4.4.1 Business and financial calculations
Mathematics is used continuously whenever business transactions take place. Using numbers and how to deal with them is central to understanding accounting and financial management. For the time being you should revise your knowledge about calculating:
- Percentage and Simple Interest
- Discounts
- Compound Interest
The following examples will help you:
Percentages and simple interest
The word percentage means by the hundred or per hundred . Most of you understand the idea and how percentages are used in news reports. For example, 'as many as 80% of the population have been vaccinated against smallpox' means that 80 out of 100 people have received the vaccine.
Mathematically the concept of percentage and the term per cent refer to the relationship between the number preceding the per cent symbol and 100. This means that we should look at fractions with the denominator of 100.
Example
When converting a fraction to a percentage, multiply the fraction by 100 and put the per cent symbol after the answer:
|
1/2 = 1/2 x 100 = 50% |
2/5 = 2/5 x 100 = 40% |
|
2/7 = 2/7 x 100 = 28.6% |
3/10 = 3/10 x 100 = 30% |
|
1/8 = 1/8 x 100 = 12.5% |
1/5 = 1/5 x 100 = 20% |
Similarly, when converting a percentage to a decimal, simply move the decimal point from the right of the last digit two places to the left and drop the % sign:
83% = 0.83; . 77.7% = 0.775; 5% = 0.05; 127% = 1.27; 0.24% = 0.0024
When converting a decimal to a percentage, move the decimal point two digits to the right and add the % sign:
0.5= .50%; 0.05 = 5%; . .1.75 = 175%; 0.0025 = 25%
Discounts
A reduction in the price of a product or service is called a discount. Discounts are often given to encourage a purchase, for buying in bulk, to clear slow moving merchandise and to encourage early payment. There are daily advertisements in the media that offer percentage discounts or absolute dollar savings, whichever appears more impressive.
There are two methods of determining the discount rate. The first is to calculate the discount rate by dividing the amount of the discount by the original price
discount rate (R) = discount (D )
list price (LP)
For example, a colour television normally sells for $650, but has been reduced to $575. To calculate the percentage discount, find the amount the purchaser will save; that is, $650 minus $575, which is $75. Divide this saving by the original selling price:
R = D = $75 = 11.5%
LP $650
To confirm this answer you must realise that the discount rate of 11.5% shows that the discounted price is 88.5% of the original selling price. Therefore, $650 x 88.5% = $575 (to the nearest dollar).
Example
The terms of payment to a retailer are a cash discount of 3.75% if payment is made within 7 days, and 2.5% for payment within 30 days. If Max buys footwear for his shop priced at $25000, how much should his cheque be for if he pays:
- within 7 days?
- within 30 days?
Answer:
- payment within 7 days:
discount (D) = list price (LP) x discount rate (R)
= $25 000 x 3.75%
= $937.50
Therefore, the cheque should be for $24 062.50 ($25 000 - 937.50)
payment within 30 days:
D = LP x R
= $25 000 x 2,5%
= $625
Therefore, the cheque should be for $24 375 ($25 000 - $625)
A faster way of working out the discounted price Max will pay in (a) is to recognise that a discount of 3.75% for payment within 7 days shows that Max will be paying 96.25% of the original price (100% - 3.75%). Therefore the cheque will be for:
Value = Base x Rate
V = $25 000 x 96.25%
= $24 062.50
and in (b):
Value = Base x Rate
= $25 000 x 97.5%
= $24 375.00
Businesses take advantage of discounts as often as possible, even if they must borrow to do so. For example, for Max to save the $937.50 in (a), he had to pay within 7 days. If he used his overdraft facility with his bank and borrowed $24 062.50 for the same period (7 days) at an interest rate of, say, 15% the interest for the 7 days will be:
Interest (I) = Principal (P) x Rate (R) x Time (T)
I = P x R x T = $24 062.50 x .15 x 1/52 = $69.41
The interest expense of $69.41 incurred for 7 days is much less than the $937.50 saved by paying early. If Max had, instead, waited the full 30 days he would have lost $937.50 in discount forgone.
Compound interest
People or institutions who have a savings or investment account with a financial institution make their deposits for a period of time and expect to earn an increased amount of interest as more time elapses. If the deposits carry a flat rate of interest calculated on the original amount invested for the whole period, a simple interest is being applied. Thus, the interest received for each period is based solely on the amount of the original deposit or principal .
In fact, all deposits at financial institutions accrue interest on the principal, and also on interest received in previous periods. The sum of the original principal plus total interest earned is called the accumulated value .
In this case the interest is described as compounded , since interest is paid on interest received in previous periods. Therefore a deposit earning a compound rate of interest will earn more interest than an identical simple rate of interest. So a compound rate of interest applies to the principal plus accumulated interest, whereas simple interest applies only to the original principal.
We often use the terms 'future value' and 'accumulated value' to refer to the sum of the principal plus the interest expected at the end of a period, and describe the difference between this accumulated value and the original principal as compound interest accrued. In other words, the future value of a principal or deposit is the total amount of the investment at the end of the term and includes the value of the original deposit plus all the interest that has accrued over the term of the deposit.
Comparing calculations of simple and compound interest . The following example shows the application of an identical interest rate to simple and compound interest to a principal amount.
Example
Max deposited $10 000 in an account carrying interest at 10% per annum. What will he have earned after four years if:
- he receives simple interest?
- He receives compound interest?
- The formula for simple interest is:
Interest amount (I) = Principal (P) x Rate of interest (R) x Period of time (T)
I = P x R x T
= $10 000 x 10% x 4
=$4000
So $4000 interest is received for $10 000 invested over four years.
- To work out the compound interest over the four-year period, the interest we must add the interest we get at the end of each year to the original principal. For example at the end of the first year we add the 10% interest earned, $1000, to the deposit and get an accumulated value of $11 000. At the end of the second year we add the 10% interest earned on $11 000, which is $1100, to get an accumulated value of $12 100.In effect, to calculate the compound interest we multiply the principal by 1.1 four times.
So, the interest compounded annually is equal to:
$10 000 (1.1)(1.1)(1.1)(1.1) or $10 000 (1.1) 4
If Max deposited the amount for eight years the compound interest will be calculated as
$10 000 (1.1) 8 .
Thus an appropriate formula to derive compound interest is:
S = P (1+I) n
Where S = sum of the principle and compound interest
P = principal
i = interest rate for the period
n = number of periods
Therefore to calculate the interest earned on Max's $10 000 over four years at 10% per annum:
S = $10 000 (1+0.1) 4
= (1.4641)
= $14 641
So interest received is $4641 ($14 641 - $10 000)
Simple interest and compound interest of 10% per annum on a principal of $10 000 over four years
Year |
Simple |
Total |
Compound |
Total |
1 |
10 000 x 0.1 |
1000 |
10 000 x 0.1 |
1000 |
2 |
10 000 x 0.1 |
1000 |
11 000 x 0.1 |
1100 |
3 |
10 000 x 0.1 |
1000 |
12 100 x 0.1 |
1210 |
4 |
10 000 x 0.1 |
1000 |
13 310 x 0.1 |
1331 |
Total |
4000 |
4641 |
Similarly, to work out interest received on $10 000 deposited for eight years at 10% per annum:
S = P(1+i) n
Therefore:
S = $10 000 (1+0.1) 8
= $10 000 (2.14358881)
= $21 425.89
The interest received is $11435.89 ($21 435.89 - $10 000).
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