For an introductory explanation of percentages, please visit the following page:
Topics covered on this page include the following:
We are constantly bombarded with advertisements from companies that are offering discounts as an enticement to draw us into their stores. Typically they may be offering 10 or 20 percent off selected merchandise during a slow weekend. But what exactly does that mean?
When a store offers savings of 10 percent, that means they will reduce the price of an item by 10 dollars for every 100 dollars it normally costs. Thus, if a pair of shoes usually costs $100, the store will knock off $10 and you will pay just $90 for them. A $500 jacket would be reduced by $50 for a sale price of $450. You save more money on more expensive items because percentages depend on the original price. This is different from a coupon for a fixed savings of $10, where no matter how much you bought, your savings would be limited to that amount.
Since the jacket originally cost $500 and we saved $50, the actual price we paid was $450. We used a two-step calculation to figure this out: first we computed ten percent of the original price, and then subtracted that amount. It would be faster if we could determine the discounted price directly in one step, so we'll look at how to do that.
The way to think about this is if the jacket was not on sale, we would pay 100 percent of the listed price. When the jacket is on sale, we are returned 10 percent of the original price, so we actually pay 100% − 10% = 90% of the price. Thus if we multiply the original price by 0.9, we should arrive at the sale price:
You can see that we computed the correct sale price.
If you have a calculator with a % button, it can determine both your savings and the sale price in the same calculation as seen here. Notice that when you press the % button, the calculator shows you the savings you are receiving. Then when you press the = button, the final sale price is shown:
Suppose you decide to splurge on a new stainless steel refrigerator that normally costs $1000. The clerk who is working the register tells you that the refrigerator is on sale, which you are happy to hear. She tells you that there are two offers; you can either take 20 percent off the listed price, or if you take only 10 percent off, the store will deliver the refrigerator to you for free. Which offer would you choose?
The problem with that question is you don't yet have all the information you need to make a smart choice. In the first offer, you save 20 percent or $200, but will need to pay to have the refrigerator delivered to your home. In the second offer, you save only $100, but shipping is included. You need to know if the free delivery is worth more or less than the savings you are giving up.
So when the clerk tells you that delivery costs $50, you realize that the offer of free delivery is actually a bad deal and you take the 20 percent discount instead. After paying the additional $50 for delivery, you still have saved $150 on your purchase, which is more than the $100 savings you would have gotten with the second offer.
The most important thing you can do with your money is invest it. Investment options vary in their level of risk and their reward potential. Safer investments are more likely to increase in value, but are generally limited in how quickly they can grow. Riskier investments offer the potential of higher returns, but also have an increased potential to decrease in value. More will be said later about choosing appropriate investments, but for now we will look at how to measure their performance.
A well-known investment option is to purchase stock of a public company. Each share of stock you buy represents a small sliver of ownership of that company. If the company does well and makes a decent profit, the price at which you could sell your stock may increase above the price you paid for it. However, if the company does not make as much money as expected, the price people would be willing to pay you for your stock might drop below what you bought it for. Obviously you hope that the stock price goes up over time, bringing you closer to retirement.
To gauge how well your investments are performing, you will typically receive a monthly or quarterly statement showing the change in value of each of them. The dollar amount of each change will usually be accompanied by the percentage change. The percentage tells you how much the value of an investment has increased or decreased independent of the total amount of money you have in the investment. This is important information if you want to compare the performance of different assets.
For example, suppose you bought 100 shares of stock at $20 per share. The total amount you invested in the company was 100 × $20 = $2,000. When you received your first monthly statement, it showed a total gain for the month of $100, which works out to $1 per share. The share price is now $21, and you now have $2,100 worth of stock.
To figure out the percentage gain of your investment, the gain of $100 is divided by the starting value of the asset, which was $2,000 at the beginning of the month:
|=||0.05 × 100|
Note that you always divide the gain by the starting value, not the ending value. This tells you the performance relative to the previous month's value. This equation also works if money is lost during the month; the gain would be a negative number and would result in a negative percentage.
You can verify that this equation is correct by adding 5 percent to $2,000:
Now if you only knew that your starting balance was $2,000 and were told that your stock increased in value by 5 percent, how would you calculate your current balance? You could figure out what 5 percent of $2,000 is and then add that to the starting balance of $2,000, but that is a two-step procedure; we'd like to compute the new balance in one step.
It is better to think of it this way: if the stock price did not change during the month, you would have exactly 100 percent of your starting balance at the end of the month. But the stock went up by 5 percent, so you end up with an additional 5 percent at month's end for a total of 105 percent of your beginning balance. So to find your final balance, you simply multiply by 105 percent:
|Final Balance||=||$2,000 × (100% + 5%)|
|=||$2,000 × 105 ÷ 100|
|=||$2,000 × 1.05|
If instead of a gain of 5 percent, your stock dropped in value by 5 percent, you would be left with only 100% − 5% = 95% of your beginning balance:
|Final Balance||=||$2,000 × 95%|
|=||$2,000 × 0.95|
Now let's see what happens over longer time periods. Suppose in the first month, your stock posts a gain of 5 percent, and then in the second month drops 5 percent in value. Your first impression might be that you've broken even, but in fact money was lost. The reason for this is that after the 5 percent gain, your account is worth more money, so the 5 percent loss on this larger balance is a larger decrease than the 5 percent gain.
|Final Balance||=||$2,000 × 105% × 95%|
|=||$2,000 × 1.05 × 0.95|
|=||$2,000 × 0.9975|
Over the two months, your account has lost 1/4 percent. You can verify this effect on a calculator:
Looking back at the equation, you can see that switching the order of the multiplication by 1.05 and 0.95 doesn't affect the result, so your account would have the same 2-month performance if it first experienced a 5 percent loss followed by a 5 percent gain:
In fact, a loss is always more significant than a gain of the same percentage.
There are many ways to obtain stock quotes, which tell you the price of a stock at a time in the past. Newspapers will show the last price a stock traded at during the previous business day. Television shows will usually show prices at least 15 minutes in the past, and if you have an account with an online broker, you may be able to obtain real-time quotes, which show the most recent sale price of a stock.
A stock quote will contain the current market price of the stock, along with the difference between the current price and the price from the previous trading day, often accompanied by more information. Sometimes the quote will contain just two pieces of information: the current price, and the percentage change from the previous day. A quote for Amazon.com near the end of March 2009 looks like:
These two numbers tell us quite a bit, actually. With a little effort, we can determine the closing price from the previous day as well as the dollar amount of the change. The quote tells us that the current price for Amazon stock is 4.3 percent lower than that of the previous day. We want to determine the previous day's price, so we need to undo the 4.3 percent drop. We saw that we can't just add 4.3 percent to the current price, since that gain would not quite cancel out the loss, so how do we do it?
We need to write an equation which relates the current price to the previous day's price:
|Previous Price − 4.3%||=||Current Price|
|Previous Price × (100% − 4.3%)||=||$70.52|
|Previous Price × 0.957||=||$70.52|
So to undo the change, we need to use division instead of multiplication; the previous day's closing price is therefore:
|Previous Price||=||$70.52 ÷ 0.957|
Let's check our work to make sure that this is correct:
We see that there is a little bit of error in the calculation, but that it is within a penny of the correct value. If you run through the calculation again, but stop after pressing the % button, you will see the dollar amount of the loss in the stock price; it dropped $3.17 from the previous day.
We've covered all of the calculations you need to understand when working with percentages, so next we will see how to apply that knowledge to interest.