Math Lesson 2.1: Simple Interest PDF

Sorry eto lang nakayanan ko 😣 MATH LESSON 2.1: SIMPLE INTEREST Key Terms and Meanings: 1. Principal (P): The initial amount of money borrowed or invested. It is also called the face value or present value. 2. Rate (r): The annual interest rate, usually written as...

Sorry eto lang nakayanan ko 😣 MATH LESSON 2.1: SIMPLE INTEREST Key Terms and Meanings: 1. Principal (P): The initial amount of money borrowed or invested. It is also called the face value or present value. 2. Rate (r): The annual interest rate, usually written as a percentage, charged by the lender or earned from an investment. Note: Always convert the percentage to decimal form (e.g., 8% = 0.08). 3. Time (t): The length of time the money is borrowed or invested, usually measured in years. ○ If time is in months: Divide by 12 (e.g., 6 months = 6/12 = 0.5 years). ○ If time is in days: Use 360 days for ordinary interest and 365 days for exact interest. 4. Interest (I): The amount paid or earned for using money. It depends on the principal, rate, and time. 5. Maturity Value (A): The total amount to be paid or received at the end of the loan or investment period. It includes both the principal and the interest. 6. / =over or divide or fraction Formulas to Remember: For Simple Interest: Interest: I=Prt For Maturity Value: Maturity Value: A=P+I  OR A=P(1+rt) Other Derived Formulas: 1. Principal: P=A/1+rt 2. Rate: r=A−P/Pt 3. Time: t=A−P/Pr Ordinary vs. Exact Interest: 1. Ordinary Interest: Based on a 360-day year (used by banks). Formula for time: t=days/360 2. Exact Interest: Based on a 365-day year. Formula for time: t=days/365 Examples for Practice: Problem 1: Anne borrowed ₱120,000 at 8% annual interest for 1 year. How much is the interest? Given: Principal (P) = ₱120,000 Rate (r) = 8% = 0.08 Time (t) = 1 year Solution: I=Prt=(120,000)(0.08)(1)=₱9,600  Problem 2: Eunice lent ₱3,000 at 14% for 6 months. How much interest did she earn? Given: Principal (P) = ₱3,000 Rate (r) = 14% = 0.14 Time (t) = 6 months = 6/12=0.5 Solution: t=6/12=0.5 years I=Prt=(3,000)(0.14)(0.5)=₱210 Problem 3: Anna invested $2,500 at 5% annually. How long until she earns $1,125 in interest? Given: Principal (P) = $2,500 Rate (r) = 5% = 0.05 Interest (I) = $1,125 Solution: t=I/Pr=1,125/(2,500)(0.05)=9 years Problem 4.1 (Ordinary Interest): You get a 180-day ₱200,000 loan from a bank at a 10.5% interest rate. Calculate the interest using ordinary interest (360-day year). Given: Principal (P) = ₱200,000 Rate (r) = 10.5% = 0.105 Time (t) = 180 days = 180/360=0.5 Solution: t=180/360=0.5 years I=Prt=(200,000)(0.105)(0.5)=₱10,500 Problem 4.2 (Exact Interest): You get a 180-day ₱200,000 loan from a bank at a 10.5% interest rate. Calculate the interest using exact interest (365-day year). Given: Principal (P) = ₱200,000 Rate (r) = 10.5% = 0.105 Time (t) = 180 days = 180/365≈0.493 Solution: t=180/365≈0.493 years I=Prt=(200,000)(0.105)(0.493)=₱10,356.16 Problem 5 (Maturity Value): Eunice lent ₱3,000 at 14% for 6 months. How much will she pay at the end of the term? Given: Principal (P) = ₱3,000 Rate (r) = 14% = 0.14 Time (t) = 6 months = 6/12=0.5 Solution: A=P(1+rt)=(3,000)[1+(0.14)(0.5)]=₱3,210 Summary of Tips: 1. Converting Rates and Time: ○ Convert percentages to decimals for rates. ○ Convert months/days to years. 2. Deciding Between Ordinary and Exact Interest: ○ Use 360 days for ordinary interest and 365 days for exact interest. 3. Using Formulas for Different Scenarios: ○ To find interest: I=Prt ○ To find maturity value: A=P(1+rt) ○ For principal, rate, or time, rearrange formulas based on the given values. LESSON 2.6 Propositions and Logic What is a Proposition? A proposition is a statement that can either be true or false, but not both at the same time. Example: ○ True Proposition: "The sun rises in the east." ○ False Proposition: "2 + 2 equals 5." Types of Propositions: 1. Simple Proposition: A simple statement that expresses one clear idea. There are no connecting words like "and," "or," or "if." ○ Example: "She is happy." (This is a simple proposition because it states a single fact). 2. Compound Proposition: A statement made up of two or more simple propositions that are joined together using connecting words like AND, OR, IF...THEN, etc. ○ Example: "It is raining, and I am going to the store." (This is a compound proposition because it combines two simple propositions: "It is raining" and "I am going to the store" with the word "and".) Connecting Simple Propositions (Compound Propositions) 1. Conjunction (AND): ○ The two propositions must both be true for the whole statement to be true. ○ Key word: AND ○ Symbol: ^  ○ Example: "It is sunny and I am going to the park." Both parts of this statement must be true for the whole thing to be true. If either "It is sunny" or "I am going to the park" is false, the entire statement is false. 2. Disjunction (OR): ○ At least one of the two propositions must be true for the whole statement to be true. ○ Key word: OR ○ Symbol: ∨ ○ Example: "I will go to the beach or the pool." If either of the two options (beach or pool) happens, the statement is true. It doesn’t matter if one is false. 3. Conditional (If...Then): ○ If the first proposition (if part) is true, then the second part (then part) will also be true. ○ Key words: If...then ○ Symbol: → ○ Example: "If you study hard, then you will pass the exam." If the first part (studying hard) is true, then the second part (passing the exam) will also happen. But if the first part is false, the whole statement can still be true (this is a tricky part of conditional statements). 4. Biconditional (If and Only If): ○ The two propositions must either both be true or both be false. If one is true and the other is false, the whole statement is false. ○ Key words: If and only if ○ Symbol: ↔ ○ Example: "You will pass the exam if and only if you study." This means you need to study to pass (if you study, you will pass), and if you don’t study, you won’t pass. 5. Negation (Not): ○ This means denying or saying that something is false. It turns a true statement into a false one, and a false one into a true one. ○ Key word: Not ○ Symbol: ~  ○ Example: "It is not raining." If the statement "It is raining" is true, then "It is not raining" will be false. If "It is raining" is false, then "It is not raining" will be true. Logical Symbols & Their Meanings: Connective Symbol What it Means Example And (Conjunction) ^ Both parts must be true. "It is sunny and I am going to the park." Or (Disjunction) ∨ At least one part must be "I will eat pizza or pasta." true. If...Then (Conditional) → If the first part is true, the "If it rains, then we stay second part follows. indoors." If and Only If ↔ Both parts are either true or "You will pass if and only (Biconditional) false. if you study." Not (Negation) ~ Makes a statement false. "It is not true that I am hungry." Examples with Symbolic Representation: 1. Example 1: "He has a green thumb and he is a senior citizen." ○ Let pp = "He has a green thumb" ○ Let qq = "He is a senior citizen" ○ Symbolic Representation: p∧q (This means both p and q must be true.) 2. Example 2: "He does not have a green thumb or he is not a senior citizen." ○ Let p = "He has a green thumb" ○ Let q = "He is a senior citizen" ○ Symbolic Representation: ∼p∨∼q (This means at least one part must be false.)  3. Example 3: "It is not the case that he has a green thumb or he is a senior citizen." ○ Symbolic Representation: ∼(p∨q) (This means neither p nor q is true.) 4. Example 4: "If he has a green thumb, then he is not a senior citizen." ○ Symbolic Representation: p→∼q (This means if p is true, q must be false.) 5. Example 5: "If he has a green thumb, then he is a senior citizen; and, if he is a senior citizen, then he has a green thumb." ○ Symbolic Representation: p↔q (This means both p and q are either true or false together.) Quick Review of the Key Connectives: Conjunction (AND): Both parts must be true. ○ Example: "I am studying and I am learning." Disjunction (OR): At least one part must be true. ○ Example: "I will have tea or coffee." Conditional (If...Then): If the first part is true, the second part happens. ○ Example: "If it rains, then I will take an umbrella." Negation (Not): Saying something is false. ○ Example: "It is not true that I am hungry." Biconditional (If and Only If): Both parts must either be true or false together. ○ Example: "You will pass the test if and only if you study." Practice Problems: 1. Write the following in symbols: "If it rains, then we will stay indoors." Answer: p→q 2. Write the following in symbols: "He has a green thumb but he is not a senior citizen."  Answer: p∧∼q 3. Write the following in symbols: "It is not the case that he has a green thumb or he is a senior citizen." Answer: ∼(p∨q) Summary of Key Points: Conjunction (AND): Both parts are true (^). Disjunction (OR): One or both parts are true (∨). Conditional (If...Then): The first part leads to the second part (→). Negation (Not): Saying something is false (~). Biconditional (If and Only If): Both parts must be equal (↔).