5. Exponential Functions

Lesson

It costs money to borrow money. The extra money that banks and other lenders charge us to borrow money is called interest. Interest may also refer to the additional money that is earned from investing money, such as into a savings accounts. There are different types of interest and in this lesson we are going to talk about **simple interest**.

Simple interest, or **flat rate interest**, describes a method of calculating interest where the interest amount is **fixed **(i.e. it doesn't change). The interest charge is always based on the original amount borrowed (or invested), and does not take into account any interest earned along the way (that is, interest on interest is not included).

Many financial institutions express their interest rates per year. For example, an interest rate might be given as $3%$3% per year.

To calculate simple interest, three quantities are involved: the principal amount $P$`P` that is invested (or borrowed), the number of time periods $t$`t`, and the interest rate of $r$`r` per time period. Simple interest is then calculated using the formula

$I=Prt$`I`=`P``r``t`

Note that sometimes $n$`n` is used instead of $t$`t` to represent the number of time periods.

To find the **total value** of the investment or loan after a given time period, we add the interest to the principal amount $P$`P`.

Summary

Simple interest is calculated as

$I=Prt$`I`=`P``r``t`

where $P$`P` is the principal amount invested (or borrowed), $r$`r` is the interest rate per time period, and $t$`t` is the number of time periods.

The total amount or value $A$`A`, earned after $t$`t` interest periods, is then calculated as

$A=P+I$`A`=`P`+`I`

a) Calculate the simple interest on a loan of $\$8580$$8580 at $2%$2% per year for $10$10 years. Round your answer to the nearest cent.

**Think:** We can substitute the values for the principal, interest rate and time periods.

**Do:**

$I$I |
$=$= | $Prt$Prt |

$=$= | $8580\times2%\times10$8580×2%×10 | |

$=$= | $8580\times0.02\times10$8580×0.02×10 | |

$=$= | $\$1716$$1716 |

b) What is the total value of the loan after $10$10 years?

**Think**: The total value is the original loan plus the interest that has accrued over $10$10 years.

**Do**: $1716+8580=\$10296$1716+8580=$10296

The interest on an investment of $\$3600$$3600 over $10$10 years is $\$2520.00$$2520.00. If the annual interest rate is $r$`r`, find $r$`r` as a percentage.

**Think:** What values do we know that we can substitute into the formula?

**Do:**

$I$I |
$=$= | $Prt$Prt |

$2520$2520 | $=$= | $3600\times r\times10$3600×r×10 |

$2520$2520 | $=$= | $36000r$36000r |

$r$r |
$=$= | $\frac{2520}{36000}$252036000 |

$r$r |
$=$= | $0.07$0.07 |

$r$r |
$=$= | $7%$7% |

For a simple interest rate of $6%$6% per year , calculate the number of years $T$`T` needed for an interest of $\$1174.20$$1174.20 to be earned on the investment $\$1957$$1957.

Give your answer as a whole number of years.

Enter each line of working as an equation.

When calculating simple interest for time periods that are not years, such as months, weeks or days, we need to make sure the interest rate and the time periods are expressed using the same period. For example, if the rate $r$`r` is given per year then the value for $t$`t` needs to be expressed in years too.

Remember!

$1$1 year | $=$= | $12$12 months |

$=$= | $52$52 weeks | |

$=$= | $4$4 quarters (of $3$3 months each) | |

$=$= | $365$365 days |

Calculate the simple interest generated on a loan of $\$3860$$3860 at a rate of $9%$9% per year for $13$13 months.

Round your answer to the nearest cent.

Calculate the simple interest earned on an investment of $\$7000$$7000 at $1.8%$1.8% per quarter for $9$9 years.

Give your answer to the nearest cent.

Calculate the simple interest earned on an investment of $\$5320$$5320 at $6%$6% per year for $95$95 weeks.

Assume that a year has $52$52 weeks.

Give your answer to the nearest cent.

Interpret expressions that represent a quantity in terms of its context. 'Limited to linear expressions and exponential expressions with integer exponents.

Interpret parts of an expression, such as terms, factors, and coefficients. 'Limited to linear expressions and exponential expressions with integer exponents

Write a function that describes a relationship between two quantities. 'Linear and exponential (integer inputs)

Determine an explicit expression, a recursive process, or steps for calculation from a context. 'Linear and exponential (integer inputs)

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.