Question

- A two wheeler depreciates at 20% of its value every year. If present value of the same be 90000 rupees. its depreciated vealue would be 36864 after :​

Answer 1

Answer:

Solution:

➡ Original value = Rs.90000

➡ Depreciated value = Rs.36864

➡ Rate of depreciating = 20%

➡ Let's assume that the time period is x.

The formula we will be using to find is similar to that of the compound interest formula when it is compounded annually.

Formula:

$\sf{\implies\:New\:value=Original\:value\left(1-\dfrac{r}{100}\right)^x}$

Substituting the values we know in this equation:

$\sf{\longrightarrow36864=90000\left(1-\dfrac{20}{100}\right)^x}$

$\sf{\longrightarrow36864=90000\times\left(\dfrac{4}{5}\right)^x}$

$\sf{\longrightarrow\dfrac{36864}{90000}=\left(\dfrac{4}{5}\right)^x}$

Now, we need to simply the value in the LHS, preferably in simplified exponential form as shown below:

Taking the LHS:

$\sf{\longrightarrow\:\dfrac{36864}{90000}\div\dfrac{2}{2}=\dfrac{18432}{45000}}$

$\sf{\longrightarrow\:\dfrac{18432}{45000}\div\dfrac{2}{2}=\dfrac{9216}{22500}}$

$\sf{\longrightarrow\:\dfrac{9216}{22500}\div\dfrac{2}{2}=\dfrac{4608}{11250}}$

$\sf{\longrightarrow\:\dfrac{4608}{11250}\div\dfrac{2}{2}=\dfrac{2304}{5625}}$

$\sf{\longrightarrow\:\dfrac{2304}{5625}\div\dfrac{3}{3}=\dfrac{768}{1875}}$

$\sf{\longrightarrow\:\dfrac{768}{1875}\div\dfrac{3}{3}=\dfrac{256}{625}}$

Now, factorizing the numerator and the denominator of the fraction 256/625:

256 = 4 x 4 x 4 x 4

625 = 5 x 5 x 5 x 5

This means that:

$\sf{\longrightarrow\:\dfrac{256}{625}=\left(\dfrac{4}{5}\right)^4}$

Now, combining the simplified LHS and the RHS:

$\sf{\longrightarrow\:\left(\dfrac{4}{5}\right)^4=\left(\dfrac{4}{5}\right)^x}$

This means that:

$\sf{\longrightarrow\:\left(\dfrac{4}{5}\right)^4=\left(\dfrac{4}{5}\right)^4}$

So:

$\bf{x=4\:years}$

• The number of years for the value to be depreciated to Rs.36864 is 4 years.

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