PERCENTAGE Calculating percentage change by iaj67571

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									PERCENTAGE
Calculating percentage change
                                                                                                     Lesson
Learning Outcomes and Assessment Standards
 Learning Outcome 1
                                                                                                              4
 Number and Operations in Context
 The learner is able to use knowledge of numbers and their relationships to investigate a range of
 different contexts, which include financial aspects of personal, business and national issues
 Assessment Standard
 AS 11.1.1
 In a variety of contexts find ways to explore and analyse situations that are numerically based,
 by:
 • Estimating efficiently
 • Working with formulae by hand and with a calculator
 • Showing awareness of the significance of digits
 • Checking statements and results by doing relevant calculations
 (The range of problem types includes percentage, ratio, rate and proportion.)



Overview
In the previous lessons we have looked at the following calculations:
● Changed fractions or ratios into percentages;

● Calculated a percentage of an amount;

● Added a percentage of an amount to an amount; and

● Subtracted a percentage of an amount from an amount.


In this lesson we will consider another type of percentage calculation. We will:
● Calculate a percentage change.




Lesson
A common example of percentage change is the year-on-year inflation. If inflation                         DVD
is said to be 6%, this gives an indication of the percentage change in the price
of a basket of goods from one year to the next. This means that the price of the
goods has increased by 6%.The way that inflation is measured is as follows:
Every month, let’s say January 2006, statisticians find out the cost of a list of
1 500 goods and services. For our example let us say this comes to R5 000.




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     One year later, in January 2007, statisticians find out the cost of that same list of
     1 500 different goods and services. This time they find that the cost is R5 650.
     Note: they use the same list of goods and services each time to ensure that they
           are comparing the same things with one another.
     The basket of goods and services has increased by R650.
     This increase, expressed as a percentage of the original amount, is called the
     year-on-year inflation or percentage change.
     Note: when we added a percentage of an amount to an amount, in a previous
           lesson, we knew the percentage and had to find the actual amount to add
           it on. In this case we know the amount that has been added so we need to
           express it as a percentage of the original amount.

     Methods and worked examples
     For the inflation problem above we can write down the steps on how to calculate
     percentage change as follows:
     Step 1: find the difference between the final value and the original value.
     Final value – original value = R5 650 – R5 000
                                  = R650
     Step 2: find what percentage this is of the original amount.
     This takes us back to the first lesson when we learnt to express a fraction as a
     percentage. The calculator keys are: 650/5000%
     This gives an answer of 13%
     We can write the above two steps as a formula:
                final value – original value
     % change = ____             
                   original value
                                         (expressed as a %)
     The inflation example was an example of where the amount had increased. The
     following example is an example of where the amount decreases.

     Worked example:
     Last year 150 incidents of theft were reported at our school. This year 117
     incidents were reported. What is the percentage change?

     Solution:
                final value – original value
     % change = ____  (expressed as a %)
                                 
                   original value
                  117 – 150
                = __ 
                   150
                                              (expressed as a %)
                    33
                    _
                = –150                        (expressed as a %)
                = –22%
     The key strokes on the calculator are:
     117-150/150%
     The negative sign in the answer shows that the original amount decreased or got
     smaller.
     Note: this calculation reminds us of subtracting an amount from an amount. In
           this example we know the whole, i.e. the original amount. We worked out
12         what was subtracted, i.e. the difference between the final and the original
           amounts, and we express this as a percentage of the whole.
In the case of subtracting an amount from an amount, we knew the percentage
change so we first calculated what the amount to be subtracted was and then
we subtracted it from the amount. In this case we know the amount that was
subtracted and our job is to calculate what percentage it is of the original amount.
Consider the following additional examples:
Additional example 1
1.     What is the percentage change in the price of a can of cold drink if the
       price went from R2,50 to R2,75?
Additional example 2
2.     What is the percentage change in the volume of water in a tank that goes
       from 1 600 l of water to 1 150l of water?

Solutions:
                   final value – original value
(3)     % change = ____  (expressed as a %)
                                    
                      original value
                      2,75 – 2,50
                    = __ 
                        2,50
                                                (expressed as a %)
                      0,25
                      _
                    =2,50                       (expressed as a %)
                    = 10%
       The answer is positive because there was an increase in the price of the
       cold drink.
(4)     % change = final value - original valueoriginal value      (expressed as a %)
                    = 1 150 – 1 600/1 600                          (expressed as a %)
                    = – 450/1 600                                  (expressed as a %)
                    = –28,125% ≈ –28%
The answer is negative because there was a decrease in the volume of water.


Activity
                                                                                        InDIVIDual
Practice examples
1      Calculate the percentage increase in the price of bread if the price
                                                                                        summative
       increased from R5,20 to R5,90.                                                   assessment
2      Calculate the percentage of VAT on goods and services if a pair of
       sunglasses costs R250, excluding VAT, and R285, including VAT.
       (VAT: value added tax )
3      The table below shows how the population of South Africa grew from
       1996 to 2001.
        1996 40,5 million
        2001 44,8 million

       Calculate the percentage growth in population from 1996 to 2001.
4      The table below shows how the urban population of four centres in South
       Africa has grown from 1991 to 2007.
                            1991 (census)   1996 (census)       2007 (estimate)
        Johannesburg
        Cape Town
                            4 180 897
                            2 175 951
                                            4 508 915
                                            2 775 677
                                                                7 372 314
                                                                4 709 212
                                                                                             13
         George             63,895          94,121           197,056
         Vryheid            25,099          53,228           175,119
                                                             Information from World Gazetter

         4.1       Calculate the percentage increase in population for Johannesburg
                   from 1991 to 1996.
         4.2       Calculate the percentage increase in population for Johannesburg
                   from 1996 to 2007.
         4.3       Calculate the percentage increase in population for Cape Town
                   from 1996 to 2007.
         4.4       Calculate the percentage increase in population for George from
                   1996 to 2007.
         4.5       Calculate the percentage increase in population for Vryheid from
                   1996 to 2007.
         4.6       Which of the four centres had the biggest percentage growth in
                   urban population?
         4.7       Which of the four centres had the most people moving to the
                   centre since 1996?
         4.8       Calculate the percentage increase in population for Vryheid from
                   1991 to 1996.
     5   The following advertisement appeared in the local newspaper:

                       One-Year Business Computer
                                      Diploma
                          EARN WHILE YOU LEARN!!
                      Was R18 999 now only R11 999

         What percentage discount is this company offering for this course?
     6   A sheep farmer had 78 new-born lambs. After unexpected cold weather
         he had only 56 lambs left because many of them died. What percentage
         of his flock of lambs did he lose?
         7.1       Rashid’s salary went up from R4 500 a month to R4 800 a
                   month. What is his percentage increase?
         7.2       Tshapelo’s earns R5 000 a month and his salary also went up by
                   R300 a month. What percentage increase did he get?




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