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As a result of inflation, the purchasing power of a fixed quantity of money decreases by $5\%$ every year. If a certain quantity of money can buy $940$ frozen dinners this year, how many frozen dinners can it buy in $16$ years? $\newline$ If necessary, round your answer to the nearest whole number. $\newline$ ____ frozen dinners $\newline$

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Exponential growth and decay: word problems

Full solution

Q. As a result of inflation, the purchasing power of a fixed quantity of money decreases by

5\%

every year. If a certain quantity of money can buy

940

frozen dinners this year, how many frozen dinners can it buy in

16

years?

\newline

If necessary, round your answer to the nearest whole number.

\newline

____ frozen dinners

\newline

Identify initial quantity and rate: Identify the initial quantity and the rate of decrease. $\newline$ The initial quantity of frozen dinners that can be bought is $940$ . The purchasing power decreases by $5\%$ every year.
Determine type of change: Determine the type of change. $\newline$ The purchasing power decreases by a fixed percentage each year, which indicates an exponential decay.
Find values of $a$ and $r$ : Find the values of $a$ and $r$ . $a = 940$ (initial quantity) $r = 0.95$ (since there is a $5\%$ decrease, the remaining purchasing power each year is $95\%$ of the previous year)
Use exponential decay formula: Use the exponential decay formula. $\newline$ The formula for exponential decay is $P(t) = a \cdot r^{(t)}$ , where $P(t)$ is the quantity after time $t$ , $a$ is the initial quantity, $r$ is the rate of decay, and $t$ is the time in years.
Calculate in $16$ years: Calculate the number of frozen dinners that can be bought in $16$ years. $\newline$ Substitute $940$ for $a$ , $0.95$ for $r$ , and $16$ for $t$ into the formula: $\newline$ $P(16) = 940 \times 0.95^{16}$
Evaluate the expression: Evaluate the expression. $\newline$ $P(16) = 940 \times 0.95^{16}$ $\newline$ $P(16) = 940 \times (0.95^{16})$ $\newline$ $P(16) \approx 940 \times 0.481$ $\newline$ $P(16) \approx 452.14$
Round to nearest whole number: Round the answer to the nearest whole number. $\newline$ Since we cannot buy a fraction of a frozen dinner, we round $452.14$ to the nearest whole number, which is $452$ .

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