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

Answer:

Monthly percent increase in revenue = 0.4273%.

Step-by-step 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

$(1 + i_{a})^{\frac{1}{m}}= 1 + i_{m}$

$i_{m} = (1 + i_{a})^{\frac{1}{m}}-1$

$i_{a}= 5.25*1/100 = 0.0525\\m = 12$

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

$i_{m}=$ 0.4273%

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