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# It takes Abby a hours to build a birdhouse. When Becky and Caroline he

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Math Expert
Joined: 02 Sep 2009
Posts: 44636
It takes Abby a hours to build a birdhouse. When Becky and Caroline he [#permalink]

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31 Aug 2015, 10:47
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55% (hard)

Question Stats:

70% (02:12) correct 30% (02:13) wrong based on 157 sessions

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It takes Abby a hours to build a birdhouse. When Becky and Caroline help Abby, the three of them can build an identical birdhouse in half the time. If it takes Becky and Caroline b and c hours, respectively, to build a birdhouse on their own, which of the following statements is true?

(A) (bc - 4a)/(abc) = 0
(B) (b^2 - 4ac)/(abc) = 0
(C) (bc - ac - ab)/(abc) = 0
(D) (a - b - c)/(abc) = 0
(E) (a + b + c)/(abc) = 0

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Re: It takes Abby a hours to build a birdhouse. When Becky and Caroline he [#permalink]

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31 Aug 2015, 10:56
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Bunuel wrote:
It takes Abby a hours to build a birdhouse. When Becky and Caroline help Abby, the three of them can build an identical birdhouse in half the time. If it takes Becky and Caroline b and c hours, respectively, to build a birdhouse on their own, which of the following statements is true?

(A) (bc - 4a)/(abc) = 0
(B) (b^2 - 4ac)/(abc) = 0
(C) (bc - ac - ab)/(abc) = 0
(D) (a - b - c)/(abc) = 0
(E) (a + b + c)/(abc) = 0

Kudos for a correct solution.

Work done by A in 1 hour=1/a, by B=1/b, by C=1/c
Work done if all three work together in 1 hour=1/a
Now, 1/a+1/b+1/c=2/a
1/a=1/b+1/c
1/a=b+c/bc
bc=ab+ac
bc-ac-ab=0
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Re: It takes Abby a hours to build a birdhouse. When Becky and Caroline he [#permalink]

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31 Aug 2015, 13:09
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+1 C
1/a+1/b+1/c= 2/a
solve this and you get option C
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Re: It takes Abby a hours to build a birdhouse. When Becky and Caroline he [#permalink]

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31 Aug 2015, 14:14
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Bunuel wrote:
It takes Abby a hours to build a birdhouse. When Becky and Caroline help Abby, the three of them can build an identical birdhouse in half the time. If it takes Becky and Caroline b and c hours, respectively, to build a birdhouse on their own, which of the following statements is true?

(A) (bc - 4a)/(abc) = 0
(B) (b^2 - 4ac)/(abc) = 0
(C) (bc - ac - ab)/(abc) = 0
(D) (a - b - c)/(abc) = 0
(E) (a + b + c)/(abc) = 0

Kudos for a correct solution.

As usual, I try to solve with a RatexTimexWork table as shown below:

Attachment:

rtw.jpg [ 13.75 KiB | Viewed 1494 times ]

Since you have 3 given variables and no fix value at all, you can plug in freely a value for a. I used a=2.

Remember for Rate x Time x Work problems, the combined rate (in this case 2/a) is given by rate A + rate B + rate C:

$$\frac{1}{a}$$+$$\frac{1}{b}$$+$$\frac{1}{c}$$= $$\frac{2}{a}$$ |multiply by a

$$1$$+$$\frac{a}{b}$$+$$\frac{a}{c}$$ = $$2$$

It follows that a/b + a/c = 1. Since we are not concerned on any specific value (a, b, or c), we can plug in b=4 and c=4, which then equals 2/4+2/4 since a=2.

Finally we have values for a=2, b=4, c=4.

All one has to do is plug in these values in the different answer choices. Answer choice C equals 0.
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Re: It takes Abby a hours to build a birdhouse. When Becky and Caroline he [#permalink]

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01 Sep 2015, 04:04
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Bunuel wrote:
It takes Abby a hours to build a birdhouse. When Becky and Caroline help Abby, the three of them can build an identical birdhouse in half the time. If it takes Becky and Caroline b and c hours, respectively, to build a birdhouse on their own, which of the following statements is true?

(A) (bc - 4a)/(abc) = 0
(B) (b^2 - 4ac)/(abc) = 0
(C) (bc - ac - ab)/(abc) = 0
(D) (a - b - c)/(abc) = 0
(E) (a + b + c)/(abc) = 0

Kudos for a correct solution.

Bunuel wrote:
It takes Abby a hours to build a birdhouse. When Becky and Caroline help Abby, the three of them can build an identical birdhouse in half the time. If it takes Becky and Caroline b and c hours, respectively, to build a birdhouse on their own, which of the following statements is true?

(A) (bc - 4a)/(abc) = 0
(B) (b^2 - 4ac)/(abc) = 0
(C) (bc - ac - ab)/(abc) = 0
(D) (a - b - c)/(abc) = 0
(E) (a + b + c)/(abc) = 0

Kudos for a correct solution.

abby's 1 hour work = 1/a
becky's 1 hour work = 1/b
caroline's 1 hour work = 1/c
they together will take half of abby's time. So their 1 hour work will be 2/a

it is given that

$$\frac{1}{a}$$ + $$\frac{1}{b}$$+$$\frac{1}{c}$$ = $$\frac{2}{a}$$

or $$\frac{1}{b}$$+$$\frac{1}{c}$$ = $$\frac{1}{a}$$

or $$\frac{1}{b}$$+$$\frac{1}{c}$$-$$\frac{1}{a}$$ = 0

or $$\frac{(ca+ab-bc)}{abc}$$ = 0

or $$\frac{(bc-ca-ab)}{abc}$$ = 0

Math Expert
Joined: 02 Sep 2009
Posts: 44636
Re: It takes Abby a hours to build a birdhouse. When Becky and Caroline he [#permalink]

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06 Sep 2015, 05:51
Bunuel wrote:
It takes Abby a hours to build a birdhouse. When Becky and Caroline help Abby, the three of them can build an identical birdhouse in half the time. If it takes Becky and Caroline b and c hours, respectively, to build a birdhouse on their own, which of the following statements is true?

(A) (bc - 4a)/(abc) = 0
(B) (b^2 - 4ac)/(abc) = 0
(C) (bc - ac - ab)/(abc) = 0
(D) (a - b - c)/(abc) = 0
(E) (a + b + c)/(abc) = 0

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

The key to this problem is the second sentence. When Becky and Caroline help, the time it takes to build the birdhouse is cut in half. That means that Becky and Caroline, when working together, work at a rate equal to the rate Abby works at.

We can represent the rates that Abby, Becky and Caroline work as 1/a, 1/b and 1/c, respectively. Therefore:

1/a = 1/b + 1/c

All of the answer choices have an expression equal to 0, so combine fractions and move them all to the left side of the equation. The common denominator for all fractions is abc.

1/a = 1/b + 1/c

bc/(abc) = ac/(abc) + ab/(abc)

bc/(abc) - ac/(abc) - ab/(abc) = 0

(bc - ac - ab)/(abc) = 0

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Re: It takes Abby a hours to build a birdhouse. When Becky and Caroline he [#permalink]

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09 Dec 2017, 08:38
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Re: It takes Abby a hours to build a birdhouse. When Becky and Caroline he   [#permalink] 09 Dec 2017, 08:38
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