12 Days of Christmas GMAT Competition - Day 9: A plant must distribute : Two-part Analysis (TPA)

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26 Dec 2024, 09:50

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Bunuel

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A plant must distribute 45 hours weekly among three production lines: A, B, and C.

Line A: Maximum 42 hours
Line B: Maximum 20 hours
Line C: Maximum 12 hours

Each line must receive more than half its maximum, all 45 hours must be used, and the number of hours allocated to each line must be an integer.

Select for Line B the hours that could have been allocated to Line B, and select for Line C the hours that could have been allocated to Line C that would be jointly consistent with the given information. Make only two selections, one in each column.

Required hours -

A >= 22
B >= 11
C >= 7

Remaining hours = 45 - 22 - 11 - 7 = 5

Now from the option choices, B >= 11 has only 2 possibilities 16 or 17. As we have only 5 remaining hours, B can have maximum value of 16 which means C should be 7.

Answer: (B, C) = (16, 7)

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26 Dec 2024, 09:51

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A plant must distribute 45 hours weekly among three production lines: A, B, and C.

Line A: Maximum 42 hours
Line B: Maximum 20 hours
Line C: Maximum 12 hours

Total hours to allocate = 45
Line A must get more than 21 hours
Line B must get more than 10 hours
Line C must get more than 6 hours

Testing options for Line B and Line C:
Line B = 16:

Remaining hours for A and C = 45 - 16 = 29
Line A must get more than 21 hours
Line C must get more than 6 hours

If Line A gets 23 hours, then Line C gets 6 hours.
But Line C must get more than 6 hours, so Line B cannot be 16.
Line B = 17:

Remaining hours for A and C = 45 - 17 = 28
Line A must get more than 21 hours
Line C must get more than 6 hours

If Line A gets 23 hours, then Line C gets 5 hours.
But Line C must get more than 6 hours, so Line B cannot be 17.
Line B = 8:

Remaining hours for A and C = 45 - 8 = 37
Line A must get more than 21 hours
Line C must get more than 6 hours

If Line A gets 23 hours, then Line C gets 14 hours, which is more than 12.
So Line B cannot be 8.

Line C = 7:

Remaining hours for A and B = 45 - 7 = 38
Line A must get more than 21 hours
Line B must get more than 10 hours

If Line A gets 22 hours, then Line B gets 16 hours, which is more than 10.
So Line B = 16 and Line C = 7 work.

Line B = 16
Line C = 7

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26 Dec 2024, 09:55

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Constraints:
all 45 hours must be used
Num of hours is an integer
Hours (line) > its maximum/2

H(A) > 21
H(B) > 10
H(C) > 6

C can be 7,8,16,17
B can only be 16 or 17

If B is 16, (left with 45-16 hours = 29 hours)
A at the minimum has to be 22
Then we are left with 7 hours for C

If B is 17, (left with 28 hours)
A >= 22
C <=6 which is not possible (rejected)

>> Answer is B = 16, C = 7

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26 Dec 2024, 09:58

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Each line must receive more than half its maximum, all 45 hours must be used.
Calculate more than half for each -
A - 22, B - 11, C - 7
We are 5 hours short, look at the options -
6 is too low not possible
if C is 7hour then B must 16 to hit 45
if C is 8 then B must 15 (not given)

Hence only combination working is B=16 and C=7hours

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26 Dec 2024, 09:58

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Each line needs more than half its max:

A >= 22 hours
B >= 11 hours
C >= 7 hours

Total must be 45 hours

Looking at given options:
For B: 16 or 17 hours
For C: 7 or 8 hours

Testing combinations and ensuring A gets valid hours (between 22-42):
B=16, C=7: A=22 - Correct
B=17, C=8: A=20 - Incorrect

Answer:
Line B: 16 hours
Line C: 7 hours

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26 Dec 2024, 10:10

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A at least be 22 Hrs (integer>21)
B at least be 11 Hrs (integer>10)
C at least be 7 Hrs (integer>6)

Given options B can be 16 or 17
check if B=16, A=22, C =7 (full it) -Total 45, OK
B=17, A=22, C=7 Total=46 Incorrect

Ans B=16, c=7

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A plant must distribute 45 hours weekly among three production lines: A, B, and C.

Line A: Maximum 42 hours
Line B: Maximum 20 hours
Line C: Maximum 12 hours

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A plant must distribute 45 hours weekly among three production lines: A, B, and C.

Line A: Maximum 42 hours
Line B: Maximum 20 hours
Line C: Maximum 12 hours

a + b + c = 45
a belongs to [22, 42]
b belongs to [11, 20]
c belongs to [7, 12]

Now from the options only available options for b are 16, 17
b = 16 => Let's take min of others => 16 + 22 + 7 => 45, therefore others have to be min

b = 17 => Min possible is 46 which is not 45, hence b = 17 no possible

b =16 and c = 7

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26 Dec 2024, 10:12

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Bunuel

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A plant must distribute 45 hours weekly among three production lines: A, B, and C.

Line A: Maximum 42 hours
Line B: Maximum 20 hours
Line C: Maximum 12 hours

Total weekly hours: 45
Line A maximum: 42 hours
Line B maximum: 20 hours
Line C maximum: 12 hours
Each line must receive more than half its maximum hours.
Line A:
Minimum hours for Line A: 42 / 2 = 21 hours
Since the total hours is 45, Line A cannot receive its maximum of 42 hours.
Therefore, Line A must receive between 21 and 41 hours.
Line B:
Minimum hours for Line B: 20 / 2 = 10 hours
Maximum hours for Line B: 20 hours
Line C:
Minimum hours for Line C: 12 / 2 = 6 hours
Maximum hours for Line C: 12 hours
Possible Allocations:
We need to find combinations of hours for Line B and Line C that satisfy the constraints and add up to the total hours (45) when combined with the hours allocated to Line A.
Let's consider some possibilities:
Line B = 16 hours, Line C = 6 hours:
This combination satisfies the minimum and maximum hours for both lines.
Assuming Line A receives 23 hours (within its allowed range), the total hours would be 16 + 6 + 23 = 45.
Line B = 17 hours, Line C = 7 hours:
This combination also satisfies the minimum and maximum hours for both lines.
Assuming Line A receives 21 hours (its minimum), the total hours would be 17 + 7 + 21 = 45.
Therefore, the possible allocations for Line B and Line C are:
Line B: 16 hours
Line C: 6 hours
Line B: 17 hours
Line C: 7 hours

IMO
Line B: 16
Line C: 7

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26 Dec 2024, 10:13

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Let's approach this problem step by step:

1. First, let's consider the constraints for each line:

* Line A: More than 21 hours (half of 42), but no more than 42 hours

* Line B: More than 10 hours (half of 20), but no more than 20 hours

* Line C: More than 6 hours (half of 12), but no more than 12 hours

2. We need to select one value for Line B and one for Line C that, when combined with the hours for Line A, will total 45 hours.

3. Let's examine the options for Line C first

* 6 hours is not valid as it's not more than half of 12

* 7,8 are both valid options

4. Now for Line B, we have options of 16 and 17 hours, both of which are valid as they're more than 10 and not more than 20.

5. Let's check combinations:

* If Line B gets 16 hours and Line C gets 7 hours, Line A would get 22 hours (45 - 16 - 7 = 22). This is valid.

* If Line B gets 16 hours and Line C gets 8 hours, Line A would get 21 hours. This is not valid as it's not more than half of 42.

* If Line B gets 17 hours and Line C gets 7 hours, Line A would get 21 hours. This is not valid.

* If Line B gets 17 hours and Line C gets 8 hours, Line A would get 20 hours. This is not valid.

Therefore, the only valid combination is:

* Line B: 16 hours

* Line C: 7 hours

This leaves 22 hours for Line A, which satisfies all the given conditions.

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26 Dec 2024, 10:20

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Minimum hours that plant has to spend on each line:

Line A: Maximum 42 hours i.e. more than 21 hrs i.e atleast 22
Line B: Maximum 20 hours i.e. more than 10 hrs i.e atleast 11
Line C: Maximum 12 hours i.e. more than 6 hrs i.e atleast 7

if we sum the minimum hrs required we get 22+11+7 = 40 hrs, so we have 5 extra hours to distribute after this.

looking at the options for Line B only possible are 16 and 17, but also notice that we only had 5 extra hours.
So line B should be =16 hrs and hence Line C =7 hrs

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26 Dec 2024, 10:28

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According to the conditions given in the question, each line must receive more than half its maximum, all 45 hours must be used, and the number of hours allocated to each line must be an integer.

Line B has only two possible options: 16 hours or 17 hours as the number of hours needs to be an integer that is more than 50% of the total maximum hours.

On taking Line B has 16 hours, we have two possible options for Line C which is 7 or 8 hours. 6 hours will be equal to 50% of the maximum number of hours allocated to Line C & hence is not valid

On taking Line C as 7 hours, total hours allocated to B+C=23 hours. Thus out of the 45 hours to be allocated, 45-23 hours =22 hours will be allocated to Line A. This is sufficient as it is more than 50% of the max number of hours assigned to Line A (max 42 hours).

Hence the correct options are:
Line B-16 hours
Line C-7 hours

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26 Dec 2024, 10:49

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The stem tells us that;

A plant must distribute 45 hours weekly among three production lines: A, B, and C.

· Line A: Maximum 42 hours
· Line B: Maximum 20 hours
· Line C: Maximum 12 hours
Each line must receive more than half its maximum, all 45 hours must be used, and the number of hours allocated to each line must be an integer.

And we have to select;

for Line B the hours that could have been allocated to Line B
for Line C the hours that could have been allocated to Line C

We are given that;

A+B+C=45
Where,

(A>21)
(B>10)
(C>6)

Let's look at the options;
Among them only possible values of B can be 16 or 17.

Let's try B=16,
Then A+C=29
Let A = 22 (minimum integer value of A)
So, C=7

This fits and since there can be only 1 possible answer, this must be true.

B=16, C=7

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26 Dec 2024, 10:51

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We know that C will be 7, as it needs to be more than half its maximum(6)

And A needs to be 22 at least, as it needs to be more than half its maximum(21)

Now, A + C = 29

We need to use 45 hours, 45 - 29 = 16 hours which will be allocated to B.

Answer. 16 and 7

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26 Dec 2024, 10:53

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21<A<42
10<B<20
6<C<12

16<B+C<32 & A+B+C=45, B+C=45-A

-42<-A<-21
45-42<45-A<45-21
=>3<B+C<24
&&
=>16<B+C<32
16<B+C<24
B+C=17,18,19,20,21,22,23

(B,C)=(17,6) ---not possible or (16,7) possible

B=16, C=7

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26 Dec 2024, 11:02

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A = 42 hrs
B = 20 hrs
C = 12 hrs

Each line must receive more than half its maximum.
so,
A > 21
B > 10
C > 6

Minimum A can be 22.
Minimum B can be 11.
Minimum C can be 7.

A + B + C = 45

If we put in B = 16

22 + 16 + C = 45
C = 7

Thus,
B = 16, C = 7.

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A plant must distribute 45 hours weekly among three production lines: A, B, and C.

Line A: Maximum 42 hours
Line B: Maximum 20 hours
Line C: Maximum 12 hours

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26 Dec 2024, 11:04

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Bunuel

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A plant must distribute 45 hours weekly among three production lines: A, B, and C.

Line A: Maximum 42 hours
Line B: Maximum 20 hours
Line C: Maximum 12 hours

Let call the hr they operate as a,b,& c.

So as per given info a>21, b>10 & c>6.

a+b+c=45. So the only pair of b,c among the options that satisfy all these conditions is b=16,c=7 then a=22.

IMO b 16,c 7

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26 Dec 2024, 11:07

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Step 1: Constraints
Total hours distributed:
A+B+C=45.
Line A:
A≤42, and A> 42/2 =21, so A≥22.
Line B:
B≤20, and B> 20/2 =10, so B≥11.
Line C:
C≤12, and C > 12/2 =6, so C≥7.
All A,B,C must be integers.

Step 2: Solve for Possible Values
Since
A+B+C=45, we can test integer values for A, B, and C within the given ranges.

Case 1: Maximum A
If A=42, then B+C=45−42=3, but this violates the constraints
B≥11 and C≥7
Thus, A must be less than 42.

Case 2: Testing A=22 to A=41
For each feasible A, we calculate
B+C=45−A, ensuring
B and C satisfy their respective constraints.

Step 3: Test Feasible Solutions
Using the ranges
22≤A≤41, 11≤B≤20, and 7≤C≤12:
A=22:
B+C=45−22=23.

Possible pairs: B=16, C=7
A=23:
B+C=45−23 = 22

Possible pairs: B=15, C=7, or B=16, C=6 (violates C≥7)
A=24:
B+C=45−24=21

Possible pairs: B=14, C=7
Continue similar checks to identify feasible values for B and C

Step 4: Answer Choices
From the tests:
Line B possible values: 16, 17
Line C possible values: 6, 7
Final Answer
Line B: 16
Line C: 7