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605-655 (Medium)|   Word Problems|                     
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ThatDudeKnows
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This question can be solved using the weighted averages approach as well.

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Weighted averages - Approach 1

We are given the total production cost - 300
Price per roll = 300/n, where n is the number of rolls

If all the rolls were sold at a price greater than 50% of the average roll price - then the total selling price will be 450
How 450 ? - The average price of each roll is 300/n, and the total cost price is (300/n)n = 300
So, if the selling price of the roll is 50% greater than the price of roll - (1.5)(300/n)n = 450

If all the rolls were sold at a price greater than 20% less than of the average roll price - then the total selling price will be 360
(How did we come to the 20% value - Question says that the price on day 2 is 20% less than that of day 1... Let's say total production cost is x, then total selling price on Day 1 is 1.5x, so day 2 will be (0.8)1.5x = 1.2x i.e. 20% greater than the total production cost)

If all rolls sold at 1.2 times the price If all rolls sold at 1.5 times the price
360 450
1 4

The actual total selling price should be closer to the day 2, as its weightage is more. The difference, 90 is split in 5 parts, and since its closer to 450, the actual cost is 1 part less than 450 i.e. 450-18 = 432.
432-300=132

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Weighted averages - Approach 2

We can also figure out the weighted averages based on change in percentage on each day, and the fraction of rolls sold on each day will be the weights.

Day 1 Day 2
20% 50%
1 4

So the average percent increase is 44%.
44% of 300 is 132.
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I liked my logic to solving this, although it's far from any standard / best practice method. Might still help, so here goes.

As $300 worth of rolls were produced, and 4/5 of those rolls were sold, then essentially $240 worth of rolls (production cost) were sold - i.e., 4/5 of the $300. Now, if their selling price is 50% higher than the cost price, and $240 equates to the cost price for the roles sold on day 1, then those raked in $240 * 1.5 = $360. We now have $120 as profit locked in (as for end of day 1).

Next day, we will sell the remaining 1/5th of those rolls, or those worth $60. If we'd've sold them at the same 50% profit, they'd have gone for $90 ($60 * 1.5). But the question specifies this last batch was sold at a price 20% lesser than that 50% - or, if the value was $90, we minus 20% from it, to arrive at $72 ($90 * 0.8). That's a $12 profit on the cost price of this batch. Add that to the $120 and you have your answer, i.e., $132 or C.
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Solve this question using weighted average concept. The weighted average of both the days; if Cost price pf each roll taken as 100x and total 5 rolls considered.
Day 1: SP = 150x (4 rolls)
Day 2: SP = 120x (1 roll)

Weighted average SP of both days= 144x
Clearly there is average of 44% profit on each roll for both the days.
Hence total profit would be: 44% of $300 = $132.

This is the shortest way of doing this question. If you find any other short approach please let me know. And feel free to directly message me to know more about this solution.


Carcass
On a certain day, a bakery produced a batch of rolls at a total production cost of $ 300. On that day, \(\frac{4}{5}\) of the rolls in the batch were sold, each at a price that was 50 percent greater than the average (arithmetic mean) production cost per roll. The remaining rolls in the batch were sold the next day, each at a price that was 20 percent less than the price of the day before. What was the bakery's profit on this batch of rolls?

A. $ 150

B. $ 144

C. $ 132

D. $ 108

E. $ 90
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