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

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01 Nov 2010, 22:19
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Difficulty:

95% (hard)

Question Stats:

55% (02:34) correct 45% (02:44) wrong based on 468 sessions

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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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01 Nov 2010, 22:36
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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.

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01 Feb 2011, 23:35
5
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 :
$$5x+3y <= 10$$
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.

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. _________________ Karishma Veritas Prep GMAT Instructor Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options > Director Joined: 29 Nov 2012 Posts: 732 Re: At a certain bookstore, each notepad costs x dollars and [#permalink] ### Show Tags 06 Oct 2013, 03:08 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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06 Oct 2013, 03:20
1
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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06 Oct 2013, 20:19
2
1
fozzzy wrote:
(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. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Manhattan Prep Instructor Joined: 04 Dec 2015 Posts: 749 GMAT 1: 790 Q51 V49 GRE 1: Q170 V170 Re: At a certain bookstore, each notepad costs x dollars and [#permalink] ### Show Tags 03 Dec 2018, 15:49 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.

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Re: At a certain bookstore, each notepad costs x dollars and   [#permalink] 03 Dec 2018, 15:49
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