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Math  /  Algebra

QuestionIf a computer depreciates at a rate of 18%18 \% per year, what is the monthly Item 17 depreciation rate?

Studdy Solution

STEP 1

1. The annual depreciation rate of the computer is 18%18\%.
2. We are looking for the equivalent monthly depreciation rate.
3. Depreciation is compounded monthly.

STEP 2

1. Convert the annual depreciation rate to a decimal.
2. Use the formula for monthly depreciation rate based on annual rate.
3. Solve for the monthly depreciation rate.
4. Convert the monthly rate back to a percentage.

STEP 3

Convert the annual depreciation rate to a decimal.
The annual depreciation rate is 18%18\%, which is equivalent to 0.180.18 in decimal form.

STEP 4

Use the formula for monthly depreciation rate based on annual rate.
The formula to convert an annual rate rr to a monthly rate rmr_m when compounded monthly is:
(1−r)=(1−rm)12 (1 - r) = (1 - r_m)^{12}

STEP 5

Solve for the monthly depreciation rate.
Substitute r=0.18r = 0.18 into the formula:
(1−0.18)=(1−rm)12 (1 - 0.18) = (1 - r_m)^{12}
0.82=(1−rm)12 0.82 = (1 - r_m)^{12}
To solve for rmr_m, take the 12th root of both sides:
1−rm=0.82112 1 - r_m = 0.82^{\frac{1}{12}}
rm=1−0.82112 r_m = 1 - 0.82^{\frac{1}{12}}

STEP 6

Calculate the value of rmr_m.
Using a calculator, compute:
0.82112≈0.9847 0.82^{\frac{1}{12}} \approx 0.9847
rm=1−0.9847 r_m = 1 - 0.9847
rm≈0.0153 r_m \approx 0.0153

STEP 7

Convert the monthly rate back to a percentage.
The monthly depreciation rate as a percentage is:
rm×100%≈0.0153×100%≈1.53% r_m \times 100\% \approx 0.0153 \times 100\% \approx 1.53\%
The monthly depreciation rate is approximately:
1.53% \boxed{1.53\%}

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