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Depreciation: exponential decay

Finance and economics, grade 11 (3 units) Lesson 4 of 14Learning

Depreciation: exponential decay

A car’s value

A car loses each year a percent of the value it had at the start of the year – depreciation. That is exponential decay: the same formula, with a decay factor.

The Cohen family bought a new car for 120,000 ₪.
The car’s value falls by 15% every year, so q=0.85\displaystyle {q}={0.85}.

  • After a year: 120000⋅0.85=102000\displaystyle {120000\cdot0.85}={102000} ₪.
  • After five years: 120000⋅0.855≈53244.64\displaystyle 120000\cdot0.85^{5}\approx53244.64 ₪ – less than half the price.
The graph of exponential decay

In exponential decay the value falls by fewer shekels each year, since 15% is taken from a value that keeps shrinking. The graph comes nearer and nearer to the x\displaystyle x-axis and never reaches it.

The car’s value, y=120000⋅0.85x\displaystyle {y}={120000\cdot0.85^{x}}:

y=120000⋅0.85x\displaystyle y=120000\cdot0.85^{x}
  • In the first year the car lost 120000⋅0.15=18000\displaystyle {120000\cdot0.15}={18000} ₪.
  • In the second year – only 102000⋅0.15=15300\displaystyle {102000\cdot0.15}={15300} ₪.
The percent lost over several years

The percent lost over several years together is found from the factor of all the years, not by adding the percents.

In two years the Cohens’ car’s value was multiplied by 0.852=0.7225\displaystyle {0.85^{2}}={0.7225}: 72.25% of the price is left, so it lost 27.75% – not 2⋅15%=30%\displaystyle {2\cdot15\%}={30\%}.

A smartphone

A phone loses value faster than the car: about 30% a year.
A phone bought for 4,000 ₪ is worth after three years:

4000⋅0.73=4000⋅0.343=13724000\cdot0.7^{3}=4000\cdot0.343=1372

Less than 35% of the price paid for it.

Exercise 1

A car was bought for 90,000 ₪.
Its value falls by 12% every year.
What will the car be worth after four years? Round to agorot.

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    Exercise 2

    Car A costs 150,000 ₪, and its value falls by 20% every year.
    Car B costs 120,000 ₪, and its value falls by 10% every year.

    a

    What will car A be worth after three years?

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      b

      What will car B be worth after three years?

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        c

        Someone plans to sell the car after three years – how much will they lose on each car? Which should they buy?

        Car A

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Car B

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            Exercise 3

            A laptop was bought for 5,000 ₪.
            Its value falls by 25% every year.

            a

            What will the laptop be worth after two years?

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              b

              What percent of its price did the laptop lose in two years?

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                Exercise 4

                A games console was bought for 3,000 ₪, and its value falls by 40% every year.
                Write its value after x\displaystyle x years as a function of x\displaystyle x.