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Mathematics

A computer purchased for ₹72900 loses two-third of its value every year. Its value is evaluated at the end of every year.

(1) Which of the following expressions gives the value of the computer (in ₹) after n years?

  1. 72900(23)n\dfrac{72900}{\Big(\dfrac{2}{3}\Big)^{n}}

  2. 2n×279003n\dfrac{2^{n} \times 27900}{3^{n}}

  3. 729003n\dfrac{72900}{3^{n}}

  4. 729002n×3n\dfrac{72900}{2^{n} \times 3^{n}}

(2) Find the value of the computer after 3 years.

  1. ₹8100
  2. ₹2430
  3. ₹5600
  4. ₹2700

(3) In how many years will the value of the computer be less than ₹200 ?

  1. 6 years
  2. 7 years
  3. 8 years
  4. 10 years

(4) By how much will the value of the computer reduce in 4 years ?

  1. ₹24300
  2. ₹1800
  3. ₹8100
  4. ₹72000

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Answer

(1) Given:

Initial Value = ₹72,900

Loss in value every year = 23\dfrac{2}{3}

Value remaining every year = 1−23=131 - \dfrac{2}{3} = \dfrac{1}{3} of the previous year's value.

Every year, the value is multiplied by 13\dfrac{1}{3}. After n years, the value is 72900×(13)n72900 \times (\dfrac{1}{3})^n, which is 729003n\dfrac{72900}{3^n}.

Hence, option 3 is the correct option.

(2) Value of the computer after 3 years = ?

Value of the computer after n years = 729003n\dfrac{72900}{3^n} \quad [From previous step]

By replacing the value of 'n' with 3, we get:

₹ 7290033=₹7290027\dfrac{72900}{3^3} = ₹ \dfrac{72900}{27}

= ₹2700

Hence, option 4 is the correct option.

(3) We know at 3 years, value = ₹2700. \quad [From previous step]

Value remaining every year = 1−23=131 - \dfrac{2}{3} = \dfrac{1}{3} of the previous year's value. \quad [From step 1]

∴ Value after 4 years = 13×2700=27003=₹900\dfrac{1}{3} \times 2700 = \dfrac{2700}{3} = ₹ 900

Value after 5 years = 13×900=9003=₹300\dfrac{1}{3} \times 900 = \dfrac{900}{3} = ₹ 300

Value after 6 years = 13×300=3003=₹100\dfrac{1}{3} \times 300 = \dfrac{300}{3} = ₹ 100

Since 100 < 200, it takes 6 years.

Hence, option 1 is the correct option.

(4) Value of the computer reduced in 4 years = ?

Value after 4 years = ₹ 7290034=₹7290081=₹900\dfrac{72900}{3^4} = ₹ \dfrac{72900}{81} = ₹ 900

Value of the computer reduced in 4 years = Initial value - Value after 4 years

Substituting the values in above, we get:

Value of the computer reduced in 4 years = ₹72900 - ₹ 900 = ₹ 72000

Hence, option 4 is the correct option.

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