Simple Interest

Definition of Terms

• Lender/Creditor
  • person or institution who makes the funds available.
• Borrower/Debtor
  • person or institution who owes or avails funds from the lender
• Origin/Loan Date
  • date/time on which money is received by the borrower
• Maturity Date/Repayment Date
  • date/time on which the money borrowed is to be returned to the lender.
• Term
  • amount of time in years the money is borrowed or invested.
  • time elapsed (or the length of time) between the origin and maturity dates.

What is Interest

• Interest
  • It is the fee or charge that the borrower pays for borrowing money from the lender.

• Where is interest used?
  • Interest is used whenever we use borrow money from banks or financial institutions
  • Interest is also used whenever we put our money into savings accounts.
    • In this way, banks pay us interest by lending our money to them through savings accounts.

• Types of Interest
  • Simple Interest: It is interest that is computed from the principal.
  • Compound Interest: It is interest computed from the principal and also on the accumulated past interest.

Computing Simple Interest

• To compute simple interest, we use the formula:

• where:
  • Principal Value/Present Value
  • Interest
  • Interest Rate
  • Length of Term or Time (in years)
  • Future Value

Example Problems

• When computing problems involving interest, it is very crucial to locate the given and unknown values first.

Solving Problems Involving Interest

• When solving problems involving interest, we want to use the formula for the simple interest, which is:

Note

Some problems do not explicitly ask you for the interest, but remember that interest is simply the change in the initial deposited amount.

1. Mark invests ₱ in a high-yield savings account with interest rate for 3 years.
   How much interest did he gain after years?

Solution

Since Mark invests ₱50,000, this would be our initial value or the principal value.
Then, we convert our interest rate into decimal: .

We now apply the formula for the simple interest.

2. Tyrone borrowed a ₱ loan with a interest rate. He later paid the bank after 2 years. Aside from the ₱ he borrowed, how much more should he pay?

Solution

Since Tyrone made an initial borrow of ₱, this would be our principal value.
The bank has an interest rate of , which is equivalent to , and he paid the bank after years.
Since it asks us how much more should he pay aside from the ₱, this is simply asking us for the interest.

Therefore, we use the simple interest formula.

3. Sophia wants to invest ₱. A financial institution offers her two options:
   Option A has a interest rate with a maximum maturity date of years.
   Option B has a interest rate with a maximum maturity date of years.
   Which option should Sophia take and why?

Solution

We want to know which option has the best returns (interest) as much as possible.
To know this, we would need to calculate the interest for each option.
Since Sophia invested ₱, this should be her principal amount.

• If Sophia invests in Option A, she would get interest for years maximum.
• Calculating the interest for Option A alone, we get:

• On the other hand, if Sophia invests in Option B, she would get interest for years maximum.

• Calculating the interest for Option B, we get.

\begin{align*}
I &= Prt \
&= (300,000)(0.05)(8) \
&= \boxed{90,000}
\end{align*}

Solving Problems involving Principal, Rate, and Time

• When solving for the principal, rate or time, we simply just divide the interest to whatever else is given.

• In other words:

  • If we want to find the principal: divide interest by rate times time.
  • If we want to find the rate: divide interest by principal times time.
  • If we want to find the time: divide interest by principal times rate

• This gives us a formula for the principal value, rate, or the time, which is simply derived from the original interest formula.

4. I invested ₱ in a savings account with interest rate.
   How long should I keep the money if I want to gain ₱ worth of interest?

Solution

Since I invested ₱, this would be my principal.
The problem explicitly states that we earned ₱ worth of interest.
The bank’s interest rate is or .

Since the length of the term is missing, we would want to use the formula for the term, which is:

Substituting values, we get:

5. Suppose that you won ₱ and you invested it in a cooperative group for years. After years, you received an additional ₱ from your initial investment. Find the interest rate of the cooperative group.

Solution

Since you invested ₱, this will be your principal amount.
After 4 years, which is the term of the investment, you gained an additional ₱, which is your interest.

Since the interest rate is missing, we use the formula for the rate.
Substituting values, we get:

6. After investing in a company with a interest rate, Juliana had an additional ₱ of interest after 6 years. How much money did she initially invest?

Solution

First, we identify what is given in the problem.

• The interest rate is

Solving Problems Involving Future Value

• Future Value refers to the principal value plus the simple interest.
  • It is also sometimes called maturity value.

• Since we know that the simple interest is , we can simply substitute it into the formula to get an another formula for the Future Value.

• Therefore, the second formula for the future value involves direct computation without requiring us to compute the interest.

Interest vs. Future Value

Interest and Future Value is not the same!

• Interest is the change in the original/principal amount.
  • It refers to how much your original deposit/loan had grown.
  • It is often useful to think how much will you get from a savings account.
• Future Value is the amount with interest already added into it.
  • It is often useful to think of future value in terms of paying a loan.

7. Joachim took a ₱ loan from a friend. They agreed that for every year that he is due, he should pay more of the original amount. How much should Joachim pay his friend after years?

Solution

Since Joachim originally took a ₱ loan, this is his principal value.
Converting the interest rate into decimal, we get: .
Substituting the other values in our interest formula, we get:

Since this is a loan, Joachim will not only pay ₱ but he should also pay the ₱ he borrowed. Therefore:

8. Suppose that you won the lottery for ₱. If you were to place it in a high-yield savings account with interest rate per annum, how long will it take to triple the money?

Solution

Here, the principal amount would be the ₱.
Converting the interest rate yields .
To find the time it took to triple the money, we should know how much the future value is:

• Since we want to triple our money, our future is ₱ .

• To find the interest, we simply use the future value formula , but we subtract from to get the interest.

\begin{align*}
FV &= P + I \
I &= FV - I \
&= 3,000,000 - 1,000,000\
&= 2,000,000
\end{align*}

t &= \frac{I}{P \times r}\[1ex]
&= \frac{2,000,000}{(1,000,000)(0.125)}\[1ex]
&= \frac{2,000,000}{125,000}\[1ex]
&= \boxed{16}

• Sometimes, when the future value is given and the principal value is missing, it would be impossible to compute the interest using the traditional simple interest formula .
• However, we also have a second formula that computes the future value without computing the interest
  • We can use this to solve for the principal by dividing by .

9. After 20 years, Claire’s investment from a savings account with a interest rate grew to $. How much did she initially invest?

Solution

Finding the given and missing values, we see that:

• the principal value is missing,
• the length of term is years
• the interest rate is or
• the final value is .

Since the principal value is missing, the only way to solve this is to use the formula for the future value

Calculating Simple Interest in terms of Months

• When computing simple interest, we should always consider the length of time or term is always in terms of years.
• Therefore, if the length of the term is in months, we first convert it to years by dividing by 12.

10. John made a $ loan with interest as he was running short of his budget. He later paid it after only 2 months. How much did he pay all-in-all?

Solution

To find how much he paid all-in-all, we first have to determine the future value of his loan.

• His initial/principal value is $.
• His interest rate is .
• The length of term is months.
  • Convert it into years, we get

Substituting everything to our interest formula, we get:

Since he took a $ loan, he would also have to pay an extra $ more.

Note

You can also compute simple interest in terms of days rather than months.
Make sure that to convert it, you divide by instead.

Calculating Monthly Payments

• To compute the monthly payment, find the future value of the investment, then divide it how many months are there in a term.

11. Juan made a loan $ loan and plans to pay it after 2 years. If the interest rate is 2% how much will he pay all-in-all? How much will he pay if he plans to pay every month for years?

Solution

First, we calculate the future value by directly using the other formula.

• The principal is $.
• The length of term is 2 years.
• The interest rate is .
  Substituting into our formula, we get:

Now, we want to divide that based on how many months have passed over the length of the term.

• Since there are 12 months per year, and the length of the term is 2 years, then there are 24 months in total.

Dividing our future value by 24 gives us the monthly payment.

Therefore, he has to pay $195 each month, or a grand total of $4680.