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A boy working alone at 50 percent of his normal speed took 3 more days

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RenB
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A boy working alone at 50 percent of his normal speed took 3 more days [#permalink] New post   05 Aug 2023, 21:49
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A boy working alone at 50 percent of his normal speed took 3 more days than a man working alone at 75 percent of his normal speed to produce n units of a product. The boy and the man worked together at their respective normal speeds for 9 days, after which the man left the job but the boy continued working for 6 more days. If a total of 3n units of the product were produced during this period, how many days would it take the boy working alone at 75 percent of his normal speed to produce 2n units?
A. 10
B. 15
C. 20
D. 30
E. 40
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Re: A boy working alone at 50 percent of his normal speed took 3 more days [#permalink] New post   05 Aug 2023, 21:50
RenB wrote:
A boy working alone at 50 percent of his normal speed took 3 more days than a man working alone at 75 percent of his normal speed to produce n units of a product. The boy and the man worked together at their respective normal speeds for 9 days, after which the man left the job but the boy continued working for 6 more days. If a total of 3n units of the product were produced during this period, how many days would it take the boy working alone at 75 percent of his normal speed to produce 2n units?
A. 10
B. 15
C. 20
D. 30
E. 40


Hi chetan2u Bunuel
1. Is this question representative of GMAT questions?
2. Could you please share the most efficient way of solving this
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chetan2u
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Re: A boy working alone at 50 percent of his normal speed took 3 more days [#permalink] New post   06 Aug 2023, 04:38
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RenB wrote:
A boy working alone at 50 percent of his normal speed took 3 more days than a man working alone at 75 percent of his normal speed to produce n units of a product. The boy and the man worked together at their respective normal speeds for 9 days, after which the man left the job but the boy continued working for 6 more days. If a total of 3n units of the product were produced during this period, how many days would it take the boy working alone at 75 percent of his normal speed to produce 2n units?
A. 10
B. 15
C. 20
D. 30
E. 40



No, this is not representative of a GMAT question. Too many steps involved

Solution

Algebraic method

Man at 3/4 his capability takes x days to produce n products, so at his full capability, he would take 3x/4 days.
The boy at 1/2 his capability takes x+3 days to produce n products, so at his full capability, he would take (x+3)/2 days.
This means man and boy take 3x/4 and (x+3)/2 days respectively to produce n products.

9 days work of man = \(9*\frac{4}{3x}=\frac{12}{x}\), while 9+6 or 15 days work of boy = \(15*\frac{2}{x+3}=\frac{30}{x+3}\)
If the outcome of above work is 3n, that is thrice, we get the equation as \(\frac{12}{x}+\frac{30}{x+3}=3\)
\(\frac{12x+36+30x}{x(x+3)}=3………42x+36=3(x^2+3x)………x^2+3x=14x+12……..x^2-11x+12=0\)
Solving, x=12 or -1. But x>0, so x=12

Days taken by boy for n products = \(\frac{12+3}{2}\)
Days taken by boy for 2n products = \(2*\frac{12+3}{2}=15\)
If the speed was 75%, time taken = 15/0.75 or 15*4/3 or 20 days.
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Re: A boy working alone at 50 percent of his normal speed took 3 more days [#permalink] New post   17 Aug 2023, 17:54
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pretty complex question.

I have an alternative solution. let m be speed of man and b be speed of boy. we need to how much time it takes to produce 2n units in 75% speed => 2n/(3b/4) = 8n/3b . so we need to use the given info to eliminate m and find n/b ratio

from 3n units case, we get 9m + 15b = 3n . that is 3m+5b = n
from n units case, we get 2b(t+3) = 3m(t) . rearranging, this gives 1 + (t/3) = 3m/2b . now place t = 4n/3m (considering man's equation alone) and we get 1 + (9m/4n) = 3m/2b

using previous eqaution, work towards eliminating m and we get a equation in n^2, b^2 and nb. divide by b^2. we get a quadratic in (n/b). solve the quadratic and we get 2 values for n/b - 30/4 and 1. 1 doesnt make sense, so its 30/4. thus final answer is 8/3*(30/4) = 20.
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Re: A boy working alone at 50 percent of his normal speed took 3 more days [#permalink]
17 Aug 2023, 17:54
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