A group of people decided to cut 128 trees in a certain number of days

Question

Competition Mode Question

A group of people decided to cut 128 trees in a certain number of days. For the first 4 days, they were able to achieve their planned per day target. However, for the remaining days, the group was able to cut 4 more trees daily than planned. In this way, the group had cut 164 trees one day before the planned finish date. What was the number of trees the group was planning to cut per day?

A. 8
B. 12
C. 16
D. 32
E. 64

x trees per day, n total days as planned
nx = 128
4x+(n-5)(x+4)=164, [sub nx=128, n=128/x]
on solving,
x^2+56x-512=0,
x=8 trees per day.

A. 8
B. 12
C. 16
D. 32
E. 64

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nick1816 · Answer

Number of trees to cut each day =x
Total number of days reqd to cut 128 days = n

n*x=128......(1)

Also 4*x+(n-4-1)*(x+4) = 164
4x+(n-5)(x+4) = 164
4x+nx-5x+4n-20=164
4n-x+128-20=164
4n-x= 56....(2)

We can use options to find (n,x).\

Option 1- x=8; n= 128/8 = 16
4*(16)-8=56 (Answer)

OR

4n-x= 56 
n=  \frac{56+x}{4}

n*x=128

(56+x)*x =128*4

x^2+56x-512=0

x^2+64x-8x-512 =0 
x=8 or -64(Rejected)

A

eakabuah · Answer

Let  n  be the total number of days planned for cutting the trees and  x  be the number of trees planned to be cut per day.
Then  128=nx  hence  n=\frac{128}{x} ----(1) 
But we are given that the group only followed the original schedule to cut  x  trees per day for the first 4days and that they actually cut 4 extra trees for the remaining days and in the process cut 164 trees by  n-1  days
Hence  164=4x+(n-5)(x+4) -----(2)

We can solve this equation by substituting  n=\frac{128}{x}  into  (2)  but a shorter approach is to use number properties and the answer choices
From  (1)   n  and  x  can only be factors of  128  and we know from prime factorization of  128=2^7  that  3  is not a factor of  128  hence we can eliminate B which says that  x=12 .
Let's assume that  x=8  (per answer choice A), it implies  n=16 
Testing with  (2) ,  164=4*8+11*12 = 32+132 = 164.

Hence A is Answer.

freedom128 · Answer

Plan: to cut 128 trees in d days ---> (128/d) trees a d ay.

First 4 days,  4*(128/d) trees were cut. 
Then  in the next (d-4-1)=(d-5) days,  (d-5)*(128/d+4) trees were cut.

Total trees cut: 
4*(128/d)+ (d-5)*(128/d+4) = 164
-(128/d) -20 + 128 +4*d = 164
-128 + 4*d^2 - 56*d = 0
-32 + d^2 - 14*d = 0
(d-16)*(d+2)=0
d=16 days or d=-2 (NA)

The number of trees the group was planning to cut per day = 128/d = 128/16 = 8 trees per day

FINAL ANSWER IS (A)

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ArunSharma12 · Answer

A group of people decided to cut 128 trees in a certain number of days. For the first 4 days, they were able to achieve their planned per day target. However, for the remaining days, the group was able to cut 4 more trees daily than planned. In this way, the group had cut 164 trees one day before the planned finish date. What was the number of trees the group was planning to cut per day?

A. 8
B. 12
C. 16
D. 32
E. 64

x trees per day, n total days as planned
nx = 128
4x+(n-5)(x+4)=164, [sub nx=128, n=128/x]
on solving,
x^2+56x-512=0,
x=8 trees per day.

A. 8
B. 12
C. 16
D. 32
E. 64

Method 1: 
let the rate of number of trees to cut per day be x
x*d = 128

as they were able to maintain this rate for 4 days, and then increased the rate to (x+4) thereby decreasing the total number of days by one.
x*4 + (x+4)(d-4-1) = 164
x*4 + x*d - x*5 +4*d - 20 = 164 
4*d - x = 56
 4*128 - x^2 = 56x 
 4*128 = x( x+ 56) 
 8*(8+56) = x( x+ 56) 
x= 8

Method 2: 
 A. 8: initial number of days = 16, 8*4 + 12*11 = 32 + 132 = 164 
B. 12: not possible as it do not divides 128 
C. 16: initial number of days = 8, 16*4 + 20*3 = 64 + 60 = 124 less than the target.
D. 32: initial number of days = 4, not possible as they worked for 4 days and then some more 
E. 64: initial number of days = 2, not possible as they worked for 4 days and still did not reached their target

Ans: A

GMATWhizTeam · Answer

Solution:  
Let x be the number of days in which trees are to be cut.
 • Number of trees cut in one day =  \frac{128}{x}  trees
• For the first four days, they cut trees as planned.
 o Trees cut in first 4 days =  4*\frac{128}{x}  
• For the remaining days they cut 4 more trees per day, till one day before the planned day
 o Trees cut in one day =  \frac{128}{x} + 4 
o Now, trees cute in (x-5) days =  (x-5)(\frac{128}{x} + 4)=108 + 4x -5*\frac{128}{x}  
• Trees cut in x-1 days =  4*\frac{128}{x} + 108 +4x -5*\frac{128}{x} 
•  164 = 108 + 4x -\frac{128}{x} 
•  4x^2 -56x – 128 =0 
•  x^2 - 14x – 32 = 0 
•  x^2 – 16x + 2x -32 =0 
•  (x-16)(x+2) = 0 
 o  x = 16 , we can discard the case of x = -2, because number of days can’t be negative. 
• Number of trees cut per day =  \frac{128}{16} = 8 . 
Hence, the correct answer is  Option A

exc4libur · Answer

rate*time=work, rt=128
(r+4)(t-5)+4r=164
rt-5r+4t-20+4r=164
rt+4t-r=184, (128)+4t-r=184
4t-(128/t)=184-128
4t^2-128=56t, t^2-14t-32=0
(t-16)(t+4)=0, t=16
r(16)=128, r=128/16=8

Ans (A)

monikakumar · Answer

A group of people decided to cut 128 trees in a certain number of days. For the first 4 days, they were able to achieve their planned per day target. However, for the remaining days, the group was able to cut 4 more trees daily than planned. In this way, the group had cut 164 trees one day before the planned finish date. What was the number of trees the group was planning to cut per day?

A. 8
B. 12
C. 16
D. 32
E. 64

x trees per day, n total days as planned
nx = 128
4x+(n-5)(x+4)=164, [sub nx=128, n=128/x]
on solving,
x^2+56x-512=0,
x=8 trees per day.

A. 8
B. 12
C. 16
D. 32
E. 64

x trees per day, n total days as planned
nx = 128
4x+(n-5)(x+4)=164,  
on solving,
x^2+56x-512=0,
x=8 trees per day.

unraveled · Answer

A group of people decided to cut 128 trees in a certain number of days. For the first 4 days, they were able to achieve their planned per day target. However, for the remaining days, the group was able to cut 4 more trees daily than planned. In this way, the group had cut 164 trees one day before the planned finish date. What was the number of trees the group was planning to cut per day?

A. 8
B. 12
C. 16
D. 32
E. 64

x trees per day, n total days as planned
nx = 128
4x+(n-5)(x+4)=164, [sub nx=128, n=128/x]
on solving,
x^2+56x-512=0,
x=8 trees per day.

A. 8
B. 12
C. 16
D. 32
E. 64

Let be the number of days the group planned to cut these 128 trees.
Trees cut per day planned =  \frac{128}{x}  
Trees cut for first 4 days =  4*\frac{128}{x} 
Rest of the days when trees were cut = x - 4 - 1 = x - 5
Trees cut er day for rest of the (x-5) days =  \frac{128}{x} + 4 
Now,  4*\frac{128}{x} + (x-5)*(\frac{128}{x} + 4) = 164 
 x^2 - 14x - 32 = 0 
x = 16
Trees cut per day planned =  \frac{128}{x} =\frac{128}{16} = 8

Answer A.

adstudy · Answer

Let original days required be D and Trees cut per day be T

A group of people decided to cut 128 trees in a certain number of days.

This means  D * T = 128   --- (1)

For the first 4 days, they were able to achieve their planned per day target. However, for the remaining days, the group was able to cut 4 more trees daily than planned. In this way, the group had cut 164 trees one day before the planned finish date.

This makes the below equation -

(4 * T) + (D - 4 - 1) (T + 4) = 164 
 4T + (D-5)(T+4) = 164 
 4T + DT - 5T + 4D - 20 = 164 
 4D + DT = 184 + T   --- (2)

From (1) and (2), we get
 4D + 128 = 184 + T 
 4D = 56 + T   --- (3)

Now we need what was the number of trees the group was planning to cut per day, that is 'T'

Substituting D from (1) in (3);

\frac{128}{T}*4 = 56 + T 
 T^2 + 56T - 512 = 0

Solving for T,

T^2 + 64T - 8T - 512 = 0 
 (T - 8)(T + 64) = 0

Hence T = 8, (A)

DeusTacitus22 · Answer

Pretty fast way of solving -

Let the number of days be - n
Let the number of trees cut per day - x

x*n = 128 -(1)

4x + (n-4-1)(x+4) = 168 -(2)

From (1) and (2),

4x + (128/x - 5)(x + 4) = 168

Now as solving this equation can get cumbersome we start putting the options into it,

(d) and (e) can be eliminated

=> (128/x - 5)<0

=> Putting (a),(b),(c),

We find that option (a) satisfies

Answer - (a)

bumpbot · Answer

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A group of people decided to cut 128 trees in a certain number of days

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