2    › Scale Factor Method           Simple & Compound Interest
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Simple interest occurs whenever the interest is not continually added to the account balance. It may be paid directly to the account holder instead of being added to the account, or it may not be paid until the account is closed...

Generally speaking - simple interest doesn't arise a lot in REAL-LIFE (Compound Interest - the next exercise - occurs a lot more often in real-life...)

It will most likely arise in the form of a FIXED INTEREST BOND: It pays the same interest year-on-year or a fixed period of time - usually many years - so for that reason, you usually cannot add or subtract to your investment (because if you could, then if interest rates went up, you'd withdraw it all and they'd go bust; or if interest rates went down, you'd borrow more money to put in and they'd go bust)...

Let's start by understanding the terminology:

TERMINOLOGY:
  • PRINCIPAL: This is the original amount invested (so it is referred to as 100%)
  • RATE: This is the interest rate ‘per annum’ (‘p.a.’ stands for ‘per annum’; it means ‘every year’)
  • TERM: This is the length of time - IN YEARS - the money is invested for
  • FULL-TERM-RATE (or just term-rate) = rate Χ term
  • Simple Interest: This is the total interest earned over the term of the investment
  • P.I. (or final-value) = principal + simple interest

 

Simple Interest Questions are dealt with by thinking in terms of TERM-RATE:

	
                   principal  ——————————►  100% 
                                                 
                                                 
                    _______   ◄———————— term-rate 
                  ╘════╦════╛          ╘════╦════╛
total amount of ═══════╝                    ╚════════ rate Χ years
simple interest
earned over the term

 ·

e.g. If £250 in invested at 12% p.a. for 4 years, what is the simple interest earned?
	
So we know:     The principal  is £250 ( so £250 ————————► 100% )
 
                             ╖
                Rate = 12%   ║
                             ╟──────► Term-Rate = 48%
                Term = 4 yrs ║
                             ╜
 
                Simple interest = ???
       
	  
So, our calculation is:
         
                          £250 ——————————► 100%
                    Χ48                       Χ48
                    100 ____ ◄——————————— 48% 100 
                           
         
         
So, the answer is: £250 Χ  48 =  £120
                          100

·

e.g. What sum of money, invested at 6% p.a. for 5 years gives £120 simple interest?
	
So we know:     Simple interest = £120
 
                             ╖
                Rate = 6%    ║
                             ╟──────► Term-Rate = 30%
                Term = 5 yrs ║
                             ╜
         
                Principal (i.e. 100%) = ???
         
	
So, this time the calculation is:
         
                          £120 ——————————► 30%
                   Χ100                      Χ100 
                     30 ___ ◄—————————— 100%  30 
                             
         
So, the answer is: £120 Χ 100 =  £400
                           30

·

e.g. For how long must £150 be invested at 15% p.a. to earn £90 of interest?
	
So we know:     Principal (i.e. 100%) = £150
         
                Simple Interest = £90
         
                Term-Rate = ???
        
                          £150 ——————————► 100%
                    Χ90                       Χ90 
                    150  £90 ——————————► ____ 150 
                                                            
         
Rate Years = 100% Χ  90  =  60%
                    150 
 
                             ╖
                Rate = 15%   ║
                             ╟──────► Term-Rate = 60%
                Term = ? yrs ║
                             ╜        
         
So: Term = 60 χ 15 = 4-years

·

·

·

·

Question 1: So, we know:

	
        Principal (i.e. 100%) = £200
        
                     ╖
        Rate = 8%    ║
                     ╟──────► Term-Rate = 16%
        Term = 2 yrs ║
                     ╜
          
         
        Simple Interest = ???
         
                          £200 ——————————► 100%
                    Χ16                       Χ16 
                    100 ___ ◄———————————  16% 100
                             
         

So the answer is: \(\color{#2b83c3}{£200} \color{#e54239}{\times} \frac{\color{#557D55}{16}}{\color{#2b83c3}{100}} \;=\; \color{#bd398c}{£......}\)

 

 

 

Question 2: We know:

	
        Principal (i.e. 100%) = £160
        
                     ╖
        Rate = 14%   ║
                     ╟──────► Term-Rate = 70%
        Term = 5 yrs ║
                     ╜
          
        Simple Interest = ???
         
                          £160 ——————————► 100%
                    Χ70                       Χ70
                    100 ___ ◄———————————  70% 100 
                             
         

So the answer is: £160 Χ E100 = £...

 

 

 

Question 3: So: ... is the Principal (i.e. 100%). And the Term-Rate is 39%:

	
        Principal (i.e. 100%) = £ ⋯⋯
        
                     ╖
        Rate = 13%   ║
                     ╟──────► Term-Rate = ⋯⋯% 
        Term = 3 yrs ║
                     ╜
         
        Simple Interest = ???
         
                          £⋯⋯ ——————————► 100%
                    Χ⍰	                     	Χ⍰
                    100	 ___ ◄—————————— ⋯⋯%	100 
                            
         

So the answer is: £⋯ Χ ⋯/100 = £...

 

 

 

Question 4: It's the same method again:

	
        Principal (i.e. 100%) = £ ⋯⋯
        
                     ╖
        Rate =  8%   ║
                     ╟──────► Term-Rate = ⋯⋯% 
        Term = 7 yrs ║
                     ╜
         
        Simple Interest = ???
         
                          £⋯⋯ ——————————► 100%
                    Χ⍰	                     	Χ⍰ 
                    100	 ___ ◄—————————— ⋯⋯%	100 
                            
         

So the answer is ...

 

 

 

Question 5: If you invest money (earning ‘Simple-Interest’) , then the interest adds on to your principal to to create the Final-Value of your investment...

On the other hand, if you borrow money (and are lucky to be charged ‘Simple-Interest’, rather than ‘Compound-Interest’), then the interest adds onto your principal borrowed to create the Final-Value Owed.

Mathematically, it's exactly the same thing - and worked out in exactly the same way...

(Although in real-life, having lots of money invested is NOT the same thing as owning lots of money - some people don't realise that!)

 

 

 

Question 7: There are two ways you can approach this type of question:

1) Using the method above to find the SIMPLE INTEREST, then adding the Principal to that:

	
Principal (i.e. 100%) = £280
        
             ╖
Rate = 12%   ║
             ╟──────► Term-Rate = 72%
Term = 6 yrs ║
             ╜
             
Simple Interest = ???
         
          £280 ——————————► 100%
    Χ72                       Χ72 
    100 ___ ◄———————————  72% 100 
                          
        └──┬──┘
           └───────────── add £280 to this!
                        to get the Final Value

 

2) By adding 100% to the Term-Rate to get the Tol-Term-Rate:

	
Principal (i.e. 100%) = £280
        
             ╖
Rate = 12%   ║
             ╟──────► Term-Rate =   72%
Term = 6 yrs ║      + Principal   +100%
             ╜                   ======
                  Tol-term-rate =  172%
         
The FINAL-VALUE (or P.I.) is then given by:
         
          £280 ——————————► 100%
   Χ172                       Χ172 
    100 ___ ◄——————————— 172%  100 
                             
 
This is slightly quicker, innit?

 

 

 

 

Question 11: If the interest payments need to total £8.75 million over the next 5-yeas:

	
        Simple interest = £8.75 million
   
                     ╖
        Rate = 7%    ║
                     ╟──────► Term-Rate = ⋯%
        Term = 5 yrs ║
                     ╜
         
        Principal (i.e. 100%) = ???
         
So, this time the calculation is:
         
                         £8.75m ——————————► ⋯%
                                             	
                       ____ ◄—————————— 100%	 
                 
         

So, the answer is ...

 

 

 

Question 12, part (a): So easy!

 

 

Question 12, part (b): So: £370 is the principal (i.e. 100%). And £233.10 is the Simple Interest earned in ⋯ years

	
        Principal (i.e. 100%) = £370
         
        Simple Interest = £233.10
         
                      ╖
        Rate = 9%     ║
                      ╟──────► Term-Rate = ⋯%
        Term = ⋯ yrs	║
                      ╜
         

        
                         £370   ——————————► 100%
                                               
                       £233.10 —————————► ____ 
                                                     
         
Term-Rate = 100% Χ ⋯⋯  =  ⋯⋯%
                   ⋯⋯ 
          

So: Term-rate = ...%

But, Term-Rate = Rate Χ Term

So: Term = ⋯ χ ⋯ = ... years

 

 

Question 13, part (a): This is easy - its just asking us, if we got £416 interest in 8-years; how much is that per year?

 

 

Question 13, part (b): Well - because we already worked out (in part a) that the interest per year - we can just use normal percentages (rather than simple-interest) terminology:

Original amount = £800. Interest = £52:

	
                         £800 —————————► 100% 
                                             
                       £52 ——————————► ___%   
     

 

 

 

Question 14: There are two ways we can work this out:

1) By working out the interest in 1-year and using normal percentages (like we did in Qu 13)

OR

2) By using Simple-Interest methodology:

	
        Simple interest = £312
         
                     ╖
        Rate = 18%   ║
                     ╟──────► Term-Rate = ⋯%
        Term = 8 yrs ║
                     ╜
         
        Principal (i.e. 100%) = ???
         
So, this time the calculation is:
         
                         £312 ——————————► ...%
                                             
                       ___ ◄——————————  100% 
                      
         

So the answer is ... 

 

 

 

Question 15, part (a): Ah -Brexit...

This is easy if you've been following the method we've been using for all of the previous question - but if you haven't, then now is the time to start doing so...

                     ╖
        Rate = 10%   ║
                     ╟──────► Term-Rate = 40%
        Term = 4 yrs ║
                     ╜
         

 

Question 15, part (b): So, we know:

	
        P.I. (i.e. final value) = £16,800    ───────────────────┐
                                                                │
                     ╖                                          │
        Rate = 10%   ║                                          │
                     ╟──────► Term-Rate = 40%                   │
        Term = 4 yrs ║                                          │
                     ╜                                          │
                                                                ├──┤ Equate these
        PercenToL: principal (100%) + Term-rate (40%) = 140% ◄──┘
         
        Principal (i.e. 100%) = ???
 
                       £16,800 ——————————► 140%
                                               
                     ______ ◄——————————  100% 
                             
         

So the answer is: ....

 

 

 

Question 16, part (a): Using the same method as Question 15:

	
        P.I. (i.e. final value) = £254       ───────────────────┐
                                                                │
                     ╖                                          │
        Rate = 9%    ║                                          │
                     ╟──────► Term-Rate = ⋯%                   	│
        Term = 3 yrs ║                                          │
                     ╜                                          │
                                                                ├──┤ Equate these
        PercenToL: principal (100%) + Term-rate (⋯%) = ⋯% 	◄───┘
         
        Principal (i.e. 100%) = ???
 
                         £254 ——————————► ⋯⋯%
                                            	 
                       ____ ◄—————————— 100%	
                             
         

So the answer is: ....

 

 

Question 16, part (d): Treat this question in exactly the same way is if the money had been INVESTED rather than BORROWED:

	
        P.I. (i.e. final value) = £2250 
        
                     ╖  
        Rate = 8%    ║ 
                     ╟──────► Term-Rate = ⋯%  
        Term = 10yrs ║  
                     ╜ 
      
        PercenToL: principal (100%) + Term-rate (⋯%) = ⋯% 
         
        Principal (i.e. 100%) = ???

 

 

 

 

Question 17, part (a): Using the same method as Question 12:

So: £500 is the principle (i.e. 100%). And £135 is the Simple Interest earned in ⋯ years

	
        Principal (i.e. 100%) = £500
         
        Simple Interest = £135
         
                     ╖
        Rate = 9%    ║
                     ╟──────► Term-Rate = ⋯%
        Term = ⋯yrs	 ║
                     ╜
         

        
                         £500 —————————► 100%
                                            
                       £135 —————————► ____ 
                                                     
         
Term-Rate = 100% Χ ⋯⋯  =  ⋯⋯%
                   ⋯⋯ 
          

So: Term-rate = ...%

But, Term-Rate = Rate Χ Term

So: Term = ⋯ χ ⋯ = ... years

 

So the answer is: ....

 

 

 

Question 18: This is an easy question - disguised as a difficult question simply by making it wordy and long.

If you don't get it, you'll kick yourself - so try and think and if that fails, leave it and try again a day later. Betcha get it then!

 

 

 

 

Question 20: INTEREST-ONLY mortgages were all the rage in the 1980s: Your repayments were less, because you were only paying off the interest - not the principal:

For instance, based on a mortgage-interest-rate of 10% (rates were high back then), with a £500,000 mortgage:

For a REPAYMENT mortgage, you'd have to have to make payments of ~£55,000 per year

But on an INTEREST-ONLY mortgage, your repayments would be £50,000 per year

The difference being, with the repayment mortgage, at the end of the 25-years, the mortgage is paid off and you own the house... But with the interest-only mortgage, at the end of 25-years, you then you still have to pay back the £500,000

It sounds crazy eh?

The point was, the £5,000 you were saving every year, you were supposed to invest in Google (it didn't even exist then...) and then you'd be richy-rich-rich and could easily payback the principal. But a lot of people ‘invested’ the £5,000 yearly saving on holidays and partying and then at the end of 25-years, the bank took their house away!

Okay - let's break it down:

1) They paid the simple-interest on the mortgage for 25-years

2) They put their £100,000 bonus into a saving scheme

3) They had to pay the remaining 40% of the £500,000 (by selling a kidney each - you are kidin-me ?!)

So, we a easily work out (3), since we know what 40% of £500,000 is. And (2) is well - £100,000 - obviously.

But what about (1)?

Well that would be easy if we knew the interest rate of the mortgage - but we don't!

But we do know one thing - we know that, when they got the £100,000 bonus, instead of using it to pay-down the principal of the mortgage, they chose to invest it in a fixed rate bond: Now with simple interest, its very easy to compare different rates - so we know that, unless the bond pad a higher rate of interest than they were paying on the mortgage, they would have just used the money to pay off part of the mortgage...

So, if we can work out the rate of interest of the bond, then we know the mortgage interest was less than that. PHEW!