Passy's World of Mathematics

Simple Interest – Part Two

Posted on March 1, 2013 by Passy

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In this lesson we look at Six Monthly, Quarterly, Monthly, and Daily Simple Interest.

In particular we look at Simple Interest Calculated on Bank Accounts.

It is recommended that you have done our previous Part 1 lesson, (at the link below), before attempting this lesson.

Basic Simple Interest Calculations

Simple Interest Formula

The following is the mathematical formula we use for all Simple Interest calculations:

When Six Monthly (or “Half Yearly”), Quarterly, Monthly, or Daily Simple Interest are involved we need to use the fractions shown below.

This is because Time must always be in Years when we use the I = PRT formula.

Example One – Half Yearly Interest

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Tahli borrows $2000 to buy a new TV at 18.5% pa Simple Interest charged 6 monthly, to be paid back over 2 years.

In this example Interest is calculated twice a year, or every six months.

This means the first thing we have to do is calculate how many dollars there are in one lot of six monthly interest.

The calculation is as shown below.

Now that we know what the Interest is every 6 months, all we need to do is work out how many lots of six months there are in the loan term of 2 years.

We can then multiply out and calculate the Total Interest to be paid over the loan as shown below.

Note that we can also calculate this same Total Interest value, using I = PRT and substitute the total time of 2 years:

I = PRT = 2000 x 0.185 x 2 = $740

It is an excellent idea to do this quick calculation as a check of the work we have done getting our previous answer of $740

The Total Amount to be paid back is always the Total Interest plus the original Loan Amount:

Example Two – Quarterly Interest

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Alexandra borrows $20 000 to set up a Beautician business at 16% pa Simple Interest charged Quarterly, to be paid back over 5 years and 3 months.

Because the Interest is paid Quarterly, the first step is to calculate the Interest each Quarter as shown below:

The next step is to now multiply out the Quarterly Interest over the full term of the Loan.

Finally we calculate the Total Cost of the Loan by adding together the Total Interest plus the original Loan Amount.

Example Three – Monthly Interest

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Joyce Deposits $2 000 into an Internet Saver Account that gives her 5.4% pa Interest paid Monthly.

She leaves the money there for 2 ½ years.

Because the Interest is paid Monthly, the first step is to calculate the Interest for one Month as shown below:

( There are 12 months in a year, so one month = 1/12 th of a Year. )

We now need to work out how many Months there are in the 2 and one half year loan.

Each year = 12 months, and one half of a year = 6 months.

So in 2 and 1/2 years we have 12 months + 12 months + 6 months = 30 months,

We now multiply this 30 months x the Interest for one month, as shown below:

Finally we calculate the Total Cost of the Loan by adding together the Total Interest plus the original Loan Amount.

Example 4 – Minimum Monthly Balance

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Some Bank Accounts give Daily Interest fr the total days in a month, by taking the Account’s lowest monthly balance as the Principal $ value to use in I = PRT

The following example shows a very simple Bank Statement:

Once we have located the Minimum Balance value, and realise that June is a 30 day month, the Interest Calculation is quite straight forward:

Example 5 – Daily Interest on Bank Accounts

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In the previous example we saw how Bank Interest is calculated on some Bank Account using the “Minimum Monthly Balance” method.

It is much better to have a Bank Account which calculates Interest based on the Daily Balance in the account. This will type of account will pay far more interest money than a Minimum Monthly Balance account.

Calculating Daily Interest that is paid at the end of each month is a bit more complicated as shown in the example below.

The first step is to add a “Number of Days” Column onto our table, and work out how many days the account was at each of the balances.

Once we have our Days column worked out, we can calculate the I = PRT interest for each of the balances, and then add them up to get the total interest for the month.

This is shown below:

Compound Interest

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Although not covered in this lesson, Compound Interest is the most powerful way of getting money to grow in value.

With Compound Interest, the Interest money is left in the account to also earn extra ongoing interest.

Unlike Simple Interest where the interest money is paid out at the end of each accumulation.

Here is a video about the Power of Compounding Interest.

Simple Interest – Part One

Posted on February 24, 2013 by Passy

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In this lesson we are looking at Interest Money associated with Loans and Investments.

When money is Borrowed or Invested, INTEREST is paid – to the Investor – or – by the Borrower.

We will only be looking at “Simple Interest” where the money is paid out at regular times, and is not left to “compound” and grow.

This means that the Interest does not earn any extra interest on itself, it is simply generated one time only.

Just to repeat, this lesson is on “Simple Interest” and does not teach “Compound Interest”.

In addition, this lesson only covers Interest being paid Yearly.

We will be covering Half-Yearly, Quarterly, Monthly, and Daily Interest payment in our “Simple Interest – Part II” lesson.

Video About Simple Interest

The following six minute Discovery Channel video gives an introduction to Simple Interest.

Simple Interest Formula

The following Mathematical Formula is used for calculating “Simple Interest”, (symbol “I”).

Let’s go through these variable letters in the formula one by one in more detail.

The “P” Principal

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The Principal is the original amount of money invested, or borrowed.

For example, if Jodie gets a $5000 loan for a car, then the “Principal” is
P = 5000.

And if Luke invests $2000 at 3% Interest per annum for three years, then the Principal is P = 2000.

The “R” Interest Rate

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If you have a lot of borrowed money, like for a housing mortgage, then even a small change in Interest Rate will cause a significant financial burden.

This happens because a small percent like 0.5%, of a big number like a mortgage for $400 000 will require the owner to pay an extra $2000 Interest in the next 12 months.

Percent Rates usually have a “pa” after them, such as 3% pa or 16% pa.

The “pa” is a short hand form for “per annum”, which means “per year”.

10% pa means that each year there is 10% of the Principal as extra Interest money created.

For the I = PRT formula, the “R” must be converted to a decimal value before putting R into the formula.

The Percent value of R is converted to a decimal by dividing by 100.

The “T” Time Value

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Over time the original “Principal” amount of money grows as Interest is added each year. Unfortunately this growth rate is in Years, rather than in hours and minutes.

The Time value in the I = PRT formula must always be entered in years.

Howver not all Interest payments are organised as Yearly. This adds a little bit of complication which involves the use of fractions.

Interest can be calculated to be paid Half-Yearly, Quarterly, Monthly, or even Daily.

For these situations, “T” needs to be expressed as a fraction of a year because the Interest Rate will usually be an Annual yearly Rate.

This is shown in the diagram below.

We will be covering Half-Yearly, Quarterly, Monthly, and Daily Interest payment in our “Simple Interest – Part II” lesson.

The Interest Rate on a Loan may not seem to be all that high, (eg. less than 20%).

However it is very important to realise how this Rate multiplied by the Time in the I = PRT formula leads to much higher than expected sums of money being involved.

This is explained in the following two minute video.

Video About Doing Simple Interest Calculations

The following fourteen minute video shows how to do common Simple Interest Calculations.

Example 1 – Calculating Interest

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“Roxanna borrows some money to buy a new car.

The loan is for $6000 at 18.5% pa Simple Interest, to be paid back over 5 years.

Calculate the amount of Interest she will have paid by the end of the 5 years.”

We need to use I = PRT to answer this question. But the “R” value of 18.5% needs to be converted to a Decimal by doing 18.5 divided by 100 to get 0.185

The working out is shown below.

The Total Amount Roxanna has to repay is all of the $5550 Interest, plus the original $6000.

Example 2 – Calculating Time

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Excel Spreadsheets with the I = PRT formula written into them can be used for doing Simple Interest Calculations.

However, we will not be working out our answers with Excel. We will use good old fashioned pen and paper.

In some situations we need to calculate the Time that money needs to be invested to earn a certain amount of Interest.

We can use one of two methods:

Method 1 – Use I = PRT, substitute in the known values, and then solve the resulting Algebra Equation for T.

This was method was demonstrated in a previous Video in this lesson.

Method 2 – Rearrange I = PRT to become the T = I / (RxT) formula and use this “T” formula directly.

We suggest trying hard to use Method 1, as this will help consolidate Algebra Equations.

However if Method 1 is too difficult, then use Method 2.

Our calculating Time example is as follows:

“How long does Keith need to invest $4000 at 3.33% pa, to earn $800 of Interest money?”

We have used Method 2 in the working out shown below.

Example 3 – Calculating Rate

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To calculate an unknown Rate, we can use one of two methods:

Method 1 – Use I = PRT, substitute in the known values, and then solve the resulting Algebra Equation for R.

This was method was demonstrated in a previous Video in this lesson.

Method 2 – Rearrange I = PRT to become the R = I / (PxT) formula and use this “R” formula directly.

We suggest trying hard to use Method 1, as this will help consolidate Algebra Equations.

However if Method 1 is too difficult, then use Method 2.

Our calculating Rate example is as follows:

“What Interest Rate do we require to get $10 000 of interest money generated from investing $20 000 for 3 years ?”

We have used Method 2 in the working out shown below.

Example 4 – Calculating Principal

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In this example we are finding the Principal Amount.

“Tayla borrowed money on 22% Simple Interest for 1 year to pay for some expensive Christmas Presents.

When she payed back the Loan a year later, the Interest that was charged was $440.

What was the original Principal Amount she borrowed ?”

The solution (using Method 2 – Direct Formula) is shown below.

Mathematics of Earning Money

Posted on February 19, 2013 by Passy

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It would be great if money just fell out of the sky like rain whenever we needed it; but unfortunately this is not the case! Most people have to get jobs and earn money.

People work, (or are “employed”), in certain jobs so they can earn money which is called an “Income”.

“Income” basically means incoming money for doing paid work during the week.

Payment for jobs can occur in different ways, depending on the agreed working conditions for the type of job.

Types of Payments

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There are four main types of Payment for jobs:

Wages, Piece Work, Salary, and Commission.

Often the type of payment depends on the job type.

For example, Trades workers are usually on “Wages”, while Sales people work on “Commission”.

Wages

Payment by wages involves a fixed Hourly Rate of Pay for a standard set number of hours. Hours worked beyond these set hours, or beyond set times of the day (eg. Night shift), are paid as “Overtime” at a higher than normal daily rate.

“Overtime usually involves “Time and a Half”, (where 1.5 times the normal hourly rate is paid), or “Double Time”, (where twice the normal hourly rate is paid for each hour worked.

Types of jobs where people are paid Wages are Trades Jobs (Plumber, Carpenter, Electrician, Welder, Mechanic, Hairdresser, etc).

Wages are also generally paid for Retail Jobs (Shop Assistant, Shelf Stacker, Cleaner, Fast Food worker, etc)

Wages are also often paid for Factory Jobs (Assembly Line Worker, Fork Lift Driver, etc)

Examples of Payment involving Wages are given later in this lesson.

Piece Work

In this type of work, people are paid for each item that is produced or processed, rather than for the number of hours worked.

Types of jobs where people can be paid by “Piece Work” are in manufacturing (dress maker, furniture maker, fruit and vegetable picker, footwear maker, hand bag or hat maker, dog groomer, dog walker, etc)

Couriers and Truck Drivers can also be paid by “Piece Work” based on how many items they transport or deliver.

Examples of Payment involving Piece Work are given later in this lesson.

Salary

This type of Income involves Weekly, Fortnightly, Monthly, or Annual rates of pay. Workers are often required to work overtime, but it is rarely paid. Instead time off may be given (if you are lucky enough to have it written into your employment conditions).

People who work on Salary are often Professionals such as Doctors, Lawyers, Engineers, Scientists, Accountants, Politicians, and Teachers.

Commission

For income earned by Commission, workers are paid a small “Retainer” or “Flat Fee” plus either a percentage of the dollar value of the Sales they make, or else a certain amount for each sale they make.

Jobs which have Commission pay are usually Sales jobs such as Real Estate, Car Sales, Sales Reps for Manufacturing companies, Book Sellers, Mortgage Brokers, and Insurance Consultants.

Examples of Payment involving Commission are given later in this lesson.

There are no fixed rules set in concrete for how some occupations are paid. For example someone might be a Computer Programmer paid an hourly rate, or maybe they are only paid for building one particular App, no matter how many hours it takes them.

Somebody else might be a Physiotherapist and either paid based on how many patients they see, or instead they may be paid a fixed hourly or daily rate.

Usually what determines how an employee is actually paid is a set of agreed working conditions that exist between “Employer” (the Boss) and the “Employee” (the worker).

Agreed Working Conditions

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There are three main types of agreed working conditions :

Awards, Agreements, and Contracts

These documents spell out quite clearly what work needs to be performed, the usual hours this work is done in, and the basic pay structure for this work.

Awards

An award is a legal document that sets out minimum wages and conditions for an industry or occupation.

Awards cover things as rates of pay, overtime, penalty rates, and allowances for travel, meals, etc.

The conditions in awards apply on top of the minimum conditions in the Australian National Employment Standards.

For more information about Awards in the Australian workplace, (especially to find out if you are being paid correctly), check out the following web page:

http://www.fairwork.gov.au/awards/pages/default.aspx

Agreements

An agreement is like an award created specifically for the employees of a particular company.

An agreement must be at least equal to the corresponding award.

A typical “Work Agreement” includes:

The names of the parties involved in the agreement, (eg. Employer and Employee)

The start and end dates of the agreement

The articles to which each party agrees, (eg. the work that is to be done by the employee, and the ways in which the employer will provide equipment and facilities to help this work to get done)

The details of the agreement such as compensation (pay), benefits (sick leave, holidays, etc) and repercussions (poor work performance, unapproved absences, etc).

Contracts

People doing very specialised work (like a programmer building an App or Web Pages, or a builder organising the construction of a home, or a Surveyor mapping out a new housing estate, a band making an album), will sign a contract for the work they do.

The Contract will link to a Project Plan Timeline, and detail what work needs to be completed by what dates, and how much will be paid upon completion of each stage of the project.

People on Contract often have to organise their own equipment and materials, working hours, sick pay, and holidays, as these are not usually part of the signed Contract.

Calculating Wages Videos

The following set of videos, (watch them one after each other), show detailed examples of how to calculate Wages and Overtime.

Example 1 – Earning Wages

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Laura is a forklift driver at a fruit farm and gets paid $22 an hour weekdays, and time and a half on weekends.

What is her pay for a week where she works a total of 20 hours for Monday to Friday, and then works 10 hours on the weekend ?

Break the question up into normal pay and time and a half. Work out each component, and then total everything up for the final answer.

Example 2 – Earning Wages

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Liam works on an assembly line making cars.

He gets paid a combination of Normal Pay, Time and a Half, and Double Time, as shown in the following example week’s pay.

Example 3 – Earning Piece Rates

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Ciara is working as a Make Up Artist for Weddings and other formal functions.

She gets paid $50 for each person she does make up for.

How much does she earn for in a week where she does two Weddings, and a School Formal, consisting of doing make up for a total of 22 people.

Example 4 – Earning Salary

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Ebony is a Maths Teacher with a yearly Salary of $48 000, which is paid to her fortnightly over the whole year.

How much does Ebony get paid every two weeks ?

Example 5 – Earning Commission

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Tom is starting out as a Car Salesman and earns a Base Salary of $400 a week. He then gets 5% Commission on any Cars that he sells.

What is his weekly pay for a week in which he sells 3 cars totalling $120 000 in value ?

Calculating Commission Videos

The following video includes several examples including a multi-part graduated commission situation.

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Posted in Math in the Real World, Percentages | Tagged business mathematics, calculate commission, calculate pay, calculate piece rates, calculating earnings, calculating overtime, calculating wages and overtime, earning commission, earning money math, earnings math, financial mathematics, how to earn money, income math, maths of income, maths of jobs, piece rate, piece rate math, Retail Mathematics, salary calculation, wages math | Leave a comment

Applied Percents – Loss

Posted on February 18, 2013 by Passy

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In a previous lesson we looked at how Retailers place “Mark Ups” on items so that they can make a Profit on the sale.

It is important that you have viewed this lesson before doing the contents of our Loss Lesson.

The Mark Up and Profit Lesson can be viewed by clicking the link below:

http://passyworldofmathematics.com/cost-price-mark-up-and-profit/

Definitions for Items in Formulas

Cost Price

The Cost Price is how much it costs a retailer to buy items they are going to sell from suppliers or from overseas.

Mark Up

The Mark Up is extra money that the retailer adds on before putting a final price on the item.

Marked Price

This is the final price of the item. The Marked Price is then written onto the tag or sticker that is put onto the item in the shop.

This is the price the customer has to pay for the item and is also often called the full “Recommended Retail Price” or “RRP”.

If the item is NOT on discounted sale, then the customer has to pay the full RRP that is on the item’s tag or sticker.

Discount

If the shop is having a Sale, you do not have to pay the full Marked Price on the item. This is because the retailer gives you some money off the full normal price to create a lower than normal “Sale Price”.

Sale Price

The Sale price is the price the customer ends up paying for the item.

If the item is on sale for a discounted price, the Sale Price (sometimes called the “Selling Price”), will always be lower than the full normal Marked Price.

But if the item is not on sale, the customer will have to pay the full normal Marked Price.

Profit

Retailers sell items for more than it cost to get them, (even when the items are on Discount Sale). This allows the Retailer to make some money.

The money that the retailer makes from selling the item is called the “Profit”.

Loss

Sometimes a retailer will get into money problems, or have more items than they can fit in their shop or warehouse.

When this happens they have to sell goods at desperately low prices to get money to pay for shop rent, wages, electricity, transport, etc.

If they do not take this desperate step, they will get into huge debt and go out of business.

If they have to sell an item for less money than it originally cost them, then they will lose money on that item.

Losing money on a sale because the Sale Price is really really low, is called “Making a Loss”.

Dollar Loss

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Tommy and Tamara buy 100 “Retro TVs” at a Cost Price of $600 each.

They thought these TVs would be a real hit with young people and priced them with a Marked Price of $1000 each.

However, after two months they had only sold one TV set, and the other 99 TVs were taking up valuable space in their store room and gathering dust.

They then had to make the sad decision to sell the TV’s in a clearance sale at below cost with a Sale Price of $450.

This means that they do not make any Profit on each TV, instead they make a loss.

Percentage Loss

Just as we covered with “Profit” in a previous lesson, there are two ways to calculate Percentage Profit and Percentage Loss.

There are two ways businesses use to express Percentage Loss:

1) Percentage Loss based on the Cost Price

and

2) Percentage Loss based on the Selling Price

This can be very confusing for people new to Financial Mathematics. The trick is to read the given question or situation very carefully.

If the question just asks for “Percentage Loss” (and does not specify based on Selling or Cost), then calculate the Percentage Loss based on the Cost Price.

Shown below is how we calculate the Percentage Loss based on the Cost Price:

Shown below is how we calculate the Percentage Loss based on the Selling Price

Selling Price with Loss Formulas

We calculate the Selling Price for items which are being sold at a Loss as follows:

Note that the above formulas are written in a format which should aloow direct entry of values into pocket calculators.

Loss Formulas Summary

Here are the main formulas we use for working with Losses.

These need to be copied down into the maths notes for Losses in your workbook.

Profit and Loss Video

Here is a very comprehensive video about Profit and Loss with plenty of example questions in it.

Applied Percents – Discounts

Posted on February 16, 2013 by Passy

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In our previous lesson we looked at how Retailers place “Mark Ups” on items so that they can make a Profit on the sale.

It is important that you have viewed this lesson before doing the contents of our Discount and Loss Lesson.

The Mark Up and Profit Lesson can be viewed by clicking the link below:

http://passyworldofmathematics.com/cost-price-mark-up-and-profit/

Definitions for Items in Formulas

Cost Price

The Cost Price is how much it costs a retailer to buy items they are going to sell from suppliers or from overseas.

Mark Up

The Mark Up is extra money that the retailer adds on before putting a final price on the item.

Marked Price

This is the final price of the item. The Marked Price is then written onto the tag or sticker that is put onto the item in the shop.

This is the price the customer has to pay for the item and is also often called the full “Recommended Retail Price” or “RRP”.

If the item is NOT on discounted sale, then the customer has to pay the full RRP that is on the item’s tag or sticker.

Discount

Sale Price

The Sale price is the price the customer ends up paying for the item.

If the item is on sale for a discounted price, the Sale Price (sometimes called the “Selling Price”), will always be lower than the full normal Marked Price.

But if the item is not on sale, the customer will have to pay the full normal Marked Price.

Profit

Retailers sell items for more than it cost to get them, (even when the items are on Discount Sale). This allows the Retailer to make some money.

The money that the retailer makes from selling the item is called the “Profit”.

Loss

Sometimes a retailer will get into money problems, or have more items than they can fit in their shop or warehouse.

When this happens they have to sell goods at desperately low prices to get money to pay for shop rent, wages, electricity, transport, etc.

If they do not take this desperate step, they will get into huge debt and go out of business.

If they have to sell an item for less money than it originally cost them, then they will lose money on that item.

Losing money on a sale because the Sale Price is really really low, is called “Making a Loss”.

GST – Goods and Services Tax

Most goods in Australia are subject to a 10% “GST” Tax, except for some basic grocery and food items.

This means that 10% tax is included in the price the customer pays.

When a Retailer gets the original item at Cost Price, then does the Mark Up, then might give a sale Discount, there is a resulting final price the customer is going to pay.

In most cases, 10% of the money which the Retailer gets from the Sale has to be forwarded by the Retailer to the Goverment as GST.

Retailers need to be aware in their Financial Planning that 10% of all their profits are taken away by the Government as GST Tax.

This means that Retailers need to average more than 10% Profit on their Sales. Otherwise they will not make any money and will go out of business.

Discount and Selling Price

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Tommy and Tamara’s TV Shop are having a “10% off” sale.

This means that customers can get any TV on sale for 10% cheaper than the normal Marked Price that is on the price label on the TV.

We say that customers are getting a “10% Discount” or that they are making a “Saving of 10%”.

The following digram shows the mathematical formula we use for calculating the sale price of a TV which is discounted by 10% during the sale.

Discount Dollars

We can calculate the Discount amount (which is the amount saved) using the formula:

Discount Dollars = Marked Price – Selling Price

This is shown in the following diagram:

Working Backwards to Find Marked Price

If we know the Sale Price, and the Percentage Discount, we can work backwards to find the original “Marked Price” by using the following Formula:

Note that we have written the formula in a form which shows how it must be typed into a Calculator to get the correct value for Marked Price.

Price the Customer Pays

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The discounted price which the customer pays is called the “Selling Price”.

Discount Formulas Summary

Here are the main formulas we use for working with Discounts.

These need to be copied down into the maths notes for Discounts in your workbook.

How to Select the Correct Formula

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To choose the best formula to use for a financial mathematics word problem, we do these steps:

1) Identify the unknown – eg. what is it we need to find the missing value of (Marked Price, Selling Price, Discount $, %Discount, etc).

Make sure that the unknown item is on the left hand side of our formula.

2) Identify what is given to us in the problem out of Marked Price, Selling Price, Discount $, %Discount, etc.

Make sure the given items are on the right hand side of the formula.

Videos About Doing Discounts

Here is a quite detailed Video about Discounts and Sale Price.

Discounts Online Calculator

We can use this calculator to get discounts. But this calculator is extra nice, because we can also enter the “Reduced” discounted price, and will it will calculate the original price.

So the calculator works in both directions.

Note that we always have to enter the Percent Value, but we only have to enter one of the three items on the top input line.

http://discount.miniwebapps.net/

Passy's World of Mathematics

Simple Interest – Part Two

Simple Interest – Part One

Mathematics of Earning Money

Applied Percents – Loss

Applied Percents – Discounts

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