Simple Interest PDF

Simple Interest Prepared by: Ms. Vanessa A. Torres  Learning Objectives In this module, you will spend at most 1 hour and 30 minutes to: a. Explain the concept of simple interest; b. Define the different terms used in computing simple interest; c. Compute for the s...

Simple Interest Prepared by: Ms. Vanessa A. Torres  Learning Objectives In this module, you will spend at most 1 hour and 30 minutes to: a. Explain the concept of simple interest; b. Define the different terms used in computing simple interest; c. Compute for the simple interest, and the maturity value of a loan; d. Differentiate and compute ordinary and exact interest; e. Differentiate and compute actual time and approximate time; and f. Compute for the principal, interest and rate.  Key Concepts In this module, you will meet the following concepts: a. Simple Interest b. Principal, Interest and Rate c. Ordinary Interest vs. Exact Interest d. Banker’s Rule vs. Calendar Year Rule e. Actual Time vs. Approximate Time f. Maturity Value  Table of contents Special Introduction to 01 Simple Interest 03 Considerations as to Time Finding the Simple Interest 02 Proper 04 Principal, Rate and Time 01 Introduction to Simple Interest What is Simple Interest? The charged amount by the lenders. This is the amount charged on the principal amount being lent (or borrowed) for a specified period of time. - When interest is computed only once regardless of the duration of the loan. - The amount charge on the entire principal for the entire length of time, regardless of that length of time.  Simple Interest and Maturity Value The formula for Simple Interest is expressed as follows: 𝐼 = 𝑃𝑅𝑇 Where: I = Simple Interest P = Principal R = Rate T = Time Principal is the amount borrowed or lent. Rate is the annual interest rate charged on the principal. This means that if the given interest rate is a monthly, bi- monthly, semi-monthly, quarterly or semi-annual rate, the rate must be first converted into an annual rate before computing the interest. When the problem is silent, the rate given is an annual rate. Time is the length or period of borrowing. Time is expressed in years or as a fraction of a year, if less than one year. After computing the interest on any given loan, the next step is to compute for its Maturity Value. This is the total amount that the borrower must pay (or the lender will receive) after adding interest on the principal. Hence: 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 (𝑀𝑉) = 𝑃 + 𝐼 OR 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 (𝑀𝑉) = 𝑃 + (𝑃𝑅𝑇) Where: P = Principal I = Interest The second formula of Maturity Value is derived by substituting the formula for simple interest. You may use either of the two formulas.  Getting the Annual Rate As mentioned, simple interest requires the use of an annual interest rate. Thus, if the interest rate given is for any equal period less than one year, it must be converted into an annual rate. Rate Periods in One Year Example Convert 1% monthly 12 – there are 12 into an annual rate: Monthly months in one year 1% x 12 = 12% Convert 1% semi- 24 – semi-monthly is monthly into an annual Semi-Monthly one-half of a month rate: 1% x 24 = 24%  Rate Periods in One Year Example Convert 1% bi- 6 – bi-monthly means monthly into an Bi-Monthly every two months annual rate: 1% x 6 = 6% 4 – one quarter of a Convert 1% quarterly Quarterly year is equal to 3 into an annual rate; months 1% x 4 = 4% Convert 1% semi- 2 – semi-annual annual into an annual Semi-Annual means half of the rate: year 1% x 2 = 2% Note: See example 4 on the next section for an illustration.  02 Simple Interest Proper  Finding the Simple Interest and Maturity Value There are other things to consider in computing the simple interest, as will be discussed in the next sections, aside from the basic formula. Here, we will tackle the basic formula of simple interest. Example 1: Where the period is one year Gorgon borrowed P25,000 at 6% simple interest for 1 year. How much will he pay at the end of 1 year?  Finding the Simple Interest and Maturity Value Solution: First, compute for the simple interest. 𝐼 = 𝑃𝑅𝑇 𝐼 = 𝑃25,000 𝑥 6% 𝑥 1 𝐼 = 𝑷𝟏, 𝟓𝟎𝟎 Observe that T or Time is expressed in years. This is to match the interest rate which is also expressed as an annual rate (or that rate applicable to a whole year). Since the amount was borrowed for 1 year, T = 1.  Finding the Simple Interest and Maturity Value Solution: Next, get the maturity value of the given loan. 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 (𝑀𝑉) = 𝑃 + 𝐼 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 𝑀𝑉 = 𝑃25,000 + 𝑃1,500 𝑴𝒂𝒕𝒖𝒓𝒊𝒕𝒚 𝑽𝒂𝒍𝒖𝒆 𝑴𝑽 = 𝑷𝟐𝟔, 𝟓𝟎𝟎 For emphasis, to get the maturity value, simply add the principal and interest. Finding the Simple Interest and Maturity Value Example 2: Where the period is less than one year Jaylen borrowed P3,000 at 14% for 6 months. How much will be the interest charged for this borrowing? What is the maturity value of the loan? Finding the Simple Interest and Maturity Value Example 2: Where the period is less than one year Solution: First, get the simple interest. 𝐼 = 𝑃𝑅𝑇 6 𝑚𝑜𝑛𝑡ℎ𝑠 𝐼 = 𝑃3,000 𝑥 14% 𝑥 12 𝑚𝑜𝑛𝑡ℎ𝑠 𝐼 = 𝑷𝟐𝟏𝟎 If Time is less than one year, it should be expressed as a fraction of a year. Since the time given is in months, we also need to look for the number of months in a year, which is 12. Thus, time should be proportional to the number of months in one year, which is then 6 expressed as 12 above. Finding the Simple Interest and Maturity Value Example 2: Where the period is less than one year Solution: Then, get the maturity value. 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 (𝑀𝑉) = 𝑃 + 𝐼 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 𝑀𝑉 = 𝑃3,000 + 𝑃210 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 𝑀𝑉 = 𝑷𝟑, 𝟐𝟏𝟎 Finding the Simple Interest and Maturity Value Example 3: Where the period is more than one year Jayson lent P600 to Brad at 12% interest for 2 years and 6 months. Find the interest due at the end of the period.  Finding the Simple Interest and Maturity Value Example 3: Where the period is more than one year Solution: Solution: You can also express I = PRT 𝐼 = 𝑃𝑅𝑇 T = 2.5, since 6 6 𝐼 = 𝑃600 𝑥 12% 𝑥 2 months is equal to 12 half a year or 0.5. 𝐼 = 𝑷𝟏𝟖𝟎 Finding the Simple Interest and Maturity Value Example 4: Where the rate is not an annual rate Marcus decided to borrow P10,000 from Tyler for a period of 2 years and 6 months. The prevailing interest rate was 3% per quarter. How much will Marcus have to pay at due date? Finding the Simple Interest and Maturity Value Example 4: Where the rate is not an annual rate Solution: First, convert the quarterly interest rate into an annual rate: 3% → 3% x 4 = 12% Then, compute the simple interest: 𝐼 = 𝑃𝑅𝑇 𝐼 = 𝑃10,000 𝑥 12% 𝑥 2.5 𝐼 = 𝑷𝟑, 𝟎𝟎𝟎 Finding the Simple Interest and Maturity Value Example 4: Where the rate is not an annual rate Solution: Finally, get the maturity value of the loan: 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 (𝑀𝑉) = 𝑃 + 𝐼 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 𝑀𝑉 = 𝑃10,000 + 𝑃3,000 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 𝑀𝑉 = 𝑷𝟏𝟑, 𝟎𝟎𝟎 03 Special Considerations as to Time The discussion in this section only affects T or Time in the simple interest formula. For the succeeding discussion, consider the following: 𝑝𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑜𝑎𝑛 𝑇𝑖𝑚𝑒 (𝑇) = 𝑑𝑎𝑦𝑠 𝑖𝑛 𝑎 𝑦𝑒𝑎𝑟 Where: a. Period of the loan can either be actual time or approximate time b. Days in a year can either be ordinary interest (banker’s rule) or exact interest (calendar year rule) Actual Time and Approximate Time If the problem does not state the period of the loan in terms of days, months or years, such as when the problem only gives a specific date, you have to compute for the T on your own. Actual Time counts the exact number of days in one year and in every month, which should be based on 365 days in one year (366 for leap year). On the other hand, Approximate Time assumes that each month only has 30 days each including February, for a total of 360 days in one year. Actual Time and Approximate Time Thus, it is important to know the number of days in each month for actual time. Here are the number of days in each month: January = October = April = 30 days July = 31 days 31 days 31 days February = August = November = 28 days (29 for May = 31 days 31 days 30 days leap years March = September = December = June = 30 days 31 days 30 days 31 days Actual Time and Approximate Time Again, you can use the “knuckle technique.” Actual Time and Approximate Time Example 1: Finding the Actual Time and Approximate Time Jimmy borrowed money from Kemba from April 22, 2019 up to August 25, 2019. Find the: actual time; and and approximate time. Actual Time and Approximate Time Example 1: Finding the Actual Time and Approximate Time Solution: To get the actual time: Actual Time and Approximate Time Example 1: Finding the Actual Time and Approximate Time Solution: To get the approximate time:  Actual Time and Approximate Time Example 2: Finding the Actual Time and Approximate Time Goran borrowed money from Duncan for the period October 18, 2017 and March 2, 2019. Find the actual time; and approximate time.  Actual Time and Approximate Time Example 2: Finding the Actual Time and Approximate Time Solution: To get the actual time:  Actual Time and Approximate Time Example 2: Finding the Actual Time and Approximate Time Solution: To get the approximate time: To emphasize, this is only relevant in determining the length of the borrowing, or the period of the loan. We are referring to the numerator of Time.  Actual Time and Approximate Time Example 2: Finding the Actual Time and Approximate Time Solution: To get the approximate time: To emphasize, this is only relevant in determining the length of the borrowing, or the period of the loan. We are referring to the numerator of Time. Ordinary Interest and Exact Interest Time is always measured in years or a fraction of a year. When the period of the loan is expressed in either days or a time between two specific dates, the year can only be expressed as a number of days. To convert T or Time into a fraction of a year, the denominator can either be 360 days or 365 days. Ordinary Interest, also known as the Banker’s Rule, assumes that each year has 360 days. This is used by banks and other financial institutions where they measure a business’s performance in terms of months and not in days. 𝑃𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑜𝑎𝑛 𝒊𝒏 𝒅𝒂𝒚𝒔 𝐼 = 𝑃𝑥𝑅𝑥 𝟑𝟔𝟎 𝒅𝒂𝒚𝒔 Ordinary Interest and Exact Interest Exact Interest, also known as the Calendar Method or Calendar Year Method, assumes that each year has 365 days. Hence, it uses the exact number of days in any given year. 𝑃𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑜𝑎𝑛 𝒊𝒏 𝒅𝒂𝒚𝒔 𝐼 = 𝑃𝑥𝑅𝑥 𝟑𝟔𝟓 𝒅𝒂𝒚𝒔 Ordinary Interest and Exact Interest Example 1: Finding the Ordinary Interest and Exact Interest Benimaru borrowed P1,000 from Joker at 8% interest for 90 days. Compute for the simple interest using ordinary interest; and exact interest. Ordinary Interest and Exact Interest Example 1: Finding the Ordinary Interest and Exact Interest Solution: To get the ordinary interest: Ordinary Interest and Exact Interest Example 1: Finding the Ordinary Interest and Exact Interest Solution: To get the exact interest: Ordinary Interest and Exact Interest Example 2: Complex Problem Reigen borrowed P283,000 on May 12, 2020, due on August 27, 2020, with interest at 11.5%. Find the interest on the loan using Banker’s Rule; and Calendar Rule. Ordinary Interest and Exact Interest Example 2: Complex Problem Solution: To get the simple interest under the Banker’s Rule or Ordinary Interest: Comprehensive Problem On March 15, 2020, Elfie borrowed from Grey P500 at an interest rate of 7%. The due date is on May 15, 2020. Compute the following: a. Ordinary interest using ordinary time b. Ordinary interest using exact time c. Exact interest using exact time d. Exact interest using ordinary time Comprehensive Problem Solution: a. Ordinary Interest and Ordinary Time Comprehensive Problem Solution: b. Ordinary Interest and Exact Time Comprehensive Problem Solution: c. Exact Interest and Exact Time Comprehensive Problem Solution: d. Exact Interest and Ordinary Time  04 Finding the Principal, Rate and Time A. Finding the Principal When the Principal is missing, the simple interest formula can be modified to get the derived formula to get the principal: 𝐼 𝑃 = 𝑅𝑇 Use either of the OR formulas 𝑀𝑉 depending on 𝑃 = what is given. 100% + 𝑅𝑇 A. Finding the Principal Example 1: Sayonara paid P108 in interest on a loan that she had for 6 months from Gomenasai. The interest rate was 12%. How much was the principal? A. Finding the Principal Solution: 𝐼 𝑃 = 𝑅𝑇 𝑃108 𝑃 = 6 𝑚𝑜𝑛𝑡ℎ𝑠 12% 𝑥 12 𝑚𝑜𝑛𝑡ℎ𝑠 𝑃 = 𝑷𝟏, 𝟖𝟎𝟎 A. Finding the Principal Example 2: A loan has a maturity value of P3,657.50. Interest rate charged was 9% with a term of 6 months. Look for the principal. A. Finding the Principal Solution: 𝑀𝑉 𝑃 = 100% +𝑅𝑇 𝑃3,657.50 𝑃 = 6 100% +(9% 𝑥 ) 12 𝑃 = 𝑷𝟑, 𝟓𝟎𝟎 B. Finding the Principal The formula to get the interest rate derived from the simple interest formula is as follows: 𝐼 𝑅 = 𝑃𝑇 B. Finding the Principal Example 1: Shaun wanted to borrow P1,500 from Adrian for 15 months. He was told that he would have to pay an interest of P225. What is the rate Shaun is being charged with? B. Finding the Principal Solution: 𝐼 𝑅 = 𝑃𝑇 𝑃225 𝑅 = 15 𝑃1,500 𝑥 12 𝑅 = 𝟏𝟐% B. Finding the Principal Example 2: Momo lent P8,000 to Jojo. She received P9,000 at the end of one year. What was the interest rate of the loan? B. Finding the Principal Solution: You can get the simple interest as the difference between the Maturity Value and Simple Interest: 𝐼 = 𝑃9,000 − 𝑃8,000 𝐼 = 𝑃1,000 Now, compute for the interest rate: 𝐼 𝑅 = 𝑃𝑇 𝑃1,000 𝑅 = 𝑃8,000 𝑥 1 𝑅 = 𝟏𝟐. 𝟓% C. Finding the Time The formula to get the time or period of the loan can be derived from the simple interest formula: 𝐼 𝑇 = 𝑃𝑅 C. Finding the Time Example: Akira borrowed P10,000 from Shiba at 8%, for which he paid P1,600 as interest at the end of the loan. What was the length of the loan? C. Finding the Time Example: Akira borrowed P10,000 from Shiba at 8%, for which he paid P1,600 as interest at the end of the loan. What was the length of the loan? C. Finding the Time Solution: 𝐼 𝑇 = 𝑃𝑅 𝑃1,600 𝑇 = 𝑃10,000 𝑥 8% 𝑇 = 𝟐 𝒚𝒆𝒂𝒓𝒔  That’s all for Today! CREDITS: This presentation template was created by Slidesgo, and includes icons by Flaticon, and infographics & images by Freepik