I29-210

Question

The table provides data about 15 different pigment batches currently in stock at a paint manufacturer. For each batch, the data include volume (liters), pigment concentration (grams per liter, g/L), and price per batch.
      Batch   Series   Volume (L)   Concentration (g/L)   Price       U1   Ultramarine   18   270   $1,420       U2   Ultramarine   20   260   $1,360       U3   Ultramarine   16   240   $1,320       U4   Ultramarine   26   280   $1,580       C1   Cadmium   15   310   $1,590       C2   Cadmium   12   290   $1,520       C3   Cadmium   28   250   $1,480       K1   Cobalt   14   240   $1,260       K2   Cobalt   16   245   $1,600       V1   Viridian   30   230   $1,330       V2   Viridian   10   235   $1,180       T1   Titanium   32   180   $1,050       T2   Titanium   21   185   $1,080       O1   Ochre   22   200   $1,200       M1   Magenta   19   260   $1,380      
For each statement, select  True  if it can be verified to be true from the table, otherwise select  False.

Bunuel · Answer

Official Solution:

• The batch with the greatest concentration has the greatest total pigment content (in grams). 
 
First, calculate the total pigment content in the batch with the greatest concentration. That batch is C1, with 15 liters and 310 g/L, giving 15*310 = 4650 g.
 
To compare, try a batch with considerably larger volume and comparable concentration. U4 has 26 liters and 280 g/L, giving 26*280 = 7280 g. Since 7280 g is greater than 4650 g, the batch with the greatest concentration does not have the greatest total pigment.
 
Answer:  False 
 
 • The median concentration of the 4 highest-volume batches is less than the median concentration of the 4 lowest-volume batches. 
 
For four values, the median is the average of the two middle values after sorting.
 
Highest volumes: T1 (32 L, 180), V1 (30 L, 230), C3 (28 L, 250), U4 (26 L, 280). Concentrations: 180, 230, 250, 280. The median is (230 + 250)/2 = 240.
 
Lowest volumes: V2 (10 L, 235), C2 (12 L, 290), K1 (14 L, 240), C1 (15 L, 310). Concentrations: 235, 240, 290, 310. The median is (240 + 290)/2 = 265.
 
Since 240 is less than 265, the statement holds.
 
Answer:  True 
 
 • The average price per liter for Cobalt batches is more than $95. 
 
First, calculate the price per liter for each Cobalt batch.
 
K1: 1260/14 = 90 dollars per liter.
 
K2: 1600/16 = 100 dollars per liter.
 
Since there are 14 liters of the first and 16 liters of the second, the overall average will be closer to the 16-liter batch’s value of 100. The midpoint between 90 and 100 is 95, so the weighted average must be greater than 95.
 
Answer:  True

Correct answer:

The batch with the greatest concentration has the greatest total pigment content (in grams).  "False"

The median concentration of the 4 highest-volume batches is less than the median concentration of the 4 lowest-volume batches.  "True"

The average price per liter for Cobalt batches is more than $95.  "True"

Adit_ · Answer

damn 3rd one is tricky! Enough to push it to 700+

SachinNayak · Answer

Bunuel  This statement -"Since there are 14 liters of the first and 16 liters of the second, the overall average will be closer to the 16-liter batch’s value of 100. The midpoint between 90 and 100 is 95, so the weighted average must be greater than 95." had there been 18liters in thr first average would have been closer to 90 instead?
I did total Price by total liters and got 95.33 so thought thats how its greater than 95. I think all this is one and the same conecpt?

Bunuel · Answer

Yes.

95 is the midpoint between 90 and 100.
If the two batches had equal liters, the average would be exactly 95.

Here, they are not equal:
14 L at 90
16 L at 100

Since the larger amount is attached to 100, the weighted average must be above 95.

If it were:
18 L at 90
16 L at 100

then the larger amount would be attached to 90, so the weighted average would be below 95.

And yes, your method total price / total liters = 95.33 is the exact calculation of that same weighted-average idea.

I29-210

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My Rewards

Batch	Series	Volume (L)	Concentration (g/L)	Price
U1	Ultramarine	18	270	$1,420
U2	Ultramarine	20	260	$1,360
U3	Ultramarine	16	240	$1,320
U4	Ultramarine	26	280	$1,580
C1	Cadmium	15	310	$1,590
C2	Cadmium	12	290	$1,520
C3	Cadmium	28	250	$1,480
K1	Cobalt	14	240	$1,260
K2	Cobalt	16	245	$1,600
V1	Viridian	30	230	$1,330
V2	Viridian	10	235	$1,180
T1	Titanium	32	180	$1,050
T2	Titanium	21	185	$1,080
O1	Ochre	22	200	$1,200
M1	Magenta	19	260	$1,380

True	False
		The batch with the greatest concentration has the greatest total pigment content (in grams).
		The median concentration of the 4 highest-volume batches is less than the median concentration of the 4 lowest-volume batches.
		The average price per liter for Cobalt batches is more than $95.