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# During a sale, a clothing store sold each shirt at a price

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During a sale, a clothing store sold each shirt at a price [#permalink]

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30 Aug 2009, 07:01
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During a sale, a clothing store sold each shirt at a price of $15 and each sweater at a price of$25. Did the store sell more sweaters than shirts during the sale?
(1) The average (arithmetic mean) of the prices of all the shirts and sweaters that the store sold during the sale was $21. (2) The total of the prices of all the shirts and sweaters that the store sold during the sale was$420.

In the xy-plane, does the line with equation y=3x+2 contain the point(r,s)?
(1) (3r+2-s)(4r+9-s)=0
(2) (4r-6-s)(3r+2-s)=0

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Manager
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30 Aug 2009, 08:05
Let number of shirts he bought = Sh
and,number of sweaters he bought = Sw

total price of all shirts and sweaters = Sh*15 + Sw*25

1.) Average price = 21

=> (15*Sh + 25*Sw)/(Sh + Sw) = 21 ----[ Av = total price/total quantity]

=> Sw = 3Sh/2

=> Sh is less than Sw( 3/2 is greater than 1)

Hence, sufficient..

2.) total price = 420

=> Sh*15 + Sw*25 = 420

=> 3Sh + 5Sw = 84

This is possible when

Sh = 18 , Sw = 6

and

Sh = 8 , Sw = 12

In one case, Sh is greater than Sw and vice versa..

Hence, insufficient..

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Manager
Joined: 25 Aug 2009
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30 Aug 2009, 08:11
2) In the xy plane, does the line with the equation y = 3x + 2 contain the point (r,s)?
1) (3r + 2 - s)(4r + 9 - s) = 0
2) (4r + 6 - s)(3r + 2 - s) = 0

The line y = 3x + 2 contains the point means it passes through (r,s)

=> it should satisfy the equation.

=> s = 3r +2
or, 3r + 2 - s = 0 --- equation 1st...

1.) (3r + 2 - s)(4r + 9 - s) = 0

It is possible when
(3r + 2 - s) = 0 or (4r + 9 - s) = 0
if (3r + 2 - s) = 0, then , (r,s) satisfies the equation ( from 1st)
but if, (4r + 9 - s) = 0 then, we can not be sure that (r,s) satisfies the equation ( from 1st)
Hence, insufficient..

2.) Same as 1

Combining both,

both are possible only when
(3r + 2 - s) = 0 ..Hence, (r,s) lies on the line..

Hence, sufficient.. C is the answer..

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Manager
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30 Aug 2009, 09:08
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First question.

Without any equations and thinking logically, it can be shown that the stmt 1 is sufficient.
The average price is 21. Price per shirt =15, price per sweater = 25. If the equal quantity of shirts and sweater was sold, then the average would be 20. Since average is greater than 20, the store sold more items with higher price - sweaters.

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Manager
Joined: 24 Aug 2009
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22 Nov 2009, 18:54
gmate2010 wrote:
2) In the xy plane, does the line with the equation y = 3x + 2 contain the point (r,s)?
1) (3r + 2 - s)(4r + 9 - s) = 0
2) (4r + 6 - s)(3r + 2 - s) = 0

The line y = 3x + 2 contains the point means it passes through (r,s)

=> it should satisfy the equation.

=> s = 3r +2
or, 3r + 2 - s = 0 --- equation 1st...

1.) (3r + 2 - s)(4r + 9 - s) = 0

It is possible when
(3r + 2 - s) = 0 or (4r + 9 - s) = 0
if (3r + 2 - s) = 0, then , (r,s) satisfies the equation ( from 1st)
but if, (4r + 9 - s) = 0 then, we can not be sure that (r,s) satisfies the equation ( from 1st)
Hence, insufficient..

2.) Same as 1

Combining both,

both are possible only when
(3r + 2 - s) = 0 ..Hence, (r,s) lies on the line..

Hence, sufficient.. C is the answer..

I have the question

the question stem itself gives us the info that 3r+2-s = 0

so from statement 1, applying question stem, the equation statisfies irrespective of of what (4r + 9 - s) is.

same for statement 2 as well, so i chose D

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Manager
Joined: 24 Aug 2009
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14 Dec 2009, 20:17
ISBtarget wrote:
gmate2010 wrote:
2) In the xy plane, does the line with the equation y = 3x + 2 contain the point (r,s)?
1) (3r + 2 - s)(4r + 9 - s) = 0
2) (4r + 6 - s)(3r + 2 - s) = 0

The line y = 3x + 2 contains the point means it passes through (r,s)

=> it should satisfy the equation.

=> s = 3r +2
or, 3r + 2 - s = 0 --- equation 1st...

1.) (3r + 2 - s)(4r + 9 - s) = 0

It is possible when
(3r + 2 - s) = 0 or (4r + 9 - s) = 0
if (3r + 2 - s) = 0, then , (r,s) satisfies the equation ( from 1st)
but if, (4r + 9 - s) = 0 then, we can not be sure that (r,s) satisfies the equation ( from 1st)
Hence, insufficient..

2.) Same as 1

Combining both,

both are possible only when
(3r + 2 - s) = 0 ..Hence, (r,s) lies on the line..

Hence, sufficient.. C is the answer..

I have the question

the question stem itself gives us the info that 3r+2-s = 0

so from statement 1, applying question stem, the equation statisfies irrespective of of what (4r + 9 - s) is.

same for statement 2 as well, so i chose D

I think from statement one 3r+2-s may be 0 or second part may be zero...similarly for statement 2, combining both will prove that 3r+2-s is zero

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Manager
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28 Mar 2012, 07:53
from statement 2,

shirts x
sweaters y

15x +20y = 420

Can we say that since 60 is a common multiple between the 15 and 20, there will be more than one answer that can satisfy the equation?
If there is a common multiple for

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Math Expert
Joined: 02 Sep 2009
Posts: 41891

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28 Mar 2012, 09:04
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dchow23 wrote:
from statement 2,

shirts x
sweaters y

15x +20y = 420

Can we say that since 60 is a common multiple between the 15 and 20, there will be more than one answer that can satisfy the equation?
If there is a common multiple for

Not sure that I understand your question correctly but generally for such kind of question it's always better to check whether an equation whose solution must be integers only has one or more set of variables satisfying it.

For more on this type of questions check:
eunice-sold-several-cakes-if-each-cake-sold-for-either-109602.html
martha-bought-several-pencils-if-each-pencil-was-either-a-100204.html
a-rental-car-agency-purchases-fleet-vehicles-in-two-sizes-a-105682.html
joe-bought-only-twenty-cent-stamps-and-thirty-cent-stamps-106212.html
a-certain-fruit-stand-sold-apples-for-0-70-each-and-bananas-101966.html
joanna-bought-only-0-15-stamps-and-0-29-stamps-how-many-101743.html

As for the question.

During a sale, a clothing store sold each shirt at a price of $15 and each sweater at a price of$25. Did the store sell more sweaters than shirts during the sale?

(1) The average (arithmetic mean) of the prices of all the shirts and sweaters that the store sold during the sale was $21 --> since the average sale price of$21 is closer to $25 than to$15 then store sold more sweaters than shirts (sweaters have more weight in the weighted average and thus pulled it closer to $21). Sufficient. (2) The total of the prices of all the shirts and sweaters that the store sold during the sale was$420 --> 15x + 25y =420 --> 3x+5y=84. If x=8 and y=12 then x<y but if x=13 and y=5 then x>y. Not sufficient.

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Math Expert
Joined: 02 Sep 2009
Posts: 41891

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Re: During a sale, a clothing store sold each shirt at a price [#permalink]

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28 Mar 2012, 09:04
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In the XY plane, does the line with equation y=3x+2 contain the point (r,s)?

Line with equation $$y=3x+2$$ contains the point $$(r,s)$$ means that when substituting $$r$$ ans $$s$$ in line equation: $$s=3r+2$$ (or $$3r+2-s=0$$) holds true.

So basically we are asked to determine whether $$3r+2-s=0$$ is true or not.

(1) $$(3r+2-s)(4r+9-s)=0$$ --> either $$3r+2-s=0$$ OR $$4r+9-s=0$$ OR both. Not sufficient.

(2) $$(4r-6-s)(3r+2-s)=0$$ --> either $$3r+2-s=0$$ OR $$4r-6-s=0$$ OR both. Not sufficient.

(1)+(2) Both $$4r+9-s=0$$ and $$4r-6-s=0$$ can not be true (simultaneously), as $$4r-s$$ can not equal to both -9 and 6, hence $$3r+2-s=0$$ must be true. Sufficient.

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Re: During a sale, a clothing store sold each shirt at a price [#permalink]

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09 Sep 2013, 07:58
Dear Bunuel,

For following statement, you provided two different sets of values for x,y. I tried to identify these values by simply plugging multiples of one variable, but this approach is taking more than 2min, is there any method or trick related to lcm or any thing which we can use to quickly identify all possible values for the equation? I am really struggling with this hit and trial approach, what we should look for while selecting these values.

(2) The total of the prices of all the shirts and sweaters that the store sold during the sale was \$420 --> 15x + 25y =420 --> 3x+5y=84. If x=8 and y=12 then x<y but if x=13 and y=5 then x>y. Not sufficient.

Thanks
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Re: During a sale, a clothing store sold each shirt at a price [#permalink]

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02 Jul 2017, 17:19
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