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Calculate the annual rate of depreciation if an item's value depreciates from shs. 800,000 to shs. 550,000 in 4 years.

Mathematics

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Answer

8.94%8.94\%

Step-by-step solution

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3. Depreciation Rate
The formula for depreciation is A=P(1−r)nA = P(1 - r)^n, where AA is the final value, PP is the initial value, rr is the annual rate of depreciation, and nn is the number of years.

  1. Identify the given values.

    Initial price, P=shs. 800,000P = \text{shs. } 800,000
    Final price, A=shs. 550,000A = \text{shs. } 550,000
    Number of years, n=4n = 4

  2. Substitute the values into the formula.

    550,000=800,000(1−r)4550,000 = 800,000(1 - r)^4

  3. Isolate (1−r)4(1 - r)^4.

    550,000800,000=(1−r)4\frac{550,000}{800,000} = (1 - r)^4 5580=(1−r)4\frac{55}{80} = (1 - r)^4 1116=(1−r)4\frac{11}{16} = (1 - r)^4

  4. Take the fourth root of both sides to solve for (1−r)(1 - r).

    1−r=(1116)141 - r = \left(\frac{11}{16}\right)^{\frac{1}{4}} 1−r=1141641 - r = \frac{\sqrt[4]{11}}{\sqrt[4]{16}} 1−r=11421 - r = \frac{\sqrt[4]{11}}{2}

    Calculate 114\sqrt[4]{11}:

    114≈1.82115\sqrt[4]{11} \approx 1.82115 1−r≈1.8211521 - r \approx \frac{1.82115}{2} 1−r≈0.9105751 - r \approx 0.910575

  5. Solve for rr.

    r=1−0.910575r = 1 - 0.910575 r=0.089425r = 0.089425

  6. Express the rate as a percentage.

    r=0.089425×100%r = 0.089425 \times 100\% r≈8.94%r \approx 8.94\%

    The annual rate of depreciation is 8.94%\boxed{8.94\%}.

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