Mathematics In The Modern World - Module 7 - PDF

MATH 111 – MATH. IN THE MODERN WORLD 1  MATH 111 – MATH. IN THE MODERN WORLD Introduction Mathematics is not just about numbers, much of it is problem solving and reasoning. Problem solving and reasonin...

MATH 111 – MATH. IN THE MODERN WORLD 1  MATH 111 – MATH. IN THE MODERN WORLD Introduction Mathematics is not just about numbers, much of it is problem solving and reasoning. Problem solving and reasoning are basically inseparable. The art of reasoning is very important in Mathematics. This is the skill needed in exemplifying the critical thinking and problem-solving ability. Logic and reasoning are very useful tools in decision making. People also do deductive reasoning extensively to show that certain conjectures are true as these follow the rules of logic. Chapter 7 MATHEMATICS OF FINANCE Learning Outcomes At the end of this topic, the students must have: 1. formulated the given in the if – then statements 2. gave examples in the different kinds of reasoning 3. solved problems using different strategies Content Simple and Compound Interest Interest is the amount paid for the use of money. It is also paid for the money borrowed from a bank. The money deposited or borrowed is called the principal (P), the money paid for its use is called interest (I), and how much is to be paid per P100 is called the rate of interest (r) including the time (t) involved usually given in years. Simple Interest It is the interest that does not become part of the principal. The formula is I = Prt or A = P(1 + rt) where P is the Principal amount of money to be invested at an Interest Rate r% per period for t, Number of Time Period 𝑰 = 𝑷𝒓𝒕 Example 1. A 2-year loan of $500 is made with 4% simple interest. Find the interest earned. Solution: Always take a moment to identify the values given in the problem. Here are given: Time is 2 years: t = 2 Initial amount is Php500: P = 500 The rate is 4%. Write this as a decimal: r = 0.04 Now apply the formula: 𝑰 = 𝑷𝒓𝒕 = (500) (0.04) (2) = 40 2  MATH 111 – MATH. IN THE MODERN WORLD Example 2. A business takes out a simple interest loan of Php10,000 at a rate of 7.5%. What is the total amount the business will repay if the loan is for 8 years? Solution: The total amount they will repay 1s the future value, A. we are also given that: t=8 r = 0.075 P = 10000 Using the simple interest formula for future value: A = P(l+rt) = 10000(1+0.075(8)) = Php16000. Compound Interest It is the interest that the sum becomes the new principal amount The formula for annual compound interest, including principal sum, is: 𝒓 𝒏𝒕 𝐀 = 𝑷 (𝟏 + ) 𝒏 Where: A = the future value of the investment/loan, including interest P = the principal investment amount (the initial deposit or loan amount) r = the annual interest rate (decimal) n = the number of times that interest is compounded per year t = the number of years the money is invested or borrowed for Example 1. If an amount of Ᵽ5,000 is deposited into a savings account at an annual interest rate of 5, compounded monthly, the value of the investment after 10 years. This can be calculated as follows. Solution: Given: P = 5000 r = 5/100 = O.05 (decimal) n = 12 t = 10 If we plug those figure into the formula, we get the following (note that ^ indicates to the power of) 𝟎. 𝟎𝟓 (𝟏𝟐)(𝟏𝟎) 𝐀 = 𝟓𝟎𝟎𝟎 (𝟏 + ) 𝟏𝟐 A=5235.05 So, the investment balance after 10 years is Ᵽ5,235.05. The theories and concepts on simple and compound interest are applicable in business transactions when people are in to credit and loans. The charging of interest may vary according to the terms and conditions specified and agreed by both parties. The following are some of the transactions where a type of interest is applied. Credit Cards and Consumer Loans According to Latoya Irby (2019), a credit card is plastic card that lets you access the credit limit your credit card issuer gives you. A credit limit is like a loan. However, instead of giving you the full 3  MATH 111 – MATH. IN THE MODERN WORLD loan in cash, the bank lets you take a much of the credit as you want as a time and allows you 20 reuse she loan over and over as long as you pay what you’ve borrowed. Janet Berry Johnson (2019) discussed that APR (annual percentage rate) is one of the factors you should consider when shopping for a credit card. Finding the lowest rate available to you means comparing offers and card terms carefully. Here's what you should look for. Introductory/promotional APR. Many cards offer an introductory APR. usually 0 percent on balance transfers or purchases for anywhere from a few months to a year. This can be super helpful, but make sure you read the terms and conditions and may off your balance before the APR jumps up to its regular rate Regular APR. After the introductory period, most cards offer a range of variable APRs depending on your creditworthiness. Generally speaking, the lower end of the APR range is reserved for consumers with good to excellent credit. On the other side of the token, the higher APRs are for consumers at the lower end of eligible credit scores. Your actual rate will be determined by the issuer when you apply, but looking at your credit scores before applying may give you a better idea of what to expect Cash advance APR. Banks and issuers typically charge a higher rate for cash advances, and interest accrues the moment you take the advance - sorry, no grace period here. For this reason, we recommend avoiding credit card cash advances whenever possible. Penalty APR. If you miss a payment, the credit card company may raise your rate in addition to charging you a late fee. Talk about adding insult to injury. Consumer loan means a secured or unsecured loan given to customers for personal family, or household purposes, or for consumable items such as a car, boat manufactured home, home equity loan, home equity line of credit, signature loan, signature line of credit, and recreational vehicle. It is usually given on the basis of borrower's integrity and ability to pay. It is also called consumer lending, consumer credit, or retail lending. In general, the higher your credit score, the lower the rate will be. Individuals with excellent credit, which is defined as any FICO credit score between 720 and 850, should expect to find rates at about 10% to 12%, and many of these individuals may even qualify for lower rates Lender's interest rates on unsecured personal loans typically range between 5% and 36%. Consumer loans must comply with the consumer protection regulation and they are monitored by government regulatory agencies Consumer loan does not include mortgage loans used for home purchases and commercial loans. Stocks Bonds, and Mutual Funds From Brittney Castro (2013), a stock is ownership in a company. When you buy a stock, you buy a piece of the company. So if the company does well, you do well. Congruently, if the company tanks, your stock tanks. Just like bonds, there are many types of stocks because there are many different types of companies out there. Large company stocks (large cap), mid cap stock, small cap stock, international stock, emerging stock, tech stock. The best way to describe a bond is to think of it like a loan. You loan your money to the government of a company, and in return they pay you interest for the term of that loan. Typically, bonds are considered conservative types of investments because you can choose the length and term of the bond and know exactly how much money you will get back at the end of the form or maturity There are many types of bonds; government bonds, corporate bonds, short-term bonds, long term bonds, municipal and inflation protected bonds, etc. For example, the trade in the country was referring to the March 12 auction of reissued five years T-bonds, where the Bureau of the Treasury raised PIS billion as planned from the securities with a remaining life of four years and 10 months. The bonds fetched an average rate of 4.132%. Mutual funds represent another way to invest in stocks, bond, or cash alternatives. You can think of a mutual fund like a basket of stocks or bonds Basically, your money is pooled, along with the money of other investors, into a fund, which then invests in certain securities according to a stated investment strategy. The fund is managed by a fund manager who reports to a board of directors Let's assume you wish to invest P100,000 in a Mutual Fund You checked and saw that the mutual fund's NAVPS (Net Asset Value per Share) price is currently P1.75. Given this NAVPS value, if you invest P100,000 you will receive 57,142 shares of this mutual fund, computed as follows: P100,000 divided by P1.75 = 57,142 shares. Your total investment value that day is P99,995 50. computed as follows: 57,142 shares x P1.75 NAVPS = P99,998.50 Home Ownership 4  MATH 111 – MATH. IN THE MODERN WORLD Owner-occupancy or home ownership (Wikipedia) is a form of housing tenure where a person called the owner occupier, owner-occupant, or home owner owns the home in which he/she lives. This home can be house, apartment condominium, or a housing cooperative. In addition to providing housing owner occupancy alto functions as a real estate investment. Owner-occupancy or home ownership is a form of housing tenure where person called the owner-occupier, owner-occupant, or home owner, owns the home in which he/she lives. This home can be house, apartment, condominium, or a housing cooperative. In addition to providing housing, owner-occupancy alto functions as a real estate investment. For example, basically, the key rates of Bangko Sentral ng Pilipinas influence housing loan interest rates in the Philippines. Currently, the Base Lending Rate is recorded at 6% which means your interest will be 3.5% on a (BLR-2.5%) loan. Additional Information Simple Interest Interest is often associated to loan. However, interest is not only for loans it is also for investments gain. Interest is the payment made by the person or entity for the use of borrowed money. Money can be from a loan or an investment. When only the principal bears the interest for the entire term is called simple interest. Compute using the formula: 𝑰 = 𝑷𝒓𝒕 where I is interest P is principal r is rate t is time Derivation of Formulas 𝑰 𝑷= for Principal value 𝒓𝒕 𝑰 𝒕 = 𝑷𝒓 for time 𝑰 𝒓 = 𝑷𝒕 for rate The following terms will be used in the discussion: 1. Principal- the amount receive by the debtor 2. Time/term- number of unit of time for which the interest is computed 3. Rate of interest- the fractional part of the principal that is paid on the loan. 4. Final amount or maturity value- the sum of the principal and interest, which is accumulated at a certain time. 5. Lender- the person who loans the money. 6. Debtor- the person who borrows money for any purpose. Example 1: Find the interest of ₱25,300 investment at 6% interest rate for 1 year. Solution: Given: t=1 year P= ₱25,300 5  MATH 111 – MATH. IN THE MODERN WORLD r= 6% Solve for: I (interest) Formula: 𝑰 = 𝑷𝒓𝒕 = (25,300) (.06) (1) = ₱1,518 Example 2: 3 Find the interest of A ₱65,850 investment at 6.72% interest rate for 2 4 years. Solution: 3 Given: t=2 4 year P= ₱65,850 r= 6.72% Solve for: I (interest) Formula: 𝑰 = 𝑷𝒓𝒕 3 = (65,5850) (.0672) (2 4 ) = ₱12,169.10 Example 3: 1 Find the interest of A ₱ 21, 320 investment at 7.5% interest rate for 1 years. 2 Solution: 1 Given: t= 12 year P= ₱ 21, 320 r= 7.5% Solve for: I (interest) Formula: 𝑰 = 𝑷𝒓𝒕 1 = (21,320) (.075) (12 ) = ₱ 2,398.50 6