Chapter 5.7, Problem 62AYU

Inflation Problems 57-62 require the following discussion. Inflation is a term used to describe the erosion of the purchasing power of money. For example, if the annual inflation rate is $3\%$, then $\$1000$ worth of purchasing power now will have only $\$970$ worth of purchasing power in 1 year because $3\%$ of the original $\$1000\left(0.03\times 1000=30\right)$ has been eroded due to inflation. In general, if the rate of inflation averages $r$ per annum over $n$ years, the amount $A$ that $\$P$ will purchase after $n$ years is $A=P·{(1-r)}^n$ where $r$ is expressed as a decimal.

Inflation If the average inflati on rate is $4\%$, how long is it until purchasing power is cut in half?

To determine

To find: Inflation is a term used to describe the erosion of the purchasing power of money. For example, if the annual inflation rate is $3\%$, then $\$1000$ worth of purchasing power now will have only $\$970$ worth of purchasing power in 1 year because $3\%$ of the original $\$1000$ $\left(0.03\times 1000=30\right)$ has been eroded due to inflation. In general, if the rate of inflation averages $r$ per annum over $n$ years, the amount A that $\$P$ will purchase after $n$ years is $A=P\cdot {(1-r)}^n$ where $r$ is expressed as a decimal. Inflation If the average inflation rate is $4\%$, how long is it until purchasing power is cut in half?

Answer to Problem 62AYU

Solution:

$16.98$ years

Explanation of Solution

Given:

$A=\frac{P}{2}, r=4\%=0.04$

Calculation:

$\frac{P}{2}=P{\left(1-0.04\right)}^n$

$\frac{1}{2}={\left(1-0.04\right)}^n$

${\left(0.96\right)}^n=0.5$

Taking $\text{log}$ on both sides

$\log {\left(0.96\right)}^n=\log 0.5$

$n\log 0.96=\log 0.5$

$n=\frac{\log 0.5}{\log 0.96}$

$n=16.98$

Therefore, the purchasing power is cut in half after $16.98$ years.