Question

The cost of conventional medications for diabetes has risen by an average of 13.1% per year over the past four years. During this same period of time, the cost of a specialty drug for rheumatoid arthritis (RA) has increased 22.5% per year. If these increases continue into the foreseeable future, in how many years will an RA medicine now costing $24 per dose be at least five times as expensive as a diabetes inhaler that now costs $11 per use? The minimum number of years required is(Round up to the nearest whole number.)

Answer 1

Let n be period in consideration.

Case of RA medicines

Current price of RA medicine=Po=$24

Growth rate=g=22.5%

Period=n

Future price of RA medicine=P1=Po*(1+g)^n=24*(1+22.5%)^n=24*(1.225)^n

Case of diabetes inhaler

Current price of diabetes inhaler=Po=$11

Growth rate=g=13.1%

Period=n

Future price of diabetes inhaler=P1=Po*(1+g)^n=11*(1+13.1%)^n=11*(1.131)^n

We are given

24*(1.225)^n=5*11*(1.131)^n

(1.225/1.131)^n=2.291667

Take natural log both sides

n*Ln(1.225/1.131)=ln(2.291667)

n=ln(2.291667)/Ln(1.225/1.131)=10.39 years

It will take 11 years.

(If we round the time to 10 years, objective is not met. So, we have taken time as 11 years)