Questions on Algebra in Finance answered by real tutors!

let the bill amount be 	x							

    Plan A =  70 + 0.35*(x - 70).

    Plan B = 230 + 0.3*(x - 230).					


Plan B < Plan A	
						
    230 + 0.3*(x-230) < 70 + 0.35*(x-70)

    230 + 0.30x - 69 < 70 + 0.35x - 24.5

    230 - 69 - 70 + 24.5 < 0.35x - 0.3x	

    115.5 < 0.05x 

    X > 115.5/0.05 = 2310


ANSWER.  Plan B is cheaper if the bill is greater than $2310.

Use the standard formula for the loan quarterly payment

    P = 


where L is the loan amount; r =  is the effective interest rate per quarter;
n is the number of payments (same as the number of quarters); P is the quarterly payment.


In this problem  L = $80000;  r = ;  n = 5*4 = 20.


Substitute these values into the formula and get for quarterly payment

    P =  = $4775.30.


Thus the quarterly payment is $4775.30.


In total, a developer will pay  5*4*4775.30 = 95,506 dollars in 5 years.


The difference $95,506 - $80,000 = $15,506 is the interest the developer pays to financial company.

How bonds work (from the Google Artificial Intelligence OVERVIEW).

Bonds work like an IOU: you lend money to a government or company (the issuer) for a set period, and in return, they 
promise to pay you regular interest (coupons) and return your original investment (principal) on a specific maturity 
date, making them a fixed-income loan where you earn predictable payments and get your initial sum back, with market 
prices fluctuating inversely to interest rates. [1, 2, 3, 4, 5]  

How a Bond Works 

1. Lending Money: When you buy a bond, you're essentially giving a loan to the entity that issues it (e.g., a company, 
   city, or country). [3, 6]  
2. Coupon Payments (Interest): The issuer agrees to pay you fixed interest payments, called coupons, at regular intervals   
   (like semi-annually). [1, 7]  
3. Face Value/Principal: This is the original amount of the loan, which is repaid to you on the bond's maturity date. 
   [2,4]  
4. Maturity Date: The date when the issuer must repay the principal to you, ending the bond. [3, 4]  

Example: 

You buy a $1,000 bond with a 5% coupon rate maturing in 2 years. [1]  

• You receive $25 every six months ($1,000 x 5% / 2). 
• After two years, you get your original $1,000 back, plus the total interest earned. 

Key Concepts 

• Issuer: The borrower (government, corporation). [3, 4]  
• Coupon Rate: The fixed interest rate paid to you. [2, 4]  
• Yield: The actual return you get, which changes with market prices. [8]  
• Interest Rate Risk: Bond prices move inversely to market interest rates; if rates rise, your existing bond's price 
  falls (and vice versa) because its fixed payments become less attractive. [2, 3, 9]  

Why People Buy Bonds [1, 10, 11]  

• Regular Income: For steady interest payments. 
• Capital Preservation: For the return of principal at maturity. 
• Diversification: To balance riskier stock investments in a portfolio. 

Types of Bonds 

• Government Bonds: Issued by national governments, generally considered safer (e.g., U.S. Treasuries). [4, 12]  
• Corporate Bonds: Issued by companies, carrying higher risk (credit risk) but potentially higher yields. [4, 12]  
• Municipal Bonds (Munis): Issued by state and local governments, often offering tax advantages. [6]  

AI responses may include mistakes.

[1] https://investor.vanguard.com/investor-resources-education/understanding-investment-types/what-is-a-bond
[2] https://dfi.wa.gov/financial-education/information/basics-investing-bonds
[3] https://www.investopedia.com/terms/b/bond.asp
[4] https://www.youtube.com/watch?v=qQAxLF_bJDc
[5] https://www.merrilledge.com/article/understanding-bonds-and-their-risks
[6] https://www.schwab.com/learn/story/what-are-bonds-understanding-bond-types-and-how-they-work
[7] https://www.nerdwallet.com/investing/learn/what-is-a-bond
[8] https://www.bbc.co.uk/news/10093437
[9] https://www.getsmarteraboutmoney.ca/learning-path/videos/bonds-101-what-is-a-bond-2/
[10] https://www.youtube.com/watch?v=vAdn7aLHpO0
[11] https://www.fidelity.com/learning-center/investment-products/fixed-income-bonds/what-is-a-bond
[12] https://www.youtube.com/watch?v=Zn3Ixr-OiF4

Face value (Principal) is $24000.

They want you find the semiannual interest payment at the annual rate 6.875% during 10 years.


It means that after bying their bond, you get from Proctor & Gamble  

 = $825 every 6 months during 10 years.


At the end of 10 years, you get $24000 back from Proctor and Gamble.


So, the answer to question (a) is THIS: you get $825 dollars from Proctor & Gamble semiannually during 10 years.


    The answer to question (b) is THIS: total amount of interest earned on this bond in 10 years is 

        2*10*825 = 16500 dollars.

Principal P = 2500
Interest rate = 4.5% = 0.045 compounded semiannually.
years = 8.


Use the standard formula for the future value

    FV =  =  = 3569.05 dollars.


ANSWER.  In 8 years, the account will be $3569.05.

This problem is to solve it mentally in one line 

    profit =  =  = 54*40 - 1800 = 2160 - 1800 = 360 naira.    ANSWER

First, obviously, we need to use two pounds of ingredient 2 to provide 2 ounces of potassium.


After that, we need to provide as minimum 20-2*2 = 16 ounces of nitrogen and 36-2*6 = 24 ounces of phosphate.


Next, it is also obvious, that for it, we should use as many pounds of ingredient 1 as possible, because 

    (a) ingredient 1 is cheaper, and 

    (b) each pound of ingredient 1 contains the same 6 ounces of phosphate as each pound of ingredient 2.


After these notices, it is clear that 24/6 = 4 pounds of the ingredient 1 should be used.


Thus we get this 


ANSWER.  2 pounds of ingredient 2  and 4 pounds of ingredient 1 should be used to satisfy the imposed conditions.

First machine caps   = 100  bottles per minute.


Second machine caps   = 125 bottles per minute.


Working together, the two machines cap  100 + 125 = 225 bottles per minute.


Working together, the two machines need   = 4.444... minutes to cap 1000 bottles.


ANSWER to question (a).  Working together, the two machines need  about 4.444 minutes, or 4 minutes 27 seconds, to cap 1000 bottles.