Understanding the real nature of percents

$Shapes$

~First of all, here are some bascis for the non-math person

~To calculate a percentage, place the number in question over the base amount representing the total or starting point.

~This will give you a fraction. It could be improper (top bigger than bottom).

~Then, divide the top by the bottom. This result may be a whole number or a decimal.

~Convert this result to a percent by moving the decimal point two places to the right. Only now, we use the % symbol.

~Example: 5 is what % of 17?

~Solution: 5/17 = .2941 (four places) = 29.41 %

~Example: 18 is what % of 6?

~Solution: 18/6 = 3 = 300% (decimal is to the right of a whole #)

~To convert a % to a decimal, move the decimal point two places to the left & DROP the % symbol.

~IMPORTANT: A very common mistake beginning students make using percentages in calculations is that they fail to convert the percentage to a decimal during the calculation. So, for all operations dealing with percentages (adding, subtacting, multiplying, dividing, & powers), percentages must be converted to a real number (without the % symbol). That means, it must be in decimal form.

~For example, for a problem dealing with compound interest, if the percentage is 6.7%, it must be expressed as .067 during the calculation.

~Note: If no decimal is visible, it is understood to be at the far right of a number.

~Example: 37 % = 37.% = .37

~To convert a % to a fraction, place the percentage (WITHOUT THE % SYMBOL)over 100.

~Example: 22.5 % = 22.5/100 = 225/1000 = 9/40 (divide top & bottom by 25 to get the reduced form).

~Note: percentages can be probabilities but not always. If they are between 0 % and 100 %, they could represent a probability since probabilities are always numbers between 0 and 1.

~However, 230 % = 2.30 can not represent a probability since this number is greater than 1.

~You should be aware of how percentages are computed & how they can be very deceiving & misleading.

~Always remember that percentages are computed on a base amount.

~If the base amount is large, a modest increase will be a very small percentage increase. However, if the base amount is very small, a modest increase will be a large percentage increase.

~For example, an amount increases from 500 to 550. That's a 50 point increase. The % increase is based on the original amount of 500, so (increase divided by the starting amount) 50/500 = 0.1 = 10 % increase.

On the other hand, if the amount is 2 and increases to 7, that's a 5 point increase but represents 5/2 =2.5= 250% increase.

In dollar amounts, the 10% increase in the first calculation ($50) is 10 times the 250% increase in the 2nd calculation ($5). So, if someone asks, "which is greater, a 10% increase or a 250% increase?", that cannot be answered unless you know the base amounts.

~% increases & % decreases are used extensively in the business world (stock market & others reporting statistics) & can be very deceiving.

~If you buy a stock at $100 a share & hold on to it & it declines to $50, you have just suffered a $50 dollar loss per share or 50% decrease in your investment (a decrease 50 divided by a starting amount of 100 = 0.5 or 50%).

~That same stock would have to increase 100% from $50 to get back to the price you originally paid (50 divided by a starting amount of 50 = 1 or 100%).

~So, a 50% loss followed by a 50% gain does not get you back to the original point.

~A common mistake made by students on exams is the following:
Take a stock that decreses from 100 to 50. That's a 50% decrease.
To get back to 100 from 50, some students figure that since 100 is 2 times 50, the % increase must be 2=200%. They figure that 200% of 50 is 100, so give the required % increase as 200%. This is in error. The increase is not 200% of the starting point of 50. That would be an increase of 100. That would take us to 150. The proper way of looking at this problem is: an increase of 50 represents what % of the starting point of 50. That would be 50/50 = 1 = 100%. A bit confusing to say the least. One problem deals with the increase as a percentage of the starting point while the other deals with the percentage of the starting point to get you where you want to go.

~The tech crash of 2000-2001 was worse than the Dow Jones crash of 1929 for many tech investers. On Oct. 21st, 1929, we had a market crash that sent the Dow Jones average from 400 to 145 (by Nov.). That's a 64% decrease. At the absolute bottom, the Dow had lost 89% of its value a few years later. It took approximately 25 years for investors to get their money back. Many think that was the worse in history. Not so. During the tech crash of 2000-2001, many popular technology stocks fell over 95%. For example, one of my favorite stocks (at that time) was AMCC (Applied Micro Curcuits). It's price went from $108/share to $3/share (97% decrease). It now trades at about $2.80/share. It would have to increase 3,757% to get back to its Oct. 2000 price (not likely). Many other tech stocks suffered similar losses. Many tech investors lost hundreds of thousands of dollars. So, for most tech investors, this was much worse than the '29 crash.

~Penny stocks (stocks that usually sell for under $1) are notorious for huge percentage increases & decreases. (since base amounts are relatively small)

~A stock selling for $0.25 could go to $3.00 for a $2.75 increase (2.75/.25=11= 1,100% increase), (but only $2.75 in gain), while another stock at $450 goes to $600 for a $150 gain or 150/450= 0.33 = 33% gain.

~Generally speaking, traders buy large amounts of low priced stocks to capture huge % gains (or suffer huge % losses). Small gains in penny stocks can give huge returns on your investment since thousands of shares are usually purchased. However, most traders lose all or most of their money.

~Stores often offer sales after secretly inflating their prices. After a quiet markup of 20%, they will offer a 15% sale price. Not a good deal. Other stores have signs that state "save 50%". Well, you don't save a penny, you never save when you spend.