Find the number of smokers in that city 3 years before.

Question

As a result of publicity on smoking, the number of smokers is decreased by $6\frac{1}{4}\%$ every year in comparison to its previous year. If the number of smokers at present in a city is $33750$ then find the number of smokers in that city 3 years before.

Answer 1

• Recall: The formula for compound interest, including principal sum, is:

  $$A=P\left(1+\frac{r}{n}\right)^{nt},$$

  where 
• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• r = the annual interest rate (decimal)
• n = the number of times that interest is compounded per unit t
• t = the time the money is invested or borrowed for

Let the total number of the smoker was in that city 3 years before is $x$. That is $P=x$

The declined rate is $6\frac{1}{4}\%$, that is, $$r=-6\frac{1}{4}\%=-\frac{25}{4}\times \frac{1}{100}=-\frac{1}{16}$$

We use the minus sign in r to denote that the rate is declined. 

We are counting a number of smoker after the end of each year, that means, once in a year. In this case, $n=1$. After $3$ years, that is, for $t=3$, the current population is, $33750$. That means, $A=33750$. Plugging all value in the Compound interest formula we get

$$\begin{aligned}A=P\left(1+\frac{r}{n}\right)^{nt}&\implies 33750=x \left(1-\frac{1/16}{1}\right)^{1\cdot 3}\\ &\implies x\left(1-\frac{1}{16}\right)^{3}=33750\\ &\implies x=33750\times \frac{16\times 16\times 16}{15\times 15\times 15}=40960.\end{aligned}$$

Hence the total number of smokers in that city 3 years before is $40960$.