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# Amy's sells kale at x dollar per pound for the first 20 pounds and .8x

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Math Expert
Joined: 02 Sep 2009
Posts: 64070
Amy's sells kale at x dollar per pound for the first 20 pounds and .8x  [#permalink]

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13 Feb 2020, 06:46
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Difficulty:

75% (hard)

Question Stats:

62% (02:12) correct 38% (03:16) wrong based on 52 sessions

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Amy's sells kale at x dollar per pound for the first 20 pounds and 0.8x for every subsequent pound. Lucia's price is x per pound for the first 14 pounds and 0.9x for subsequent pounds. What is the minimum number of pounds over 15 for which Amy's becomes an equal or better deal?

A. 24
B. 25
C. 26
D. 27
E. 28

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Re: Amy's sells kale at x dollar per pound for the first 20 pounds and .8x  [#permalink]

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13 Feb 2020, 07:01
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For the six pounds between the 14th and 20th pound, you save 0.1x dollars by buying from Lucia. But after the 20th pound, you save 0.1x dollars by buying from Amy. So for the prices to be equal buying from either person, you'd need to buy six pounds beyond the first 20, and the answer is 26 pounds.

The wording of the question is confusing though -- when it asks for "the minimum number of pounds over 15", I think you could legitimately interpret the question to be asking how many pounds we would need to add to a 15 pound purchase to arrive at equal prices (in which case the answer would be 11). That's not the intended meaning though.
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Amy's sells kale at x dollar per pound for the first 20 pounds and .8x  [#permalink]

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13 Feb 2020, 08:42
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Amy’s price — 20x+ 0.8x*a
Lucy’s price — 14x + 0.9x*b
—> 20x+ 0.8x*a ≥ 14x+ 0.9x*b

Let’s find the minimum number of (20 + a ) :

20x + 0.8x*a = 14x + 0.9x*b
6x = x( 0.9b —0.8a)
—> $$a = \frac{(9b —60)}{8}$$— must be integer

If $$b= 12$$ , then $$a = \frac{(108–60)}{8 }= 6$$
—> $$20 + a = 26$$

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Re: Amy's sells kale at x dollar per pound for the first 20 pounds and .8x  [#permalink]

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14 Feb 2020, 10:47
1
Bunuel wrote:
Amy's sells kale at x dollar per pound for the first 20 pounds and 0.8x for every subsequent pound. Lucia's price is x per pound for the first 14 pounds and 0.9x for subsequent pounds. What is the minimum number of pounds over 15 for which Amy's becomes an equal or better deal?

A. 24
B. 25
C. 26
D. 27
E. 28

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Solution

• Let us assume that the number of pounds over 15 is n.
o So, the total number of pounds $$= 15+ n$$
• So, the total price at which Amy's sells (15 +n) pounds of kale $$= 20*x + (15+n-20) *0.8*x …. Eq.(i)$$
• Now, the total price at which Lucia's sells (15 +n) pounds of kale $$= 14 *x + (15+n -14) *0.9*x …………. Eq. (ii)$$
• From Eq.(i) and Eq.(ii) we can say that deal offered by Amy’s to become a equal or better than Lucia’s:
o $$20x + (15+n-20) * 0.8x ≤ 14 x + (15+n -14) *0.9x$$
o Or, $$20x + (n-5) *0.8x ≤ 14x + (1+n) *0.9x$$
o Or, $$6x + 0.8nx – 4x ≤ 0.9nx+ 0.9x$$
o Or, $$1.1x ≤ 0.1nx$$
o Or, $$11 ≤ n$$
So, for minimum 11 pounds over 15, Amy’s will be a better deal.
Hence total number of pounds = 15 +11 = 26
Thus, the correct answer is Option C.
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Amy's sells kale at x dollar per pound for the first 20 pounds and .8x  [#permalink]

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16 Feb 2020, 00:04
The question is confusing on two counts:
The first, as explained by IanStewart, is that it can be logically interpreted to mean the solver has to find the minimum number of pounds that a buyer has to buy IN EXCESS OF 15 POUNDS so that he gets an equal or better deal from Amy than from Lucia.
The second is the condition in the stem which states that "Amy's becomes an equal OR better deal". According to this, the correct answer could be either C (26 lbs) (the buyer spends the same amount of money whether he buys from Amy or Lucia) or D (27 lbs) ( where he spends less money when he buys from Amy).

Corrigendum: Sorry, my second point is incorrect - I overlooked the "minimum number of pounds" stipulation. First point stands though.
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Re: Amy's sells kale at x dollar per pound for the first 20 pounds and .8x  [#permalink]

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21 Feb 2020, 08:04
1
1
Bunuel wrote:
Amy's sells kale at x dollar per pound for the first 20 pounds and 0.8x for every subsequent pound. Lucia's price is x per pound for the first 14 pounds and 0.9x for subsequent pounds. What is the minimum number of pounds over 15 for which Amy's becomes an equal or better deal?

A. 24
B. 25
C. 26
D. 27
E. 28

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We can create the inequality:

20x + 0.8x(T - 20) ≤ 14x + 0.9x(T - 14)

20 + 0.8(T - 20) ≤ 14 + 0.9(T - 14)

6 + 0.8T - 16 ≤ 0.9T - 12.6

2.6 ≤ 0.1T

26 ≤ T

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Re: Amy's sells kale at x dollar per pound for the first 20 pounds and .8x   [#permalink] 21 Feb 2020, 08:04