I29-170

Question

A factory is working on a production order using identical machines. The production begins with a certain number of machines operating in the first hour. At the start of the second hour, and at the start of each hour thereafter, the same fixed number of additional machines is added, thus increasing the units produced in each subsequent hour. By the end of 8 hours, the machines have produced exactly 220 units.
 
Select for  1st hour  the number of units produced in the first hour, and select for  8th hour  the number of units produced in the eighth hour that would be jointly consistent with the given information. Make only two selections, one in each column.

Bunuel · Answer

Official Solution:

In 8 hours, the machines produced a total of 220 units.
 
So the average production per hour is 220/8 = 27.5 units.
 
Because the output increases each hour,  the production in the first hour must be less than the average . Among the choices, the only value below 27.5 is  17 .
 
Thus, in the first hour, 17 units were produced.
 
Since the same number of additional machines is added each hour, the hourly outputs form an arithmetic sequence. For such a sequence, the average equals (first hour + eighth hour)/2.
 
(17 + eighth-hour output)/2 = 27.5
 
17 + eighth-hour output = 55
 
Eighth-hour output =  38

Correct answer:

1st hour  "17"

8th hour  "38"

gemministorm · Answer

I got confused, I took
1st day x, and would have worked 8 hours
2nd day 2x, and would have worked 7 hours
3rd day 3x, and would have worked 6 hours
.
.
.
.
so at end
x(1*8 + 2*7 + 3*6 + 4*5 + 5*4 + 6*3 + 7*2 + 8*1) = 220....
could someone help, where did i go wrong

Bunuel · Answer

The problem says production increases each hour, not that the number of hours worked changes.

Each hour is one fixed hour, but output per hour rises evenly forming an arithmetic sequence. Here, x is the number of units produced in the first hour, and d is the fixed increase in the number of units produced per hour.

Your setup treats it as if the first batch worked 8 hours, the second 7, etc., which is incorrect. The correct setup is
 
x + (x + d) + (x + 2d) + (x + 3d) + (x + 4d) + (x + 5d) + (x + 6d) + (x + 7d) = 220.

Karym · Answer

I like the solution - it’s helpful.

I29-170

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