This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

- Welcome
- Learning Math Strategies (Online)Toggle Dropdown
- Study Skills for MathToggle Dropdown
- Simply Math
- Business Math
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Basic Laws
- Operations on Signed numbers
- Order of Operations
- Fractions
- Decimals
- Percents
- Ratios and Proportions
- Exponents
- Statistics
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Solving Systems of Linear Equations
- Trade and Cash Discounts
- Multiple Rates of Discount
- Payment Terms and Cash Discounts
- Markup
- Simple Interest
- Compound Interest
- Nominal and Effective Interest Rates
- Ordinary Simple Annuities
- Ordinary General Annuities

- Hospitality MathToggle Dropdown
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Order of Operations
- Basic Laws
- Prime Factorisation and Least Common Multiple
- Fractions
- Decimals
- Percents
- Exponents
- Units of Measures
- Fluid Ounces and Ounces
- Metric Measures
- Yield Percent
- Recipe Size Conversion
- Ingredient Ratios
- Food-Service Industry Costs

- Engineering MathToggle Dropdown
- Basic Laws
- Order of Operations
- Prime Factorisation and Least Common Multiple
- Fractions
- Exponents
- Radicals
- Reducing Radicals
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Areas and Volumes of Figures
- Congruence and Similarity
- Functions
- Domain and Range of Functions
- Basics of Graphing
- Transformations
- Graphing Linear Functions
- Graphing Quadratic Functions
- Solving Systems of Linear Equations
- Solving Quadratic Equations
- Solving Higher Degree Equations
- Trigonometry
- Graphing Trigonometric Functions
- Graphing Circles and Ellipses
- Exponential and Logarithmic Functions
- Complex Numbers
- Number Bases in Computer Arithmetic
- Linear Algebra
- Calculus
- Set Theory
- Modular Numbers and Cryptography
- Statistics
- Problem Solving Strategies

- Upgrading / Pre-HealthToggle Dropdown
- Basic Laws
- Place Value in Decimal Number Systems
- Decimals
- Significant Digits
- Prime Factorisation and Least Common Multiple
- Fractions
- Percents
- Ratios and Proportions
- Exponents
- Radicals
- Reducing Radicals
- Metric Conversions
- Factoring
- Solving Linear Equations
- Solving Quadratic Equations
- Functions
- Domain and Range of Functions
- Polynomial Long Division
- Exponential and Logarithmic Functions
- Statistics

- Nursing MathToggle Dropdown
- Arithmetic Operations
- Order of Operations
- Place Value in Decimal Number Systems
- Decimals
- Fractions
- Percents
- Ratios and Proportions
- Nutrition Labels
- Interpreting Drug Orders
- Oral Dosages
- Dosage Based on Size of the Patient
- Parenteral Dosages
- Intravenous (IV) Administration
- Infusion Rates for Intravenous Piggyback (IVPB) Bag
- General Dosage Rounding Rules

- Transportation MathToggle Dropdown
- PhysicsToggle Dropdown

What is an **annuity**? It is a series of equal payments which occurs at equal time intervals - usually monthly, quarterly, semi-annually or annually. There are two important terms of annuities - **ordinary **and **simple**.

**Ordinary Annuity : **Has payments at the end of each time interval. There is *no payment *at the start of the term of the annuity.

**Simple Annuity : **Frequency of the payment is the same as the frequency of the compound interest.

Below are the definitions of the terms we use for ordinary simple annuities:

\( S_n \ or \ FV = \) future value at \( n \) payment

\( \ \ \ \ \ \ \ \ \ \ \ \ \ R = \) regular payment amount

\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = \frac{j}{m} = \) periodic rate of interest

\( \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j = \) nominal annual rate of interest

\( \ \ \ \ \ \ \ \ \ \ \ \ \ m = \) compounded frequency *daily: \( m = 365 \); monthly: \( m = 12 \); quarterly \(m = 4\)*

\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ n = (length \ of \ period) \times m = \) time

\(A_n \ or \ PV = \) present value at \( n \) payment

The **future value** (\( FV \) or \( S_{n} \)) of an annuity is the value of a current asset at a future date based on an assumed rate of growth. The **future value** is important to investors and financial planners as they use it to estimate how much an investment made today will be worth in the future.

\[ S_{n} = \frac{R[(1+i)^n - 1]}{i} \]

`Example 1`

Suppose you invested $1000 per quarter over a 15 year period. If money earns an annual rate of 6.5% compounded quarterly, what would be the value at the end of the time period?

`Solution`

\( \Longrightarrow S_{n} = ? \) | \( R = $1000 \) | \( i = \frac{6.5\%}{4} = 0.01625 \) | \( n = 15*4 = 60 \)

\( \Longrightarrow S_{60} = \frac{1000[(1+0.01625)^{60} - 1]}{0.01625} = $10, 033.67 \)

The **present value **of an annuity is the amount of cash today equivalent in value to a payment, or to a stream of payments, to be received in the future. It is calculated using the following formula:

\[ A_{n} = \frac{R[1- (1+i)^{-n}]}{i} \]

`Example 2`

Suppose you with to be able to withdraw $3000 at the end of each month for two years. How much money must you deposit now at 2.75% interest compounded monthly?

`Solution`

\( \Longrightarrow A_{n} = ? \) | \( R = $3000 \) | \( i = \frac{2.75\%}{12} = 0.002291666 \) | \( n = 2*12 = 24 \)

\( \Longrightarrow A_{24} = \frac{3000[1-(1+0.002291666)^{-24}]}{0.002291666} = $69, 977.66 \)

Since there are different formulas for the present and future values of annuities. The first step is to determine whether the given value of the question is the future or present value. If we rearranged the formulas to solve for the payment, we have the following.

\[R=\frac{i\cdot S_n}{\left(1+i\right)^n-1}\]

\[R=\frac{i\cdot A_n}{1-\left(1+i\right)^{-n}}\]

Let's try the following example.

`Example`

How much would you have to pay into an account at the end of every 6 months to accumulate $10,000 in eight years if interest is 3% compounded semi-annually?

`Solution`

To solve any annuity, we need to pick out the important pieces of information from the question.

- Payment is at the end of the period which implies this is an
**ordinary annuity**. - Payment is every 6 months and compounding is semi-annual. Since it is the same this is a
**simple annuity**. - The nominal interest rate, j, is 3% compounded semi-annually.
- Thus, the periodic interest rate is \(i=\frac{0.03}{2}=0.015\).
- There are payments every 6 months for eight years, so \(n=16\).
- We are trying to accumulate to $10,000. Since this is a value after all the payments, this is the
**future value.**

\begin{align}R&=\frac{i\cdot S_n}{\left(1+i\right)^n-1}\\&=\frac{0.015\cdot 10,000}{\left(1+0.015\right)^{16}-1}\\&=557.65\end{align}

Therefore, a regular payment of $557.65 every 6 months for eight years will be required to accumulate to $10,000.

Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.

- Last Updated: Sep 22, 2023 2:04 PM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
- Print Page

chat loading...