Percent change and compound interest

About this lesson

Algebra I › Introduction to exponential functions

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Percent change and compound interest

Warm-up

Three quick questions from earlier lessons.

Discover

A 50-dollar game goes up by 10%. Which calculation gives the new price in one step?

  1. 50⋅1.150 \cdot 1.1
  2. 50⋅0.150 \cdot 0.1

Key idea

Percent change is a growth factor.

An increase of rr (as a decimal) multiplies by 1+r1 + r; a decrease multiplies by 1−r1 - r. Compound interest at yearly rate rr, compounded nn times a year: A=P(1+rn)ntA = P\left(1 + \tfrac{r}{n}\right)^{nt}; yearly: A=P(1+r)tA = P(1 + r)^t.

Worked example

800 dollars at 5% per year, compounded yearly, for 3 years.

1.051.05
Factor.
800⋅1.053800 \cdot 1.05^3
Three years.
≈926.10\approx 926.10 dollars
Calculator.

Your turn, guided

Your turn: the growth factor for a 7% increase.

1.071.07
1 + 0.07.

Practice ★

  1. A quantity grows by 9% each year. By what number is it multiplied each year (the growth factor)?
  2. A quantity grows by 16% each year. By what number is it multiplied each year (the growth factor)?
  1. 882 dollars are invested at 5% per year, compounded yearly. Write the value after 4 years (an exact expression is fine).

Watch out!

Practice ★★

  1. 660 dollars are invested at 4% per year, compounded monthly. Write the value after 2 years (an exact expression is fine).
  2. 1818 dollars are invested at 7% per year, compounded monthly. Write the value after 3 years (an exact expression is fine).
  1. Boss problem. Ana puts 1800 dollars in a savings account that pays 8% per year, compounded yearly.

Exit ticket

Three questions to check you have got it.

  1. A quantity grows by 9% each year. By what number is it multiplied each year (the growth factor)?
  2. A quantity grows by 16% each year. By what number is it multiplied each year (the growth factor)?
  3. A quantity grows by 34% each year. By what number is it multiplied each year (the growth factor)?

Summary card

Percent change and interest

Increase rr: × (1+r)(1 + r); decrease: × (1−r)(1 - r); compounding: rate and exponent per period.

Watch out: 1.1, not 0.1, for a 10% increase.

End of the lesson

Back to the chapter

Answers: Percent change and compound interest

A

  1. 109100\frac{109}{100}
  2. 2925\frac{29}{25}

B

  1. 8576612180000\frac{85766121}{80000}

C

  1. 33650272028774857920304120635356211338224302652032980013192114707158941350000000000000000000000000000000000000000000000\frac{3365027202877485792030412063535621133822430265203298001319211}{4707158941350000000000000000000000000000000000000000000000}
  2. 8826389678019491028561666037929750124492491838427862676103233692368874238245665042611416984282293575685580883930139377881943616213632296905944976064512000000000000000000000000000000000000000000000000000000000000000000000000\frac{88263896780194910285616660379297501244924918384278626761032336923688742382456650426114169842822935756855808839301}{39377881943616213632296905944976064512000000000000000000000000000000000000000000000000000000000000000000000000}

D

  1. 5248825\frac{52488}{25}

E

  1. 109100\frac{109}{100}
  2. 2925\frac{29}{25}
  3. 6750\frac{67}{50}

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