Question

A computer valued $1500 loses 20% of its value each year. How do you write a function rule that models the value of the computer?

Answer 1

Hint: Here we will consider the price of the computer as P. Then after one year 20% loss will be deducted from the original price. Similarly, after one more year 20% will be deducted from the new price. This will be the new price. Now this will be the equation of the model to find the value of the computer.

Complete step-by-step answer:
Let the price of the computer be P.
 \[P = 1500\]
After one year the price will drop by 20% of the original.
Thus, after one year the price will be,
 \[{P_1} = 1500 - 1500 \times 20\% \]
 \[{P_1} = 1500\left( {1 - 20\% } \right)\]
20% can be written as,
 \[{P_1} = 1500\left( {1 - \dfrac{{20}}{{100}}} \right)\]
Taking LCM,
 \[{P_1} = 1500\left( {\dfrac{{100 - 20}}{{100}}} \right)\]
 \[{P_1} = 1500\left( {\dfrac{{80}}{{100}}} \right)\]
This is the price after one year. After one more year 20% will lose;
 \[{P_2} = {P_1} - 20\% {P_1}\]
Substituting the value of \[{P_1}\] ,
 \[{P_2} = 1500\left( {\dfrac{{80}}{{100}}} \right) - 20\% \times 1500\left( {\dfrac{{80}}{{100}}} \right)\]
Taking the common numbers,
 \[{P_2} = 1500\left( {\dfrac{{80}}{{100}}} \right)\left[ {1 - 20\% } \right] \]
20% can be written as,
 \[{P_2} = 1500\left( {\dfrac{{80}}{{100}}} \right)\left[ {1 - \dfrac{{20}}{{100}}} \right] \]
Taking the LCM,
 \[{P_2} = 1500\left( {\dfrac{{80}}{{100}}} \right)\left[ {\dfrac{{100 - 20}}{{100}}} \right] \]
 \[{P_2} = 1500\left( {\dfrac{{80}}{{100}}} \right)\left[ {\dfrac{{80}}{{100}}} \right] \]
Now this can be written as,
 \[{P_2} = 1500{\left( {\dfrac{{80}}{{100}}} \right)^2}\]
So this can be expressed in model rule as,
 \[{P_n} = 1500{\left( {\dfrac{{80}}{{100}}} \right)^n}\]
So, the correct answer is “ \[{P_n} = 1500{\left( {\dfrac{{80}}{{100}}} \right)^n}\] ”.

Note: Here note that, the price depreciates by 20% every year. So we will deduct 20% of the original from the original value and not the 20% of the price only. Again don’t find the value, keep the values as it is. Since we are going to build the rule so we will calculate the pricing for n such years. Now we can find the value of the computer after any year we want.