# 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?

Jul 10, 2018

I need some clarification.

#### Explanation:

1st month $\to$ $72 million 3rd month $\to18 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$$72 x$ million

So 3rd month is $$\left(72 x\right) x$million =$18?

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}_{0} {e}^{k t}$$18 = 72 {e}^{3 k}$$\ln \frac{\frac{1}{4}}{3} = k$A_t=72e^((ln(1/4))^(t/3) ${A}_{t} = 72 {\left(\frac{1}{4}\right)}^{\frac{t}{3}}$${A}_{t} = 72 {\left(\frac{1}{\sqrt[3]{4}}\right)}^{t}$So $\left(\frac{1}{\sqrt[3]{4}}\right)$seems to be the fraction we seek. Testing this: looking at$18 M. after 3 months.

$\frac{18}{72} = \frac{1}{4}$

We would expect after 6 months the amount to be:

$\frac{1}{4} \cdot 18 = \frac{9}{2}$

And:

${\left(\frac{1}{\sqrt[3]{4}}\right)}^{6} = \frac{9}{2}$

$\frac{1}{2}$

#### 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:

$\frac{18}{72} = \frac{1}{4}$

Since this represents the total decline over 2 months, by observation we can see the fraction is $\frac{1}{2}$:

${x}^{2} = \frac{1}{4} \implies x = \frac{1}{2}$

Testing this, we should see sales declining by $\frac{1}{2}$ each month:

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