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High School Algebra I: Help and Review25 chapters | 292 lessons

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Lesson Transcript

Instructor:
*Joseph Vigil*

In this lesson, find out what compound interest is and what makes it different from simple interest. Then learn the formula for calculating compound interest. Finally, compare both types of interest with a graph that shows the growth in interest.

I'm giving you a thousand dollars! Well, a thousand dollars of virtual money. But, I'm giving it to you on the condition that you have to keep it in a bank for six months. There are two banks nearby: Simple Savings & Loan and Compound Credit Union. They both have great customer service and similar services. The only difference is that Simple Savings & Loan offers 5% simple interest on its savings account while Compound Credit Union offers 5% compound interest.

What does that mean? Which bank should get your thousand dollars?

**Interest** is the money someone is paid at a specified rate for use of cash that has been lent. For example, say a bank advertises a 5% monthly interest rate for its regular savings account. That means that every **period**, or the amount of time in which the bank pays interest, it will pay that person 5% of the balance just for keeping their money there. Therefore, interest on a savings account increases the amount of money in the account over time. However, interest is also calculated on money that is borrowed. Individuals pay interest to lenders when they borrow money for car, home, and other types of loans.

This example will show money earned through interest by using a savings account. So to calculate the interest at Simple Savings, find 5% of $1,000. To do that, multiply $1,000 * 0.05. 5% of $1000 equals 50. This account will earn $50 every month.

Look at the table that shows the monthly interest for February through June. Simple interest would have the money growing fifty dollars per month.

Month | Previous Balance | Interest | New Balance |
---|---|---|---|

February | $1,000 | $1,000 * 0.05 = $50 | $1,050 |

March | $1,050 | $1,000 * 0.05 = $50 | $1,100 |

April | $1,100 | $1,000 * 0.05 = $50 | $1,150 |

May | $1,150 | $1,000 * 0.05 = $50 | $1,200 |

June | $1,200 | $1,000 * 0.05 = $50 | $1,250 |

Just by keeping the money in the bank, the individual earned $250 by June. That's fine. But, wait. When the Simple Savings bank calculates the interest, it keeps using the original value of $1,000. This is called **simple interest**, a kind of interest in which the bank keeps using the original balance to figure each period's interest.

In fact, the only thing happening is multiplying the original balance by the interest rate and by the number of periods that the money stays in the account. So the formula for calculating simple interest looks like:

*I* = *P* * *r* * *t*

where *P* is the **principal**, or the original balance; *R* is the interest rate percentage; and *T* is the number of periods the money stays in the account. So, at Simple Savings:

Interest = $1,000 * .05 * 5

But, wouldn't it be nice if the bank used the new, higher balance every month to calculate the interest?

Say, for example, that in March, instead of using the original balance of $1,000, the bank used the new balance of $1,050 to determine the monthly interest. To do this, multiply $1050 * 0.05. This equals $52.50. Even though this new interest is only $2.50 more than the old one, watch what happens.

March's new balance is now $1102.50. So to figure that month's interest, use that new, higher balance instead of the original $1,000 balance. See how this works in favor of the person making the interest? The interest builds up much more quickly because the bank is using a higher balance every month to calculate the new interest. This is called **compound interest**, a kind of interest in which the bank calculates interest based on the previous balance plus the last period's interest.

Look at the same table but with compound interest instead of simple interest:

Month | Previous Balance | Interest | New Balance |
---|---|---|---|

February | $1,000 | $1,000 * 0.05 = $50 | $1,050 |

March | $1,050 | $1,050 * 0.05 = $52.50 | $1,102.50 |

April | $1,102.50 | $1,102.50 * 0.05 = $55.13 | $1,157.63 |

May | $1,157.63 | $1,157.63 * 0.05 = $57.88 | $1,215.51 |

June | $1,215.51 | $1,215.51 * 0.05 = $60.78 | $1,276.29 |

So instead of $250 from the Simple Savings bank this account earned over $270 at Compound Credit Union. And since the interest snowballs with compound interest, the longer the money is kept in the account, the more quickly the interest grows.

Since the interest isn't constant, the formula for calculating compound interest is a little more complicated:

*A* = *P* (1 + (*r* / *n*) ) *nt*

In this formula *A* is the new balance, *P* is the principal, *r* is the interest rate percentage, *n* is the number of interest periods in a year, and *t* is the number of years the money stays in the account.

It's easier to understand the difference between simple and compound interests when it can be visualized. Assume an original balance of $1 at 5% interest per year. Look at a graph of simple interest versus compound interest.

A balance with simple interest shows **linear growth**, meaning it grows by the same amount each period. That's why the graph of simple interest is a straight line. A balance with compound interest, meanwhile, shows **exponential growth**, meaning it grows more quickly each period. That's why the graph of compound interest curves; it gets steeper as the money continues to stay in the account. Of course, the amount of the initial deposit also affects the rate of growth. The larger it is, the more quickly the balance will grow because there is more money there for the interest to build on.

When a bank offers** simple interest**, the interest each period is figured based on the account's original balance or principal. Therefore, the balance grows at the same rate every period. When a bank offers** compound interest**, it figures the interest each period based on the account's previous balance plus the interest gained in the last period. Therefore, the balance grows more quickly with each passing period. Accounts with compound interest always grow exponentially. This can be seen visually on a line graph where compound growth is a curved line that is steeper than the linear growth of a simple interest account. The amount of interest paid to an account is dependent on four factors:

- Initial balance
- Interest rate
- Interest type
- Time

While both kinds of interest end up increasing the balance, compound interest grows the account more quickly.

**Interest**: money someone is paid at a specified rate for use of cash that has been lent**Period**: amount of time during which the bank pays interest**Simple interest**: a kind of interest in which the bank keeps using the original balance to figure each period's interest**Linear growth**: the interest grows by the same amount each period**Exponential growth**: with compound interest, the amount grows more quickly each period

After tackling the lesson on compound interest, apply what you've learned to:

- Calculate for simple and compound interest
- Contrast simple and compound interest
- Highlight the four factors that affect the amount of interest paid to an account

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High School Algebra I: Help and Review25 chapters | 292 lessons

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