The cost per unit of electricity during a 4 year period is a

Question

The cost per unit of electricity during a 4 year period is as follows:

Year 1, $1 per unit; 
Year 2, $1.5 per unit; 
Year 3, $1.2 per unit; 
Year 4, $2 per unit.

If a factory spent equal dollars on electricity each year, what was the average cost of a unit of electricity during the 4 year period?

A. $1.333
B. $1.250
C. $1.425
D. $1.175
E. $1.500

MacFauz · Accepted Answer

Let money spent each year = $6

Units bought first year = 6
Units bought second year = 4
Units bought third year = 5
Units bought fourth year = 3

Total number of units = 18
Total money spent = 6*4 = 24

Average = 24/18 = 4/3 = 1.33

Answer is A

KarishmaB · Answer

Here is another method for the more mathematically inclined. When we talk about equal amount of money spent, the average price is the harmonic mean of all the prices.

1/1 + \frac{1}{1.5} + \frac{1}{1.2} + \frac{1}{2} = \frac{4}{H}

\frac{6}{6} + \frac{4}{6} + \frac{5}{6} + \frac{3}{6} = \frac{4}{H}

H = \frac{4}{3} = 1.333

Answer (A)

veergmat · Answer

It can be solved by  weighted average  formula, which is

= ( x1*n1+x2*n2+x3*n3+......................)/(n1+n2+n3+...............)

where, x1,x2,x3 are values assigned

and, n1,n2,n3 are the frequency or no. of repetition of x1,x2,x3 respwctively.

As problem states:
x1*n1=x2*n2=x3*n3=x4*n4

LCM of (x i )1,1.5,1.2,and 2 is 6

Therefore n1=6, n2=4, n3=5and n4= 3

Calculating average using above formula
Avg.= 4*6/(6+4+5+3)
 = 1.33

mvictor · Answer

my approach was to pick a number that would be easy to divide by all the prices given.
suppose company spent 12$ each year.
1st year - used 12 units
2nd year - 8 units
3rd year - 10 units
4th year - 6 units
total, we have 36 units.
since 12$/year was spent, in all 4 years was spent 48$.
now, avg price is total $/total # units = 48/36 = 4/3 or 1.33

A.

chetan2u · Answer

Hi all,
 you cannot apply AVERAGE straightway as the units are not the same for each year, example the expense at $1 is higher than at $2..
Also we cannot apply weighted average as we have four quantities and that too in not proper form.. 
there can be two ways...

arithmetic way

choose a number, which is LCM of all four, say 6 here..
how much can you purchase
Year 1, $1 per unit; ---- 6/1=6 units
Year 2, $1.5 per unit; -- 6/1.5=4 units
Year 3, $1.2 per unit; -- 6/1.2=5 units
Year 4, $2 per unit. ---- 6/2=3 units...
total units = 6+4+5+3=18 units..
cost incurred = 6*4=24..
avg cost= 24/18=1.333

Weighted average

lets test weighted average for two pairs separately..
also since overall expenditure is same each year, the units will be proportional to cost.. 
 1) $1 and $2..
the ratio of units is 2:1, so the weighted average will be 1+(2-1)* 1/(1+2)= 1.333
2) $1.5 and %1.2..
ratio of units = 1.5:1.2...
so weighted average= 1.2 + 1.5/(1.2+1.5)= 1.2+(1.5-1.2)*1.5/2.7= 1.2 + 3*1.5/27= 1.2+ 1/6= 1.2+ 0.1333= 1.333..

now we have two quantities with the same average rate= 1.333,
so their combined averge rate will also be 1.333

NOTE:- the answer will be lesser than the average of the cost, as while doing so we take the qty at each price to be same, whereas the quantities are higher for lower values..
so average< (1+1.5+1.2+2)/4= 1.425.. you can straight way eliminate C and E..

AndyNeedsGMAT · Answer

Hi Karishma,
Is there any mathematical principal for this ?

Abhishek009 · Answer

Since equal dollars is spent on electricity each year we take the LCM of 1, 1.2, 1.5 & 2 ; ie - 6

So, We get the factory spent $6 dollar each year ( Year 1 - Year 4 ), so during the 4 years they spent $ 24

Now, calculate the unit of electricity consumed.

Unit of electricity consumed = Total Cost /Cost per unit of electricity

Now, calculate the consumption Year wise-

Year 1 = 6/1 =>6 units

Year 2 = 6/1.5 => 4 units

Year 3 = 6/1.2 => 5 units

Year 4 = 6/2 => 3 units

Total consumption during the four years is ( 6+4+5+3) 18 units.

We are asked to find -

Average cost of a unit of electricity = Total cost of electricity during the 4 years / Total Consumption during the 4 years.

Average cost of a unit of electricity = 24 /18 =>1.33
  
Hence answer is (A)

mvictor · Answer

bumped again into this question..started to consider some coefficients for units used..but then...considered 6$ as total spent. I believe this is the LCM of 1, 1.5, 1.2, and 2.
1st year - 6 units
2nd year - 4 units
3rd year - 5 units
4th year - 3 units

total = 18 units. total $ spent = 24$. 24/18 = 4/3 = 1.33

ScottTargetTestPrep · Answer

We can assume the factory spent $18 a year on electricity. Thus, it used

Year 1: 18 units of electricity

Year 2: 12 units of electricity

Year 3: 15 units of electricity

Year 2: 9 units of electricity

Therefore, the average cost per unit is:

(4 x 18)/(18 + 12 + 15 + 9)

72/54

4/3

1.333

Answer: A

sayan640 · Answer

Since the factory spent equal dollars on electricity each year ,
let x dollar be the amount spent.
So in the first year :-
dollar spent = x $
unit bought= x

in the second year :-
dollar spent = x$
units bought = x/1.5 ( Since 1.5 dollar can buy 1 unit ; 1 dollar can buy 1/1.5 unit ; x dollar can buy x/1.5 units )

in the third year :-
dollar spent = x$
units bought = x/1.2

in the fourth year :-
dollar spent = x$
units bought = x/2

Total money spent = 4*x $

Total units bought = x + x/1.5 + x/1.2 + x/2 = 3*x units

Now,
3*x units cost you 4*x $
1 unit costs you 4*x / 3*x = 4/3 dollar = 1.33 dollar

So , the average cost of a unit of electricity during the 4 year period = 1.333 dollar
Option A is the answer.

KarishmaB · Answer

Responding to a pm:

Harmonic mean is a kind of average. In situations where same amount of money is spent on items with different costs, HM is used.

When you need to average prices a, b and c:

HM = \frac{N}{1/a + 1/b + 1/c}

N is the total number of elements for which you need to find the average.

Now consider that you spent equal amount of money $D all four years at diff rates

Average Rate = Total money/Total units consumed

Total number of units consumed = D/a + D/b + D/c

Total money = 3*D

Average rate =  \frac{3D}{D/a + D/b + D/c} = \frac{3}{1/a + 1/b + 1/c}

This is the same as the calculation of HM. Hence, we can do it directly if we know where to use it.

MathRevolution · Answer

Let the company spent $60 each year

Year 1:  \frac{60}{ 1}  = 60 units

Year 2:  \frac{60}{ 1.5}  = 40 units

Year 3:  \frac{60}{ 1.2}  = 50 units

Year 4:  \frac{60}{ 2}  = 30 units

Total units: 60 + 40 + 50 + 30 = 180

Total money spent: 60 * 4 = 240

Average:  \frac{240 }{ 180}  =  \frac{4 }{ 3}  = 1.33

Answer A

Basshead · Answer

We're told the factory spent equal dollars on electricity each year. Lets say that amount is $6 each year.

Year 1: 6 units
Year 2: 4 units
Year 3: 5 units
Year 4: 3 units

Total: 18 units
Total amount spent $24

$24/18 = 1.333

Answer is A.

ocelot22 · Answer

Pick a $$ amount for the yearly amount spent on electricity. Get used to doing this. Many many problems require this kind of thing. $100 is a good amount. So over 4 years $400 will be spent. Divide the unit costs into the $100 to get units of electricity purchased. you should get that 100+66+83+50 units of electricity were purchased, so basically 300 units. Then $400/300 units = $4/3units = $1.33 per unit A

quetzalferry · Answer

For a more mathmatical approach, took me moment to figure this out.
Let Y be the amount spent per year, and let u_i be the number of unit spend on i year.

Y = u_1 * $1/unit
Y = u_2 * $1.5/unit
Y = u_3 * $1.2/unit
Y = u_4 * $2/unit

The total number of units spent will then be
u_1 + u_2 + u_3 + u_4 = Y/1 + Y/1.5 + Y/1.2 + Y/2 = 18Y/6 units = 3Y units
Total Spent = 4Y, Since Y is the amount you spend per year.

Average Cost = Total Spent / Total number of Unit
Average Cost = 4Y / 3Y  = 1.333

Thus $1.333/unit is the average cost. This question was really confusing.

C_K · Answer

Why can't we just take the simple average cost of all four years? Why is that not the right approach?

Krunaal · Answer

Please refer to text highlighted in red in this post: https://gmatclub.com/forum/the-cost-per ... l#p1645627

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