business math problem solving

Business Math: A Step-by-Step Handbook

(0 reviews)

Jean-Paul Olivier

Copyright Year: 2017

Last Update: 2021

Publisher: Lyryx

Language: English

Formats Available

Conditions of use.

Table of Contents

PART 1: Mathematic Fundamentals for Business

• Chapter 1: How To Use This Textbook
• Chapter 2: Back to the Basics
• Chapter 3: General Business Management Applications

PART 2: Applications in Human Resources, Economics, Marketing, and Accounting

• Chapter 4: Human Resource & Economic Applications
• Chapter 5: Marketing and Accounting Fundamentals
• Chapter 6: Marketing Applications
• Chapter 7: Accounting Applications

PART 3: Applications in Finance, Banking, and Investing: Working With Single Payments

• Chapter 8: Simple Interest: Working With Single Payment & Applications
• Chapter 9: Compound Interest: Working With Single Payments
• Chapter 10: Compound Interest: Applications Involving Single Payments

PART 4: Applications in FInance, Banking, and Investing: Working With A Continuous Series of Payments (Annuities)

• Chapter 11: Compound Interest: Annuities
• Chapter 12: Compound Interest: Special Applications of Annuities

PART 5: Applications in Finance, Banking, and Investing: Amortization and Special Concepts

• Chapter 13: Understanding Amortization & Its Applications
• Chapter 14: Bonds and Sinking Funds
• Chapter 15: Making Good Decisions

Ancillary Material

About the book.

Business Mathematics was written to meet the needs of a twenty-first century student. It takes a systematic approach to helping students learn how to think and centers on a structured process termed the PUPP Model (Plan, Understand, Perform, and Present). This process is found throughout the text and in every guided example to help students develop a step-by-step problem-solving approach.

This textbook simplifies and integrates annuity types and variable calculations, utilizes relevant algebraic symbols, and is integrated with the Texas Instruments BAII+ calculator. It also contains structured exercises, annotated and detailed formulas, and relevant personal and professional applications in discussion, guided examples, case studies, and even homework questions.

About the Contributors

Jean-Paul Olivier has been teaching Business and Financial Mathematics, as well as Business Statistics and Quantitative Methods for the past 21 years. He is a dedicated instructor interested in helping his students succeed through multi-media teaching involving PowerPoints, videos, whiteboards, in-class discussions, readings, online software, and homework practice. He regularly facilitates these quantitative courses and leads a team of instructors. You may have met Jean-Paul at various mathematical and statistical symposiums held throughout Canada (and the world). He has contributed to various mathematical publications for the major publishers with respect to reviews, PowerPoint development, online algorithmic authoring, technical checks, and co-authoring of textbooks. This text represents Jean-Paul’s first venture into solo authoring

Contribute to this Page

• Math Article

Business Mathematics

Business Mathematics consists of Mathematical concepts related to business. It comprises mainly profit, loss and interest. Maths is the base of any business. Business Mathematics financial formulas, measurements which helps to calculate profit and loss , the interest rates, tax calculations, salary calculations, which helps to finish the business tasks effectively and efficiently.

Table of contents:

• Related Topics
• Important Terms
• Problems and Solutions

Business Mathematics is highly related to the Statistics concepts which give solutions to business problems. In business, we deal with the exchange of money or products, which have a monetary value. Each business leads to some profit and some loss. To identify these factors, we have to study the primary topics of Maths such as formulas to find a profit, loss, their percentages, discount, etc. Even Though, the requirement of this subject does not contain pure Maths, it needs Mathematical thinking and some Maths formulas . Here, we will discuss what is Business Mathematics, terminologies, and important formulas with problems and solutions.

What is Business Mathematics?

Business Math always deals with profit or loss. The cost of a product is fixed by taking into consideration it’s profit, margin, cash discount, trade discount, etc. Business mathematics is used by commercial companies to record and manage business works. Commercial businesses use maths in departments of accounting, inventory management, marketing, sales forecasting and financial analysis.

Business Mathematics Topics

The most important topics covered in Business Mathematics are:

• Profit and Loss
• Simple and Compound Interest
• Interest Rates
• Markups and markdowns
• Taxes and Tax Laws
• Discount Factor
• Depreciation
• Future and Present Values
• Financial Statements

Business Mathematics Basic Terms

• Selling Price: The market price is taken to sell a product.
• Cost Price: The original price of the product is the cost price.
• Profit: If the selling price is more than the cost price, the difference in the prices is the profit.
• Loss: If the selling price is less than the cost price, the difference in the prices is the loss.
• Discount: The reduced amount in the selling price of a product.
• Simple Interest: Simple interest is that interest which is counted against the capital amount or the portion of the main amount that remains unpaid.
• Compound Interest: Compound interest is the investment rate that increases exponentially.

Business Mathematics Formulas

Here, the 9 basic Business Mathematics formulas that we cannot ignore. They are:

Net Income Formula:

Net Income = Revenue – Expense

Accounting Equation:

Assets = Liabilities + Equity

Equity = Assets – Liabilities

Cost of Goods Sold Formula:

COGS = Beginning inventory + Purchase during the period – Ending inventory

Break-Even point Formula:

Break-Even Point = Fixed cost / (Sales Price per unit – variable cost per unit)

Current Ratio Formula:

Current Ratio = Current Assets / Current Liabilities

Profit Margin Formula:

Profit Margin = (Net Income/ Revenue) × 100

Return of Investment (ROI) Formula:

ROI = [(Invest gain – Cost of Investment)/ Cost of Investment] × 100

Markup Formula:

Markup Percentage = [(Revenue- COGS)/COGS] × 100

Selling Price using Markup = (COGS × markup percentage) + COGS

COGS = Cost of goods sold

Inventory Shrinkage Formula:

Inventory Shrinkage = [(Recorded Inventory – Actual Inventory)/ Recorded Inventory] × 100

Business Mathematics Example

While doing business, one can earn a good profit or face loss. The price of a product is fixed, taking into consideration it’s cost price, profit, margin, trade discount, cash discount, etc. The price marked on the commodity is called marked price or catalogue price. For trading purposes, the manufacturer proposes a discount on the MRP to the buyer. This is called a trade discount. In addition to the trade discount, if the buyer pays cash against goods, he gets another cut called cash discount. The price of the object after subtracting the trade discount and cash discount is called the selling price. Thus, we have, Selling price = Cost price – Dis counts.  Let us discuss the most important concept called “Profit and Loss” here.

Profit and Loss:

A profit is the earned amount received by a business on selling a product whereas loss is the amount which is less than the actual price of the product. The formula for profit and loss is given based on the selling price and cost price of a commodity.

Both these measures have their percentage value also and they are given by;

Business Mathematics Problems and Solutions

Question 1: A music system was bought for Rs.10,500 and sold at Rs.9,500. Find the amount of profit or loss.

Solution: Given,

Cost Price of the music system = Rs.10,500

Selling Price of the music system = Rs. 9,500

We can see here, C.P. is greater than S.P. Therefore, there is a loss in this business.

Hence, we need to calculate the loss amount.

Loss = C.P. – S.P.

Loss = 10,500 – 9,500 = Rs.1,000/-.

Question 2: A pair of shoes is bought at Rs 200 and sold at a profit of 10%. Find the selling price of the shoes.

Profit = 10% of Rs.200

P = (10/100) × 200 = Rs. 20

S.P. = C.P. + Profit

S.P. = 200 + 20 = Rs.220/-

Question 3: If the cost price of an article including 20% for taxes is Rs. 186, then find the original cost of the article excluding taxes.

Let x be the original price of an article.

Tax = 20% of x = (20/100) × x = 0.2x

According to the given statement,

Original cost + tax = New cost price

x + 0.2x = Rs. 186

1.2x = Rs. 186

x = Rs. 186/1.2

x = Rs. 155

Therefore, the cost of the article without taxes = Rs. 155

Keep Visiting BYJU’S – The Learning App and register with the app to get all the Mathematics related topics to learn with ease.

• Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Chapter 13: Solutions to Exercises

13.1: calculating interest and principal components.

• A lump sum of $100,000 is placed into an investment annuity to make end-of-month payments for 20 years at 4% compounded semi-annually. a) What is the size of the monthly payment? b) Calculate the principal portion of the 203rd payment. c) Calculate the interest portion of the 76th payment. d) Calculate the total interest received in the fifth year. e) Calculate the principal portion of the payments made in the seventh year.

a) What is the size of the monthly payment?

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 20 \times 12 = 240 \;\text{payments}[/latex] [latex]I/Y = 4[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 2[/latex] [latex]PV = -100,\!000[/latex] [latex]FV = 0[/latex] [latex]PV = -100,\!000[/latex] [latex]CPT\; PMT =\$604.2464648[/latex]

Make sure to reinput PMT =$604.25 (Input as a positive value rounded to 2 decimal places).

b)  Calculate the principal portion of the 203rd payment.

2nd AMORT P1 = 203 P2 = 203 ↓ ↓ PRN = $533.03

c) Calculate the interest portion of the 76th payment.

2nd AMORT P1 = 76 P2 = 76 ↓ ↓ ↓ INT = $253.73

d)  Calculate the total interest received in the fifth year.

Year 1: payments 1 – 12 Year 2: payments 13 – 24 Year 3: payments 25 – 36 Year 4: payments 37 – 48 Year 5: payments 49 – 60

2nd AMORT P1 = 49           (Starting with payment 49) P2 = 60          (Ending with payment 60) ↓ ↓ ↓ INT = $3,332.61

e) Calculate the principal portion of the payments made in the seventh year .

Year 1: payments 1 – 12 Year 2: payments 13 – 24 Year 3: payments 25 – 36 Year 4: payments 37 – 48 Year 5: payments 49 – 60 Year 6: payments 61 – 72 Year 7: payments 73 – 84

2nd AMORT P1 = 73 P2 = 84 ↓ ↓ PRN = $4,241.39

• At the age of 54, Hillary just finished all the arrangements on her parents’ estate. She is going to invest her $75,000 inheritance at 6.25% compounded annually until she retires at age 65, and then she wants to receive month-end payments for the following 20 years. The income annuity is expected to earn 3.85% compounded annually. a) What are the principal and interest portions for the first payment of the income annuity? b) What is the portion of interest earned on the payments made in the second year of the income annuity? c) By what amount is the principal of the income annuity reduced in the fifth year?

a) What are the principal and interest portions for the first payment of the income annuity?

Step 1:   Find [latex]FV[/latex].

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 11 \times 1 = 11 \;\text{payments}[/latex] [latex]I/Y = 6.25[/latex] [latex]P/Y = 1[/latex] [latex]C/Y = 1[/latex] [latex]PV = 75,\!000[/latex] [latex]PMT = 0[/latex] [latex]PV = 75,\!000[/latex] [latex]CPT\; FV =-\$146,\!109.88[/latex]

Step 2:  Find [latex]PMT[/latex].

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 20 \times 12 = 240 \;\text{payments}[/latex] [latex]I/Y = 3.85[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 1[/latex] [latex]PV = \$146,\!109.88[/latex] [latex]FV = 0[/latex] [latex]PV = \$146,\!109.88[/latex] [latex]CPT\; PMT =-868.83224[/latex]

Make sure to reinput PMT = -868.83 (Input as a negative value rounded to 2 decimal places).

2nd AMORT P1 = 1 P2 = 1 ↓ ↓ PRN = $408.13 INT = $460.70

b) What is the portion of interest earned on the payments made in the second year of the income annuity?

2nd AMORT P1 = 13 P2 = 24 ↓ ↓ ↓ INT = $5,250.65

c)   By what amount is the principal of the income annuity reduced in the fifth year?

2nd AMORT P1 = 49 P2 = 60 ↓ ↓ PRN = $5,796.37

• Art Industries just financed a $10,000 purchase at 5.9% compounded annually. It fixes the loan payment at $300 per month. a) How long will it take to pay the loan off? b) What are the interest and principal components of the 16th payment? c) For tax purposes, Art Industries needs to know the total interest paid for payments 7 through 18. Calculate the amount.

a) How long will it take to pay the loan off?

[latex]I/Y = 5.9[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 1[/latex] [latex]PV = 10,\!000[/latex] [latex]PMT=-300[/latex] [latex]FV= 0[/latex] [latex]CPT\; N =36.402469\;\text{rounded up to}\; 37\;\text{ monthly payments}[/latex]

[latex]\begin{align} \text{Number of years}&=\frac{37}{12}\\ &= 3.08\overline{3}\\ &=3\;\text{years and}\; 0.08\overline{3}\times 12=1\; \text{month} \end{align}[/latex]

3 years, 1 month

b) What are the interest and principal components of the 16th payment?

2nd AMORT P1 = 16 P2 = 16 ↓ ↓ PRN = $270.84 INT = $29.16

c) For tax purposes, Art Industries needs to know the total interest paid for payments 7 through 18. Calculate the amount.

2nd AMORT P1 = 7 P2 = 18 ↓ ↓ ↓ INT = $403.33

13.2: Calculating the Final Payment

• Semi-annual payments are to be made against a $97,500 loan at 7.5% compounded semi-annually with a 10-year amortization. a) What is the amount of the final payment? b) Calculate the principal and interest portions of the payments in the final two years.

Step 1 : Find the regular payment.

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 10 \times 2 = 20 \;\text{payments}[/latex] [latex]I/Y = 7.5[/latex] [latex]P/Y = 2[/latex] [latex]C/Y = 2[/latex] [latex]PV = 97,\!500[/latex] [latex]FV = 0[/latex] [latex]CPT\; PMT = -\$7,\!016.30449[/latex]

Make sure to reinput PMT = -7,016.30 (Input as a negative value rounded to 2 decimal places).

Step 2: Use the AMORT function to find the BAL on the last line (payment 20).

2nd AMORT P1 = 20 P2 = 20 ↓ BAL = $0.130277

Step 3: Find the Final Payment

Final Payment = $7,016.30 + $0.130277 = $7,016.43.

Year 9: payments 17-18 Year 10: payments 19-20

2nd AMORT P1 = 17 P2 = 20 ↓ BAL = $0.130277 (*added to payment of $7,016.30 = $7,016.43) PRN = $25,619.18861 (* add BAL:  $25,619.18861 + $0.130277 = $25,619.32) INT = $2,446.011387

Therefore, PRN = $25,619.32; INT = $2,446.01.

• A $65,000 trust fund is set up to make end-of-year payments for 15 years while earning 3.5% compounded quarterly. a) What is the amount of the final payment? b) Calculate the principal and interest portion of the payments in the final three years.

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 15 \times 1 = 15 \;\text{payments}[/latex] [latex]I/Y = 3.5[/latex] [latex]P/Y = 1[/latex] [latex]C/Y = 4[/latex] [latex]PV = 65,\!000[/latex] [latex]FV = 0[/latex] [latex]CPT\; PMT = -\$5,\!662.190832[/latex]

Make sure to reinput PMT = -5,662.19 (Input as a negative value rounded to 2 decimal places).

Step 2: Use the AMORT function to find the BAL on the last line (payment 15)

2nd AMORT P1 = 15 P2 = 15 ↓ BAL = $0.016114

Final Payment = $5,662.19 + $0.016114 = $5,662.21.

2nd AMORT P1 = 13 P2 = 15 ↓ BAL = $0.016114 (*added to payment of $5,662.19 = $5,662.21) PRN = $15,849.40807 (* add BAL:  $15,849.40807 + $0.016114 = $15,849.42) INT = $1,137.161928

Therefore, PRN = $15,849.42; INT = $1,137.16.

• Mirabel Wholesale has a retail client that is struggling and wants to make instalments against its most recent invoice for $133,465.32. Mirabel works out a plan at 12.5% compounded monthly with beginning-of-month payments for two years. a) What will be the amount of the final payment? b) Calculate the principal and interest portions of the payments for the entire agreement.

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 2 \times 12= 24 \;\text{payments}[/latex] [latex]I/Y = 12.5[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 12[/latex] [latex]PV = 133,\!465.32[/latex] [latex]FV = 0[/latex] [latex]CPT\; PMT = -\$6,\!248.793434[/latex]

Make sure to reinput PMT = -6,248.79 (Input as a negative value rounded to 2 decimal places).

Step 2: Use the AMORT function to find the BAL on the last line (payment 24)

2nd AMORT P1 = 24 P2 = 24 ↓ BAL = $0.093078

Final Payment = $6,248.79 + $0.093078 = $6,248.88.

2nd AMORT P1 = 1 P2 = 24 ↓ BAL = $0.093078 (*added to payment of $6,248.79 = $6,248.88) PRN = $133,465.2269 (* add BAL:  $133,465.2269 + $0.093078 = $133,465.32) INT = $16,505.73308

Therefore, PRN = $133,465.32; INT = $16,505.73.

13.3: Amortization Schedules

• A farmer purchased a John Deere combine for $369,930. The equipment dealership sets up a financing plan to allow for end-of-quarter payments for the next two years at 7.8% compounded monthly. Construct a complete amortization schedule and calculate the total interest.

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 2 \times 4= 8 \;\text{payments}[/latex] [latex]I/Y = 7.8[/latex] [latex]P/Y = 4[/latex] [latex]C/Y = 12[/latex] [latex]PV = 369,\!930[/latex] [latex]FV = 0[/latex] [latex]CPT\; PMT = -\$50,\!417.92645[/latex]

Make sure to reinput PMT = -50,417.93 (Input as a negative value rounded to 2 decimal places).

Step 2: Use the AMORT function to fill out the table.

Step 3: Adjust for the “missing pennies” (noted in bold italics ) and total the interest.

• Ron and Natasha had Oasis Leisure and Spa install an in-ground swimming pool for $51,000. The financing plan through the company allows for end-of-month payments for two years at 6.9% compounded quarterly. Ron and Natasha instruct Oasis to round their monthly payment upward to the next dollar amount evenly divisible by $500. Create a schedule for the first three payments, payments seven through nine, and the last three payments.

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 2 \times 12= 24 \;\text{payments}[/latex] [latex]I/Y = 6.9[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 4[/latex] [latex]PV = 51,\!000[/latex] [latex]FV = 0[/latex] [latex]CPT\; PMT = -\$2,\!280.18[/latex]

Make sure to reinput PMT = – $2,500 ($2,280.18 rounded up to $2,500).

Step 2 : Recalculate N with new PMT.

[latex]I/Y = 6.9[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 4[/latex] [latex]PV = 51,\!000[/latex] [latex]PMT = -2,\!500[/latex] [latex]FV = 0[/latex] [latex]CPT\; N = 21.753021\; \text{rounded up to}\; 22\; \text{monthly payments}[/latex]

Step 3:  Use the AMORT function to fill out the table.

Step 4: Adjust for the “missing pennies” (noted in bold italics ) and total the interest.

• Hillary acquired an antique bedroom set recovered from a European castle for $118,000. She will finance the purchase at 7.95% compounded annually through a plan allowing for payments of $18,000 at the end of every quarter. a) Create a complete amortization schedule and indicate her total interest paid. b) Recreate the complete amortization schedule if Hillary pays two additional top-up payments consisting of 10% of the principal remaining after her third payment as well as her fifth payment. What amount of interest does she save?

Step 1: Find [latex]N[/latex].

[latex]I/Y = 7.95[/latex] [latex]P/Y = 4[/latex] [latex]C/Y = 1[/latex] [latex]PMT = -18,\!000[/latex] [latex]PV = 118,\!000[/latex] [latex]FV = 0[/latex] [latex]CPT\; N = 7.076614\; \text{rounded up to}\; 8\; \text{quarterly payments}[/latex]

Step 2:  Use the AMORT function to fill out the table.

Step 1:  Use the AMORT function to fill out the table.

*The balance after the third payment was $69,918.68.  10% of this amount is $6,991.87. **The balance after the fifth payment was $29,032.76. 10% of this amount is $2,903.28.

Step 2: Adjust for the “missing pennies” (noted in bold italics ) and total the interest.

From Question 18a, amount of interest paid was $9,391.27. Interest Saved = $9,391.27 − $8,695.87 = $695.40.

13.4: Special Application – Mortgages

• Three years ago, Phalatda took out a mortgage on her new home in Kelowna for $628,200 less a $100,000 down payment at 6.49% compounded semi-annually. She is making monthly payments over her three-year term based on a 30-year amortization. At renewal, she is able to obtain a new mortgage on a four-year term at 6.19% compounded semi-annually while continuing with monthly payments and the original amortization timeline. Calculate the following: a) Interest and principal portions in the first term. b) New mortgage payment amount in the second term. c) Balance remaining after the second term.

Step 1 : Find the initial payment.

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 30 \times 12 = 360 \;\text{payments}[/latex] [latex]I/Y = 6.49[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 2[/latex] [latex]FV = 0[/latex] [latex]PV = 628,\!200 − 100,\!000 = 528,\!200[/latex] [latex]CPT\; PMT =-\$3,\!305.288742[/latex]

Make sure to reinput PMT = -3,305.29 (Input as a negative value rounded to 2 decimal places).

Step 2: Use the AMORT function to find the BAL on the after the first term (payment 1-36).

2nd AMORT P1 = 1 P2 = 36 ↓ BAL = $508,947.54

2nd AMORT P1 = 1 P2 = 36 ↓ ↓ PRN = $19,252.46 INT = $99,737.98

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 27 \times 12 = 324 \;\text{payments}[/latex] [latex]I/Y = 6.19[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 2[/latex] [latex]FV = 0[/latex] [latex]PV = 508,\!947.54[/latex] [latex]CPT\; PMT =-\$3,\!211.32429[/latex]

Make sure to reinput PMT = -3,211.32 (Input as a negative value rounded to 2 decimal places).

Use the AMORT function to find the BAL on the after the second term (payment 1-48).

2nd AMORT P1 = 1 P2 = 48 ↓ BAL = $475,372.69

• The Verhaeghes have signed a three-year closed fixed rate mortgage with a 20-year amortization and monthly payments. They negotiated an interest rate of 4.84% compounded semi-annually. The terms of the mortgage allow for the Verhaeghes to make a single top-up payment at any one point throughout the term. The mortgage principal was $323,000 and 18 months into the term they made one top-up payment of $20,000. a)What is the balance remaining at the end of the term? b)By what amount was the interest portion reduced by making the top-up payment?

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 20 \times 12 = 240 \;\text{payments}[/latex] [latex]I/Y = 4.84[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 2[/latex] [latex]FV = 0[/latex] [latex]PV = 323,\!000[/latex] [latex]CPT\; PMT =-\$2,\!094.701842[/latex]

Make sure to reinput PMT = -2,094.70 (Input as a negative value rounded to 2 decimal places).

Step 2: Use the AMORT function to find the BAL on the after the first 18 months.

2nd AMORT P1 = 1 P2 = 18 ↓ BAL = $308,009.80

  Step 3: Find New Balance after $20,000 top-up payment.

New Balance = $308,009.80 − $20,000 = $288,009.80. Reinput PV = $288,009.80.

Step 4: Use the AMORT function to find the BAL on the after the last 18 months of the first term.

2nd AMORT P1 = 1 P2 = 18 ↓ BAL = $270,417.34

Step 1: Find Original BAL paid without top-up payment (payments 1-36).

Reinput PV = $323,000

2nd AMORT P1 = 1 P2 = 36 ↓ BAL = $291,904.76

Step 2: Find Interest Difference.

[latex]\begin{align} \text{Interest Difference}&=\$291,\!904.76 − \$270,\!417.34 − \$20,\!000\\ &= \$1,\!487.42 \end{align}[/latex]

• Fifteen years ago, Clarissa’s initial principal on her mortgage was $408,650. She set up a 30-year amortization, and in her first 10-year term of monthly payments her mortgage rate was 7.7% compounded semi-annually. Upon renewal, she took a further five-year term with monthly payments at a mortgage rate of 5.69% compounded semi-annually. Today, she renews the mortgage but shortens the amortization period by five years when she sets up a three-year closed fixed rate mortgage of 3.45% compounded semi-annually with monthly payments. What principal will she borrow in her third term and what is the remaining balance at the end of the term? What total interest portion and principal portion will she have paid across all 18 years?

Step 1 : Find the initial payment for 10-year term.

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 30 \times 12 = 360 \;\text{payments}[/latex] [latex]I/Y = 7.7[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 2[/latex] [latex]FV = 0[/latex] [latex]PV = 408,\!650[/latex] [latex]CPT\; PMT =-\$2,\!879.565159[/latex]

Make sure to reinput PMT = -2,879.57 (Input as a negative value rounded to 2 decimal places).

  Step 2: Use the AMORT function to find the BAL on the after the 10-year term (payments 1-120).

2nd AMORT P1 = 1 P2 = 120 ↓ BAL = $355,303.81

Step 3 : Find the payment for 5-year term.

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 20 \times 12 = 240 \;\text{payments}[/latex] [latex]I/Y = 5.69[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 2[/latex] [latex]FV = 0[/latex] [latex]PV = 355,\!303.81[/latex] [latex]CPT\; PMT =-\$2,\!468.979621[/latex]

Make sure to reinput PMT = -2,468.98 (Input as a negative value rounded to 2 decimal places).

Step 4: Use the AMORT function to find the BAL on the after the 5-year term (payments 1-120).

2nd AMORT P1 = 1 P2 = 60 ↓ BAL = $299,756.24

Step 5 : Find the payment for 3-year term with amortization period shortened by 5 years. Number of years remaining = 15 – 5 = 10.

[latex]N = (\text{Number of Years}) \times (\text{Payments Per Year})[/latex] [latex]N = 10 \times 12 = 120 \;\text{payments}[/latex] [latex]I/Y = 3.45[/latex] [latex]P/Y = 12[/latex] [latex]C/Y = 2[/latex] [latex]FV = 0[/latex] [latex]PV =299,\!756.24[/latex] [latex]CPT\; PMT =-\$2,\!953.710318[/latex]

Make sure to reinput PMT = -2,953.71 (Input as a negative value rounded to 2 decimal places).

Step 6:   Use the AMORT function to find the BAL on the after the 3-year term (payments 1-36).

2nd AMORT P1 = 1 P2 = 36 ↓ BAL = $220,328.74

Start of 3rd term principal = $299,756.24. Remaining balance at end of 3rd term = $220,328.74.

[latex]\begin{align} \text{Total principal across all}\;18\;\text{years}&=\$408,\!650 - \$220,\!328.74\\ &= \$188,\!321.26 \end{align}[/latex]

[latex]\text{Total interest across all}\;18\;\text{years}[/latex] [latex]=(120 \times 2,\!879.57) + (60 \times 2,\!468.98) + (36 \times 2,\!953.71) - 188,\!321.26[/latex] [latex]= \$411,\!499.50[/latex]

Image Description

Figure 13.1.2: Timeline: Deferral period from age 54 until age 65 at 6.25% compounded annually. Starting at age 55, 20 years end of month payments of PMT at 3.85% compounded annually. $75,000 at age 54 brought to age 65 as FV. At age 65 the FV becomes the PV for the stream of PMT’s brought back to age 65. [ Back to Figure 13.1.2 ]

Business Math: A Step-by-Step Handbook Abridged Copyright © 2022 by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Solver Title

Generating PDF...

• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
• Pre Algebra
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

Most Used Actions

Number line.

• x^{2}-x-6=0
• -x+3\gt 2x+1
• line\:(1,\:2),\:(3,\:1)
• prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x)
• \frac{d}{dx}(\frac{3x+9}{2-x})
• (\sin^2(\theta))'
• \lim _{x\to 0}(x\ln (x))
• \int e^x\cos (x)dx
• \int_{0}^{\pi}\sin(x)dx
• \sum_{n=0}^{\infty}\frac{3}{2^n}
• Is there a step by step calculator for math?
• Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem.
• Is there a step by step calculator for physics?
• Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go.
• How to solve math problems step-by-step?
• To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
• My Notebook, the Symbolab way Math notebooks have been around for hundreds of years. You write down problems, solutions and notes to go back...

Please add a message.

Message received. Thanks for the feedback.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

4: Business Math

• Last updated
• Save as PDF
• Page ID 35415
• 4.1: Ratio and Proportion Examples and applications of ratios are limitless: speed is a ratio that compares changes in distance with respect to time, acceleration is a ratio that compares changes in speed with respect to time, and percentages compare the part with the whole. We’ve already studied one classic ratio, the ratio of the circumference of a circle to its diameter, which gives us the definition of π.
• 4.2: Introduction to Ratios and Rates We use ratios to compare two numeric quantities or quantities with the same units.
• 4.3: Introduction to Proportion In this section, we equate ratio and rates in a construct called a proportion.
• 4.4: Percent
• 4.5: Percent, Decimals, Fractions So, when you hear the word “percent,” think “parts per hundred.”
• 4.6: Solving Basic Percent Problems
• 4.7: General Applicants of Percent
• 4.8: Percent Increase or Decrease A person’s salary can increase by a percentage. A town’s population can decrease by a percentage. A clothing firm can discount its apparel. These are the types of applications we will investigate in this section.
• 4.9: Interest
• 4.10: Pie Charts