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Word Problem.........700 level

by Mo2men » Wed Dec 21, 2016 5:49 am
Alice has $15, which is enough to buy 11 muffins and 7 brownies, is $45 enough to buy 27 muffins and 27 brownies?
(1) $15 is enough to buy 7 muffins and 11 brownies.
(2) $15 is enough to buy 10 muffins and 8 brownies.
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by GMATGuruNY » Wed Dec 21, 2016 6:49 am
Mo2men wrote:Alice has $15, which is enough to buy 11 muffins and 7 brownies, is $45 enough to buy 27 muffins and 27 brownies?
(1) $15 is enough to buy 7 muffins and 11 brownies.
(2) $15 is enough to buy 10 muffins and 8 brownies.
Test EXTREMES.
Since $15 is enough to buy 11 muffins and 7 brownies, the greatest possible cost for 11 muffins and 7 brownies is $15:
11M + 7B = 15.

Statement 1:
Since $15 is enough to buy 7 muffins and 11 brownies, the greatest possible cost for 7 muffins and 11 brownies is $15:
7M + 11B = 15.

Adding together the two equations in blue, we get:
18M + 18B = 30.

Dividing by 6, we get:
3M + 3B = 5.

Multiplying by 9, we get:
27M + 27B = 45.

Implication:
The greatest possible cost for 27 muffins and 27 brownies is $45.
Thus, $45 is enough to buy 27 muffins and 27 brownies.
SUFFICIENT.

Statement 2:
Case 1: Minimize the cost for each muffin and brownie
Let the cost for each muffin = $0.01 and the cost for each brownie = $0.01.
In this case, $45 is clearly enough to buy 27 muffins and 27 brownies.

Case 2: Maximize the cost for each brownie, minimize the cost for each muffin
Since $15 is enough to buy 8 brownies -- and 15/8 = 1.875 -- it is possible that each brownie costs $1.80, while each muffin costs $0.01.
Cost for 27 muffins and 27 brownies:
(27)(0.01) + (27)(1.8) > 45.
In this case, $45 is NOT enough to buy 27 muffins and 27 brownies.

INSUFFICIENT.

The correct answer is A.
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by Brent@GMATPrepNow » Fri Dec 23, 2016 7:28 am
Mo2men wrote:Alice has $15, which is enough to buy 11 muffins and 7 brownies, is $45 enough to buy 27 muffins and 27 brownies?
(1) $15 is enough to buy 7 muffins and 11 brownies.
(2) $15 is enough to buy 10 muffins and 8 brownies.
Given: $15 is enough to buy 11 muffins and 7 brownies
Let M = the price of ONE muffin
Let B = the price of ONE brownie
So, for example, the TOTAL cost of 11 muffins and 7 brownies = 11M + 7B
So, if $15 is enough to buy 11 muffins and 7 brownies, we can say that $15 is GREATER THAN OR EQUAL TO 11M + 7B
In other words, 15 ≥ 11M + 7B

Target question: Is $45 enough to buy 27 muffins and 27 brownies?
This is a good candidate for rephrasing the target question.
Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

REPHRASED target question: Is 45 ≥ 27M + 27B?

Statement 1: $15 is enough to buy 7 muffins and 11 brownies
In other words, 15 ≥ 7M + 11B
We also know: 15 ≥ 11M + 7B
Add the inequalities to get: 30 ≥ 18M + 18B
Multiply both sides by 1.5 to get: 45 ≥ 27M + 27B
Perfect!!!
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: $15 is enough to buy 10 muffins and 8 brownies.
In other words, 15 ≥ 10M + 8B
We also know: 15 ≥ 11M + 7B
Add the inequalities to get: 30 ≥ 21M + 15B
Multiply both sides by 1.5 to get: 45 ≥ 31.5M + 22.5B
Is this enough information to answer the REPHRASED target question?
No
There are several values of M and B that satisfy the inequality 45 ≥ 31.5M + 22.5B
2. Here are two:
Case a: M = 1.4 and B = 0 (this satisfies 45 ≥ 31.5M + 22.5B). In this case, 27M + 27B = 27(1.4) + 27(0) = 37.8. In other words, 45 ≥ 27M + 27B
Case b: M = 0 and B = 1.9 (this satisfies 45 ≥ 31.5M + 22.5B). In this case, 27M + 27B = 27(0) + 27(1.9) = 51.3. In other words, 45 < 27M + 27B
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT


Answer = A

RELATED VIDEOS
- Inequalities - Part I: https://www.gmatprepnow.com/module/gmat ... /video/979
- Inequalities - Part II: https://www.gmatprepnow.com/module/gmat ... /video/982
Last edited by Brent@GMATPrepNow on Fri Dec 23, 2016 9:21 am, edited 1 time in total.
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by Mo2men » Fri Dec 23, 2016 9:14 am
Brent@GMATPrepNow wrote:
Mo2men wrote:Alice has $15, which is enough to buy 11 muffins and 7 brownies, is $45 enough to buy 27 muffins and 27 brownies?
(1) $15 is enough to buy 7 muffins and 11 brownies.
(2) $15 is enough to buy 10 muffins and 8 brownies.
Given: $15 is enough to buy 11 muffins and 7 brownies
Let M = the price of ONE muffin
Let B = the price of ONE brownie
So, for example, the TOTAL cost of 11 muffins and 7 brownies = 11M + 7B
So, if $15 is enough to buy 11 muffins and 7 brownies, we can say that $15 is GREATER THAN 11M + 7B
In other words, 15 > 11M + 7B

Target question: Is $45 enough to buy 27 muffins and 27 brownies?
This is a good candidate for rephrasing the target question.
Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

REPHRASED target question: Is 45 > 27M + 27B?

Statement 1: $15 is enough to buy 7 muffins and 11 brownies
In other words, 15 > 7M + 11B
We also know: 15 > 11M + 7B
Add the inequalities to get: 30 > 18M + 18B
Multiply both sides by 1.5 to get: 45 > 27M + 27B
Perfect!!!
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: $15 is enough to buy 10 muffins and 8 brownies.
In other words, 15 > 10M + 8B
We also know: 15 > 11M + 7B
Add the inequalities to get: 30 > 21M + 15B
Multiply both sides by 1.5 to get: 45 > 31.5M + 22.5B
Is this enough information to answer the REPHRASED target question?
No
There are several values of M and B that satisfy the inequality 45 > 31.5M + 22.5B
2. Here are two:
Case a: M = 1.4 and B = 0 (this satisfies 45 > 31.5M + 22.5B). In this case, 27M + 27B = 27(1.4) + 27(0) = 37.8. In other words, 45 > 27M + 27B
Case b: M = 0 and B = 1.9 (this satisfies 45 > 31.5M + 22.5B). In this case, 27M + 27B = 27(0) + 27(1.9) = 51.3. In other words, 45 < 27M + 27B
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT


Answer = A

RELATED VIDEOS
- Inequalities - Part I: https://www.gmatprepnow.com/module/gmat ... /video/979
- Inequalities - Part II: https://www.gmatprepnow.com/module/gmat ... /video/982
Thanks Brent.

I have question. When you use 'enough', I think we should less than or equal. i.e 15 >= 11M + 7. Similarly, in facts 2.
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by Brent@GMATPrepNow » Fri Dec 23, 2016 9:22 am
Mo2men wrote: Thanks Brent.

I have question. When you use 'enough', I think we should less than or equal. i.e 15 >= 11M + 7. Similarly, in facts.
Ahh, good catch!!
I've changed all of the >'s to ≥'s.

Cheers and thanks,
Brent
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by Mo2men » Fri Dec 23, 2016 9:46 am
Brent@GMATPrepNow wrote:

Statement 2: $15 is enough to buy 10 muffins and 8 brownies.
In other words, 15 ≥ 10M + 8B
We also know: 15 ≥ 11M + 7B
Add the inequalities to get: 30 ≥ 21M + 15B
Multiply both sides by 1.5 to get: 45 ≥ 31.5M + 22.5B
Is this enough information to answer the REPHRASED target question?
No
There are several values of M and B that satisfy the inequality 45 ≥ 31.5M + 22.5B
2. Here are two:
Case a: M = 1.4 and B = 0 (this satisfies 45 ≥ 31.5M + 22.5B). In this case, 27M + 27B = 27(1.4) + 27(0) = 37.8. In other words, 45 ≥ 27M + 27B
Case b: M = 0 and B = 1.9 (this satisfies 45 ≥ 31.5M + 22.5B). In this case, 27M + 27B = 27(0) + 27(1.9) = 51.3. In other words, 45 < 27M + 27B
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT


Answer = A
Thanks a lot.

I have other question.In Statement 2, You choose B=0 to test value but should not we choose a value other than zero? The question implies that we can buy both Muffin & Brownies.

Thanks
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by Brent@GMATPrepNow » Fri Dec 23, 2016 9:55 am
Mo2men wrote: Thanks a lot.

I have other question.In Statement 2, You choose B=0 to test value but should not we choose a value other than zero? The question implies that we can buy both Muffin & Brownies.

Thanks
There's nothing in the questions that say the price per muffin or brownie must be greater than $0.
However, we could always plug in M = 1.4 and B = 0.000000000000000001 for case a and plug in M = 0.000000000000000001 and B = 1.9 for case b

Cheers,
Brent
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