Question

Last year, the total revenue for Home Style, a national restaurant chain, increased 5.25% over the previous year. If this trend were to continue, which expression could the company’s chief financial officer use to approximate their monthly percent increase in revenue? [Let m represent months.]

Answer 1

Monthly percent increase in revenue = 0.4273%.

Explanation:

Let m = number of months in time period.

We have that: Annual increase based on annual rate = Annual increase based on monthly rate * 12.

To find the increase in the original amount, we use the formula:

`A_(0)(1 + i)^(n)`, where n = number of time periods and i = interest rate.

Therefore,

Ao(1 + ia) = Ao(1 + im)m,

`A_(0)(1 + i_(a) )^{n_(a) } = A_(0)(1 + i_(m) )^{n_(m) }`

where Ao= original amount, ia = annual interest rate, im = monthly interest rate, na = number of annual time periods, nm = number of monthly time periods.

`n_(a)=1; n_(m)=m`

`A_(0)(1 + i_(a) ) = A_(0)(1 + i_(m) )^(m)`

Divide both sides by Ao

`(1 + i_(a)) = (1 + i_(m) )^(m)`

Taking the mth root of both sides

`(1 + i_(a))^{(1)/(m)}= 1 + i_(m)`

`i_(m) = (1 + i_(a))^{(1)/(m)}-1`

`i_(a)= 5.25*1/100 = 0.0525\\m = 12`

`i_(m) = (1 + 0.0525)^{(1)/(12)}-1\\i_(m) = (1.0525)^{(1)/(12)}-1 = 1.004273 - 1\\i_(m) = 0.004273 * 100\\i_(m) = 0.4273`

`i_(m)=` 0.4273%

Thus, the monthly revenue increase of Home Style is 0.4273%.