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Percent change: An outstanding solution to a constantly changing values

The world of mathematics has been in recent times changing and evolving to some new findings such as the Twin Prime conjecture, abc conjecture, Navier stokes problem, to mention few. Percent change is another step up in the mathematical world as it has been used severally and in day to day activities, without us - the non-mathematicians - knowing about it.

Percent change is a concept in mathematics which was proposed in order to be able to describe the relationship between an old value and a new value. In case we all wonder whether or not values are the only ones subjected to this described relationship, Percent Change can also be used to describe the type of relationships that exist between two quantities.

In ancient Rome, before the existence of a decimal system, computations were made in fractions that were multiples of 1/100. Many of these arithmetic texts use the approach of profit and loss, interest rate, as well as rule of three.

A quick instance for those that are still wondering if any relationship does actually exist between two variables, values or quantities. Let’s say V1 is our old value (a.k.a. initial value) and V2 is our new value, Percent Change can be found with a specifically designed equation: {(V2 – V1)/ V1} x 100. With this breakdown, it is very possible for us to find out that relationship. In real values, assuming we substitute our V1 and V2 as $100 and $50 respectively, it then follows that the relationship between that initial $100 and the new $50 will be 50%.

From a logical point of view, as done when carrying out essay proofreading, let’s say the North Pole had 50 birth cases last year, as did Capital City which means there is no difference in birth cases between these cities. well, we go back and look at the number of births in those towns in previous years, so that we can determine a percent change. 5 years back, Capital City had 42 births while the North Pole had just 29. Subtract the old value from the new value for each city then divide by the old or previous values. For Capital City: means 50 - 42 and dividing that result by 42. For the North Pole, figure 50 - 29 and divide that result by 29. That will show you that, over a 5-year period, Capital City had a 19 percent increase in births, while North Pole's increase was more than 72 percent.

To best understand this simple calculation, we further classify percent change into: Percent increase and Percent decrease. These two are measures of percent change, which is the extent at which something gains or loses value. Percent change s are helpful in making people understand differences in the value over time.

Percent increase is calculated by first finding out the difference between the 2 sets numbers to be compared. But first, an increase will be calculated in order to calculate the percent increase. The formula to use: increase = new number - original number. The answer you get after you have subtracted original number from the new number is then divided the original number itself and finally multiplying the answer by 100. Practically, this is what it should look like:

  • % increase = increase ÷ original number x 100. A positive answer indicates percent increase.

Percent decrease is calculated using the same principle as the percent increase. We must, first of all, calculate the decrease; Decrease = original number – new number, after which the answer you obtain for decrease will be divided by the original number again, then later multiplied by 100. A negative answer indicates a percent decrease. Here is a quick view practically: % Decrease = decrease ÷ original number x 100.

Percent change types and different approaches

Unlike working on a regular grant proposal, let’s use some more realistic values and situation for better understanding. We will be using a medical doctor by the name Adam. In January, Adam a medical practitioner worked a total of 35 hours, in the month of February he worked 45.5 hours – by what percentage did Adam’s working hours increase in February? In order to solve this problem, percent change can be used but first, we need to calculate the difference in hours between the new number and the old numbers, i.e. 45.5 - 35 hours = 10.5 hours. We noticed that Adam worked 10.5 hours more in the month of February than he did in the month before (January). This is his increase in his job. By the way, Dr Adam is a good and hard working doctor. To work out the increase as a percentage, it is necessary to divide the increase by the original number (January):

    • 10.5 ÷ 35 = 0.3.

Lastly, to get the percentage we multiply the answer by 100, which simply means moving the decimal place two columns to the right:

      • 0.3 × 100 = 30.

Adam has therefore worked 30% more hours in month of February than he did in January.

Deeper into his work life, in the month of March, Adam worked 35 hours again like he did in January or say 100% of January hours. What is the percent change between Adam’s February hours 45.5 and his March hours was 35? You might think that as there was a 30% increase between Adam’s January hours (35) and February (45.5) hours there will be 30% decrease between his February and March hours. This is an incorrect assumption. First calculate the decrease in hours, i.e.:

    • 45.5 - 35 = 10.5

      Then divide the decrease by the original number which is the month of February working hours:

      • 10.5 ÷ 45.5 = 0.23

      Finally, multiply 0.23 by 100 to give 23%. Adam’s hours were 23% lower in March than in February. This clearly shows assumptions sometimes might not give the best answer.

      Most times it is easier to point out percent change in the form of percentage decrease as a negative number, to do this we use the formula above to calculate the percentage increase. The answer will be a negative number if there was really a decrease. In Adam’s case; the decrease in work was -15.5. -10.5 ÷ 45.5 = -0.23. -0.23 × 100 = -23%.

      Just like in book reviews it is very useful to be able to calculate the actual values based on the percent increase or decrease. The most common example will be a weather forecast in the media. You frequently see headlines such as: Japan rainfall was 23% above average during the summer, Unemployment figures show a 2% decline and last example could be: A Surgeon’s bonuses slashed by 45%. This kind of headlines give an idea of a trend where something is either increasing or decreasing, but often no actual data.

      In order to find out how much something has increased or decreased in real terms, actual data is needed to calculate percent change. Using the example of an accounting paper on Japan rainfall that increased 23% above average during the summer, it is now clear that Japan experienced almost a quarter (25%) more rainfall than average over the summer. However, without knowing how much rain fell over the period in question, we cannot work out how much rain actually fell.

      But it is a different situation when we actually know the amount of rainfall for that period which means percent change can be calculated. For instance, it was reported that the average rainfall over the summer period is 250 mm, then we can easily work out the amount of rainfall using the summation of 250 mm of the rainfall and 23 % increment in rainfall.

      First work out 1% of 250 i.e. 250 ÷ 100 = 2.5, then multiply the answer by 23, this is because there was a 23% increase in rainfall, therefore, 2.5 × 23 = 57.5. Rainfall for the period broadcasted was therefore 250 + 57.5 = 307.5 mm but if it states the new measurement of a percent change, “Japan rainfall was 23% more than average and the total of 320 mm of rain fell”. In this example, we know the total amount rainfall of 320mm, we also know that there is 23% above average. The average rainfall is calculated thus: we divide the total (320) by 1.23.

      Therefore, 320 ÷ 1.23 = 260.2 while the percent change between the average rainfall and the actual rainfall is now calculated thus: 320 - 260.2 = 59.8. Therefore, 59.8 is 23% of the average rainfall amount (260.2), and that in real terms 59.8 mm more rain than average fell. Percent change can also be calculated in other ways different from the earlier explained methods. Midpoint method is a new way of going around this kind of problem by using the average the two values in question to find the percent change. The formula for this method is illustrated below

        • %∆price =(P2-P1√(5(P1+P2)×100

      this formula gives us an average of the 2 values and we can use the formula for any 2 values we may need to calculate average including price, quantity, and income. Blake's income has increased from $100,000 to $108,000 while the consumption of cruises increased from one cruise to two cruises. Find the percent change in his Income Elasticity of Demand using the midpoint method. income = ($108,000 - $100,000)/.5($100,000 + $108,000) x100 = 8%. Income Elasticity = 67%/8% = 8.4.

      Percent Change can also be used to calculate the change in growth rate over a long period of time. The percent change from one period to another is calculated using the formula described below:

          • PR = (Vpresent - Vpast)/Vpast x 100


          • PR means percent rate
          • Vpresent means present or future value
          • Vpast means past or present value

      Therefore the annual percent change in growth rate is simply the percent growth which will then be divided by N, the number of years. For example, following a lab report format : In the year 2000, the population in Lane city was 250,000. This grew to 280,000 in 2001. The annual percentage growth rate for Lane City can be calculated using the above formula.

          • Vpresent = 280,000
          • Vpast = 250,000
          • N = 10 Years
          • PR = {(280,000 – 250,000)/250,000x100)/10} = 1.2%. The population of Lane City grew 12% between 1980 and 1990 or at a rate of 1.2 percent annually.
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