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# Mark and Ann together were allocated n boxes of cookies to sell for a

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Math Expert
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Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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15 Jun 2016, 01:30
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77% (01:45) correct 23% (01:53) wrong based on 1341 sessions

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Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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29 Nov 2016, 15:52
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Bunuel wrote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

We are given that Mark sold 10 boxes less than n and Ann sold 2 boxes less than n, and that together they have sold less than n boxes. We can create the following inequality:

n - 10 + n - 2 < n

2n - 12 < n

n < 12

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Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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15 Jun 2016, 05:23
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Bunuel wrote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

Mary sold n-10 boxes.
Ann sold n-2 boxes.

They have each sold at least one box.

so n-10 is more than or equal to 1. Also n-2 is more than or equal to 1
n>= 11 [2nd condition only gives us n>=3. We already have this info when we say n>=11]

Now

Total boxes are n only.

so (n-10+n-2) < n

2n-12 < n
n <12

Only value. n = 11

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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10 Oct 2016, 04:49
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Bunuel wrote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

Quote:
Mark and Ann together were allocated n boxes of cookies

M + A = n
Quote:
Mark sold 10 boxes less than n

M = n - 10
Quote:
Ann sold 2 boxes less than n.

A = n - 2
Quote:
Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n

M + A = {(n - 10) + (n - 2) } < n
Or, (2n - 12) < n
Or, n < 11

So, Answer will be (A) 11

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Joined: 29 Aug 2016
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Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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28 Nov 2016, 05:23
1
Sorry, I ment the part where

"M + A = {(n - 10) + (n - 2) } < n
Or, (2n - 12) < n
Or, n < 11"

I understand how 2n-12<n is received, but n<11? Is it a typo or am I misunderstanding the concept?

As I see it, it should be 2n-12 < n <=> 2n - n < 12 <=> n < 12. Got a little bit confused
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Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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28 Nov 2016, 06:24
1
MrBarksdale wrote:

Sorry, I ment the part where

"M + A = {(n - 10) + (n - 2) } < n
Or, (2n - 12) < n
Or, n < 11"

I understand how 2n-12<n is received, but n<11? Is it a typo or am I misunderstanding the concept?

As I see it, it should be 2n-12 < n <=> 2n - n < 12 <=> n < 12. Got a little bit confused

Its a typo. n < 12 is right.
only option A suffices the condition
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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18 Oct 2017, 05:22
1
HHXX wrote:
Oh my god... it seems that only me don't understand why mark sold n-10 and ann sold n-2... the title says that mark sold 10 boxes less than n. what does that mean? any expert can told me...

Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

Step-by-step:

1. Mark sold 10 boxes less than n --> Mark sold n - 10 boxes;

2. Ann sold 2 boxes less than n --> Ann sold n - 2 boxes;

3. Mark sold at least one box of cookies: $$n - 10 \geq 1$$ -->$$n \geq 11$$;

4. Together they have sold less than n boxes: $$(n - 10) + (n - 2) < n$$ -->$$n < 12$$.

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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18 Mar 2018, 06:19
1
niks18 Hatakekakashi
amanvermagmat chetan2u

Quote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

Can you validate my understanding?

Quote:
n - 10 + n - 2 < n

2n - 12 < n

n < 12

I know n is a positive integer and hence I can subtract n from both sides of inequality without disturbing the inequality.
and also add +12 on both sides giving n < 12. Can I reduce steps in such a manner?

This is correct but there is a flaw in your reasoning. We are concerned about the sign of a variable when multiplying/dividing an inequality by it. However we can safely add/subtract a variable from both sides of an inequality regardless of its sign.
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Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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18 Mar 2018, 06:37
1
Bunuel wrote:

This is correct but there is a flaw in your reasoning. We are concerned about the sign of a variable when multiplying/dividing an inequality by it. However we can safely add/subtract a variable from both sides of an inequality regardless of its sign.

Quote:
you are correct but the same procedure should be done even when n is negative..

change in INEQUALITY sign is when you multiply both sides by '-'..

Hi Bunuel, I thought both the quoted portion meant same but the member asking question must have felt otherwise and seems did not understand my reply..
I am off to taking VERBAL coaching classes.
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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15 Jun 2016, 04:59
if n =11
mark sold 1 box
and Ann sold 9 boxes
total 10 < 11
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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05 Apr 2017, 21:49
I got it wrong. I think that the catch in this question is to realize that it's an inequality problem.

If you make an equation out of it, you get 12. Else, if you appropriately decode the question, you set up the inequality and you get that $$n<12$$.

$$n-10+n-2=n\\ n-10+n-2<n$$
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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09 Sep 2017, 10:58
Experts please correct me if I have done this wrongly:

Both have sold at least 1 box but together they have sold less than n boxes.

2<=(n-10)+(n-2)<n
2<=2n-12<n
1<=n-6<n/2
7<=n<(n+12)/2

I thought that answer would come from: n< (n+12)/2
From the first part of the inequality, we know that n<=7

2n<n+12
n<12

Only one option fits in. (11)

My query is can we solve only one part of the inequality like I did above or it will impact the answer? Bunuel it would be great if you could give your view about the solution above.
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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18 Oct 2017, 04:59
Oh my god... it seems that only me don't understand why mark sold n-10 and ann sold n-2... the title says that mark sold 10 boxes less than n. what does that mean? any expert can told me...
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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17 Jan 2018, 13:13
Hi All,

This question can be solved by TESTing THE ANSWERS. There's a great 'logic shortcut' built into this prompt - but you have to pay careful attention to how the question is specifically phrased to catch the shortcut.

We're told that Mark sold 10 boxes LESS than N and Ann sold 2 boxes LESS than N. The prompt also states that the TOTAL of those two numbers is also LESS than N. Logically-speaking, since that pair of numbers is dependent on the value of N, the way to make the sum of those numbers less than N is to make those two numbers as SMALL as possible. Since we're given 5 possible values for N, we should start with the smallest value and see what happens...

IF.... N = 11 boxes
Mark = 11 - 10 = 1 box sold
Ann = 11 - 2 = 9 boxes sold
Total = 1 + 9 = 10 boxes sold
This matches what we were told, so this MUST be the answer.

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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18 Mar 2018, 06:13
niks18 Hatakekakashi
amanvermagmat chetan2u

Quote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

Can you validate my understanding?

Quote:
n - 10 + n - 2 < n

2n - 12 < n

n < 12

I know n is a positive integer and hence I can subtract n from both sides of inequality without disturbing the inequality.
and also add +12 on both sides giving n < 12. Can I reduce steps in such a manner?
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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18 Mar 2018, 06:19
niks18 Hatakekakashi
amanvermagmat chetan2u

Quote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

Can you validate my understanding?

Quote:
n - 10 + n - 2 < n

2n - 12 < n

n < 12

I know n is a positive integer and hence I can subtract n from both sides of inequality without disturbing the inequality.
and also add +12 on both sides giving n < 12. Can I reduce steps in such a manner?

Hi...

you are correct but the same procedure should be done even when n is negative..

change in INEQUALITY sign is when you multiply both sides by '-'..
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Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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18 Mar 2018, 11:59
Bunuel wrote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

total sold boxes=2n-12
let unsold boxes=x
n=2n-12+x
→x=12-n
x must=1 because if x>1, then n<11
substituting, 1=12-n
n=11
A
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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17 Oct 2018, 04:54
n − 10 ≥ 1 and n − 2 ≥ 1, which implies that n ≥ 11.

(n − 10) + (n − 2) < n, which implies that n < 12.

Therefore, n is an integer such that n ≥ 11 and n < 12, which implies that n = 11.
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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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02 Jan 2019, 13:11
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Bunuel wrote:
Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

A) 11
B) 12
C) 13
D) 14
E) 15

Mark sold 10 boxes less than n and Ann sold 2 boxes less than n
We can write:
n - 10 = number of boxes that Mark sold
n - 2 = number of boxes that Ann sold

Together they (Mark and Ann) have sold less than n boxes
In other words: (# boxes Mark sold) + (# boxes Ann sold) < n
Rewrite as: (n - 10) + (n - 2) < n
Simplify: 2n - 12 < n
Add 12 to both sides: 2n < n + 12
Subtract n from both sides: n < 12

What is the value of n?
We know that n < 12
Check the answer choices . . . only answer choice A (11) is less than 12

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Re: Mark and Ann together were allocated n boxes of cookies to sell for a  [#permalink]

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17 Apr 2019, 19:52
This question doesn't make sense to me...

From the question I arrived at 2n<n+12, however, this was my approach:

2<M+A<n
2<2n-12<n
14<2n<n+12
7<n<(n+12)/2

Even though the answer n=11 fits this criteria, you cannot have half a box of cookies from (n+12)/2

I don't know anymore
Re: Mark and Ann together were allocated n boxes of cookies to sell for a   [#permalink] 17 Apr 2019, 19:52
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