Consumer Arithmetic
Mathematics
Topic Thirteen

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Percentage Profit and Percentage Loss

The definition of profit is any monetary gain made from a transaction. A profit is made when the selling price of a product is greater than the cost price. 

Therefore,

Profit = Selling price – Cost price.

Percentage profit = profit/cost price x 100

A loss is made from a transaction if the selling price of a product is less than the cost price.

Therefore,

Loss = Cost price – Selling price

Percentage loss = loss/cost price x 100

Example:

A man bought 10 tools for $200 each and sold them at $400 each.

  1. Calculate his profit
  2. Calculate his percentage profit.

Solution

  1. Profit = $400 x 10 - $200 x 10 = $4000 - $2000 = $2000.
  2. Percentage profit = Profit/Cost Price x 100

    = 2000/2000 x 100 = 100 %

Finding cost price (CP) when the selling price and percentage profit or loss are given

Rule: Cost Price (CP) = 100/100 (100 plus or minus % profit or loss) times the selling price.

Example

If a store manager made a 20 percent profit by selling a product for $480, calculate the product’s cost price.

Solution

Cost Price (CP) = 100/ (100 plus 20 profit) times the selling price.

CP = 100/120 x 480 = 400.

Therefore, the cost price is $400.

Example

If a store manager made a 20 percent loss by selling a product for $800, calculate the product’s cost price.

Solution

Cost Price (CP) = 100/ (100 minus the 20 loss)/ times the selling price.

CP = 100/80 x 800 = 1000.

Therefore, the cost price is $1,000

Finding selling price (SP) when the cost price and percentage profit or loss are given

Rule: Selling Price (SP) = 100 (100 plus % profit or minus % loss)/100 times the cost price.

Example

Jack’s Pizza bought a machine for $480 and sold it for a 20 % profit.  Calculate the selling price. 

Solution

SP = 120/100 x 480 = 576.

Therefore, the selling price is $576.

Example

Jack’s Pizza bought another machine for $480 and sold it making a 20 % loss.  Calculate the selling price. 

Solution

SP = 80/100 x 480 = 384.

Therefore, the selling price is $$384.

Discount

A discount is a reduction in the selling price of a product. Discounts are usually given as a percentage of the selling price.  The discount price is the difference between the selling price and the discount.

Example:

A computer was marked at a selling price of $5000. If a 5% discount is given to a customer, what is the discounted price of the computer?

Solution

Discount = 5/100 x 5000 = 250

The discounted price = 5000 – 250 = 4750.

Example:

After a discount of 20 %, a tool was sold for $800.  Calculate the price of the tool before the discount.

Original price before discount = 100/80 x 800 = 1000.

Sales Tax

Sales tax is a fee charged by governments on products, and is calculated as a percentage of the selling price of a product. To find the selling price of a product inclusive of the sales tax amount, add the sales tax to 100% and find that percentage of the selling price. So if the sales tax is 15% , the formula would be:

Selling Price after sales tax = 115/100 x price before the sales tax.

Original Price when sales tax is included = 100/115 x selling price inclusive of sales tax

Example:

Given that the sales tax on all products is 15%, find the final selling price of a product which costs $3000.

Solution

Selling price inclusive of sales tax = 115/100 x 3000 = 3450.

Therefore, the selling price inclusive of the sales tax is $3,450.

Example:

Given that the sales tax on all products is 15% and the final selling price of a product was $3300, find the price before the sales tax.

Solution

Original price = 100/115 x 3300 = 2870

Therefore, the selling price before the sales tax is $2,870.

Simple Interest

Simple interest is the interest calculated on a principal sum. Principal is the sum of money invested in or borrowed from an institution, denoted as P. The money earned by an invested principal and the money charged for borrowing a principal amount from an institution is also called interest, denoted as I. The amount of interest earned or charged per year is expressed as a percentage and is known as the rate of interest, denoted as R.

Therefore,

SI = (P x I x T)/100

P – principal

I – interest

T - time

Example

A person invested $6000 in a bank at 20 % per annum for 5 years.

  1. What is the simple interest earned?
  2. Determine what this interest earned equals to in terms of monthly payments.

Solution

  1. Simple interest (SI) = (6000 x 20 x 5)/100 = 6000
  2. Determination of monthly interest earned = Total SI/number of months = 6000/60 = 100

Calculating Rate

The formula for finding Rate when Principal, Interest and Time are given is:

From: Interest = (Principal × Rate × Time)/100

Rate = (100 × Interest)/(Principal × Time).

Example

Calculate the interest rate required for an investment of $8000 to earn $400 in simple interest over 4 years.

Solution

R = (100 x 400)/(8000 x 4) = 40000/32000 = 1.25 %

Calculating Time

The formula for finding Time when Principal, Interest and Rate are given is:

From: Interest = (Principal × Rate × Time)/100

Time = (100 × Interest)/(Principal × Rate)

Example

What time will it take for $1200 to earn $360 in simple interest at 3 percent?

Solution

T = (100 x 360)/(1200 x 3) = 36000/3600 = 10 years.

Compound Interest

Compound interest is interest which is calculated on the principal sum for the first year and on the accruing amount for the remaining years.  The formula for annual compound interest, including principal sum, is:

A = P (1 + r/n) (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 years.

Example

A man deposited $4000 into a bank account paying 4 % interest for 4 years.

  1. Calculate his compound interest for the 4 years.

    A = P (1 + r) (n)

  2. If he puts the same amount of money paying simple interest, which is the more profitable investment?

Solution

  1. A = 4000 (1 + .04)4

    = 4000(1.04)4

    = 4000(1.16985856)

    = $4,679

  2. SI = (4000 x 4 x 4)/100 = $640

    Total investment is the simple interest (640) plus the initial investment or deposit (4000)

    = 4000 + 640 = $4640

In order to determine whether the compound interest investment or the simple interest investment is better, we need to compare the total amount accumulate at the end of the period which is 4679 for the compound interest and the 4640 for the simple interest.  Because the compound interest accumulation (4679) is greater than the simple interest accumulation (4640), the compound interest investment is better and more profitable.

Depreciation

Most fixed assets such as machinery decrease in value or depreciate over time.  This decrease in value is called depreciation.  To find the amount by which an asset depreciates over a year, the rate of depreciation (the percentage by which it decreases) is multiplied by the initial cost of the asset, the depreciation amount is then subtracted from the initial cost to give the current book value of the asset.  The following equation is used to calculate depreciation.  It is similar to the compound interest equation but instead it has a minus sign instead of positive sign. 

D = P (1 - r) (n)

Example

A company purchased a machine for $8000 which depreciated at a rate of 4%.

  1. Calculate the value of the machine after 4 years.
  2. By how much did the machine depreciate?

Solution

  1. D= 8000 (1 - .04)4

    = 8000(0.96)4

    = 8000(0.8493) = $6795

  2. By how much did the machine depreciate?

    Depreciation = original value of the machine – the value of the machine

    = 8000 – 6795

    = $1205

Hire Purchase

Hire purchase is a method of buying goods through making installment payments over time. Most hire purchase agreements require a deposit to be made and monthly installments over a specific period of time usually 2 to 5 years.

Example        

A man has an option of paying $5000 cash for a furniture set or through hire purchase with a deposit of 20 % of the cash price and 36 monthly installments of $200.  If he chooses to purchase the furniture set for cash, how much will he be saving?

Solution

Cash

Cash = $5000

Hire purchase

Deposit = 20/100 x 5000 = 1000

Installments = 200 x 36 = 7200

Total hire purchase price = 1000 + 7200 = $8,200

By paying for cash instead of hire purchase, the man will be saving = 8200 – 5000 = $3200.

Wage

There are some mathematical computations relating to wages and hours of work that students will need to become familiar with.

Example

A man works Monday to Friday at a wage rate of $8 per hour.  On Saturdays, he works overtime at double time meaning he is paid for his overtime at twice his normal wage rate of $8.  For a particular week, he worked his normal hours plus overtime of 8 hours at double time.  Calculate his total income for that week.

Solution

Normal work week:

Hours Monday-Friday = 8 x 5 = 40 hours

Normal wage = $8 per hours

Normal income = 40 x 8 = $320

Overtime work week:

Hours Saturday = 8 hours

Overtime wage = $8 per hours x 2 = $16 per hour

Normal income = 8 x 16 = $128

Total income = 320 + 128 = $448

Commission

If a salesperson is paid a certain percentage of the value of the sales, the amount received by the salesperson is called a commission.

Example

A car salesperson is paid a fixed amount of $1000 a week plus 5% commission on his sales. Calculate his total weekly earnings if his sales were $20,000 for the week?

Solution

Fixed earnings =$1000

Commission = 5 % of 20,000 = $1000

Total Earnings = Fixed Income + Commission Income = $1000 + $1000 = $2,000

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