Question

The value of purchase of a commodity is $25000\,{\text{dollars}}$. This value is decreased $12\%$ per year. Calculate the value of purchase after 1yr, 3yr, 5yr and 7yrs.

Answer 1

We are given a value of purchase of a commodity and we have to simply find the decrement it will have in certain amount of time i.e. in 1yr, 3yr, 5yr and 7yrs so we use the basic successive percentage method to find the decrement for that time period that is given by the formula ${\text{Decreased}}\,{\text{value}}\,{\text{ = }}\,{\rm P}{\left( {1 - \dfrac{{\text{R}}}{{100}}} \right)^{\rm T}}$ ; Where P = Value of commodity, R = Rate of decrement, T=time.

Complete solution step by step:
Firstly, we write down all the given information in our question
Value of purchase of a commodity (in dollars) $= 25000$
Decrement rate = $12\%$
Time = 1, 3, 5, and 7 years
So we solve the decreased amount one by one for different times i.e.
The value of purchase after 1 year

$$\Rightarrow 25000{\left( {1 - \dfrac{{12}}{{100}}} \right)^1} = 25000\left( {\dfrac{{22}}{{25}}} \right) \\\ = 1000 \times 22 \\\ = 22000 \\\$$

So after one year our value of purchase will reduce to 22000 dollars.
The value of purchase after 3 years

$$\Rightarrow 25000{\left( {1 - \dfrac{{12}}{{100}}} \right)^3} = 25000{\left( {1 - \dfrac{3}{{25}}} \right)^3} \\\ = 25000{\left( {\dfrac{{22}}{{25}}} \right)^3} \\\ = \dfrac{{40 \times 10648}}{{25}} \\\ = \dfrac{{85184}}{5} \\\ = 17036.8 \\\$$

So after three years the value of purchase will reduce to 17036.8 dollars.
The value of purchase after 5 years

$$\Rightarrow 25000{\left( {1 - \dfrac{{12}}{{100}}} \right)^5} = 25000{\left( {1 - \dfrac{3}{{25}}} \right)^5} \\\ = 25000{\left( {\dfrac{{22}}{{25}}} \right)^5} \\\ = 13193.29792 \approx 13193.298 \\\$$

So after five years the value of purchase will reduce to 13193.298 dollars.
The value of purchase after 7 years

$$\Rightarrow 25000{\left( {1 - \dfrac{{12}}{{100}}} \right)^7} = 25000{\left( {1 - \dfrac{3}{{25}}} \right)^7} \\\ = \dfrac{{1000 \times {{\left( {22} \right)}^7}}}{{{{\left( {25} \right)}^6}}} \\\ = 10216.889 \approx 10216.89 \\\$$

So after seven years the value of purchase will reduce to 10216.89 dollars.

Note: Successive decrement or increment can be used with the same formula used in the question i.e. ${\text{Increased}}\,{\text{or}}\,{\text{Decreased}}\,{\text{value = }}\,{\rm P}{\left( {1 \pm \dfrac{{\text{R}}}{{100}}} \right)^{\rm T}}$ ; Where positive sign is used for increment of value and negative is used for decrement. Successive increment is also known as Compound Interest on a sum of money.