During the first month after the opening of a new shopping home, sales were $72 million. Each subsequent month, sales declined by the same fraction. If the sales during the third month after the opening totaled $18 million, what is that fraction?

3 Answers
Jul 10, 2018

I need some clarification.

Explanation:

1st month #-># $72 million
3rd month #-># $18 million

I assume that you are saying that every month, the sales drop in this manner:

1st month is $72 million,

(Letting fraction = #x#, )

2nd month is $#72x# million

So 3rd month is $#(72x)x# million = $#18?#

If so, please give me the thumbs up to go ahead..

Because it can also mean a fraction the original $72 million was removed in the 2nd month.

Jul 10, 2018

#color(blue)(1/(root(3)(4))#

Explanation:

If we model this on an exponential function:

#A_t=A_0e^(kt)#

#18=72e^(3k)#

#ln(1/4)/3=k#

#A_t=72e^((ln(1/4))^(t/3)#

#A_t=72(1/4)^(t/3)#

#A_t=72(1/(root(3)(4)))^t#

So #(1/(root(3)(4)))# seems to be the fraction we seek.

Testing this:

looking at $18 M. after 3 months.

#18/72=1/4#

We would expect after 6 months the amount to be:

#1/4*18=9/2#

And:

#(1/(root(3)(4)))^6=9/2#

#1/2#

Explanation:

I'm reading the question this way - We have sales in Month 1 of $72M, month 3 of $18M, and an equal fraction decline from months 1 to 2 and 2 to 3. What's that fraction?

The first thing I'd do is to look at the fraction decline from 72 to 18:

#18/72=1/4#

Since this represents the total decline over 2 months, by observation we can see the fraction is #1/2#:

#x^2=1/4=>x=1/2#

Testing this, we should see sales declining by #1/2# each month:

Month 1: #$72M#
Month 2: #1/2xx$72M=$36M#
Month 3: #1/2xx$36M=$18M#