover is not specified. e.g. it is correct to say $10 is enough to buy the chocolate bar I want. The bar could cost $5 or $8 or $9.2 or $10 etc. I any case, $10 is enough.

Check out a detailed discussion of this question here: http://www.veritasprep.com/blog/2013/09 ... -of-words/
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Re: At a certain bookstore, each notepad costs x dollars and  [#permalink]

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New post 18 Jan 2016, 01:39
I am pretty sure this wasnt intended to be a hard question.
Both statements in a way give the same info. So answer should be either D or E.
Since both x and y have changed in the question (4x+4y≤10?) from the given (5x+3y≤10) the answer has to be E
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At a certain bookstore, each notepad costs x dollars and  [#permalink]

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New post 02 Dec 2018, 12:08
metallicafan wrote:
At a certain bookstore, each notepad costs x dollars and each markers 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

Do not forget that there are noninteger dollar values, but all values in cents are integers!

\(\left\{ \matrix{
\,{\rm{notepads}}\,,\,\,\$ \,n\,\,{\rm{cents}}\,\,{\rm{each}}\,\,\,\left( {n = 100x} \right) \hfill \cr
\,{\rm{markers}}\,,\,\,\$ \,m\,\,{\rm{cents}}\,\,{\rm{each}}\,\,\,\left( {m = 100y} \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,5n + 3m \le 1000\,\,\,\left[ {{\rm{cents}}} \right]\,\,\,\,\left( * \right)\)

\(4n + 4m\,\,\mathop \le \limits^? \,\,1000\,\,\,\left[ {{\rm{cents}}} \right]\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,n + m\,\,\mathop \le \limits^? \,\,250\,\,\,\left[ {{\rm{cents}}} \right]\)

\(\left( 1 \right)\,\,n < 100\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {n,m} \right) = \left( {50\,,\,200} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\left[ {\,250 + 600 < 1000\,\,\left( * \right)\,} \right]\,\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {n,m} \right)\, = \left( {50,210} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\,\left[ {\,250 + 630 < 1000\,\,\left( * \right)\,} \right]\,\,\, \hfill \cr} \right.\)

\(\left( 2 \right)\,\,11n \le 1000\,\,\,\,\mathop \Leftrightarrow \limits^{n\,\,{\mathop{\rm int}} } \,\,\,\,n \le 90\,\,\,\left\{ \matrix{
\,\left( {{\rm{Re}}} \right){\rm{Take}}\,\,\left( {n,m} \right) = \left( {50,200} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {n,m} \right)\, = \left( {50,210} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.\,\,\,\,\,\,\,\,\)

\(\Rightarrow \,\,\,\,\left( {\rm{E}} \right)\)


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

Regards,
Fabio.
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Re: At a certain bookstore, each notepad costs x dollars and  [#permalink]

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New post 03 Dec 2018, 14:49
metallicafan wrote:
At a certain bookstore, each notepad costs x dollars and each markers 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


Here's one way to visualize it. It helps to put yourself into the story when the problem asks about things like shopping and prices! Our intuitions about these topics are often better than our pure math skills.

$10 is enough to buy 5 notepads and 3 markers. Imagine that you have 5 notepads and 3 markers in your shopping cart. But, you'd rather buy 4 notepads and 4 markers. Can you safely 'swap out' a notepad for a marker without going over budget?

For starters, you can definitely do that if a marker is less expensive than a notepad. Since you were under $10 before, you'll still be under $10.

On the other hand, suppose that you're already very close to your budget when you have 5 notepads and 3 markers in your cart. If a marker is more expensive than a notepad, then switching might bring you over budget.

With that in mind, let's look at the statements.

(1) says that a notepad is inexpensive. But a marker could be even cheaper - or it could be much more expensive! Insufficient.

(2) says more or less the same thing. A marker could cost as little as $0.01, or as much as $3 (or more), as long as the notepads were cheap enough. If we switch a notepad for a very cheap marker, we'd stay under budget. If we switch for an expensive one, we could go over.

Since (1) and (2) are redundant, having both of them doesn't help answer the question, either.

The answer is E.
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Re: At a certain bookstore, each notepad costs x dollars and &nbs [#permalink] 03 Dec 2018, 14:49
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