Section 5: Percents
Percentages are an important concept because they suround us daily. Advertisements scream 30% off today only! Buy now with only 5% down!
0% interest until 2012! If we don't understand percents, we won't understand many of our daily purchasing decisions. Further, percents are important in probability, statistics and financial mathematics.
The word "percent" comes from the root "cent" for century or 100 and "per" meaning "for each" or "to each". Thus, "percent" means "for each 100". For example,
5% down means that the buyer must pay 5 for each 100 units (dollars, nickels, euros, ounces, etc.) initially and pay the rest later. Thus a $100 item would need a $5 down payment, a $150 item would need a $7.50 down payment, and a $3 item would require a $0.15 down payment.
5.1 Translating Percents to Decimals
When people work with percents mathematically, they often use the decimal
representation of the percent. Translating between decimals and percents simply
requires moving the decimal poiut two places right or left. This is based on the
"per 100" concept. If I divide or multiply by 100, I simply shift the decimal point two places. Here are the rules:
• To translate a percent to a decimal shift the decimal point two spaces to the left. (i.e. divide by 100)
• To translate a decimal to a percent shift the decimal point two spaces to the right. (Le. multiply by 100)
For example, 5% is "5 per 100" or 0.05 and 83% is "83 per 100" or 0.83. Going the other way, 0.38 is "38 per 100" or 38% and 1.74 is "174 per 100" or 174%.
5.2 Solving Percentage Problems
Solving percent problems requires the appropriate use of a single formula: y= n% × x or y = n%(x). This formula says "y is n% of x". Once you have identified the y, the n, and the x in a problem, you only need to put them into the appropriate place in the formula to get a solution. Generally, n is not too hard to identify because it's the percent in the problem. However, y and x can be a little harder. You can think of x as the "original" or "what you start with". The key word to look for is "of". Whenever you find "of" following by a value, that value is probably x. Then y is the "outcome" or "what you end up with". Here the key word is "is". Whenever you find "is" the value with it is probably y although it could be before or after" is". In most problems two of these three values will be given and the other will be what you need to find to solve the problem.
Even I have to admit that it sounds a little complicated. But it really isn't bad once you've done a few examples. After all the values have been identified and placed into the correct spot, a simple linear equation often results. If you don't know how to solve them yet, don't worry. That will be covered in a couple sections. Right now the important skill is identifying y, n,and x and placing them correctly.
Example 11: What is 15 percent of 200? Look for the word or symbol for percent. It has a 15 with it so n must be 15. Do you see "of" anywhere'? There is a 200 with that so x must be 200. The word "is" is there, but there's no number with it so that must be y. That's the value needed to solve the problem. In formula form then we have: y = 15%(200). To solve it, we will translate 15% to 0.15. Now we have: y = .15(200) and y = 30.
Example 12: 28 is 12% of what number? Again, the % symbol helps identify 12 as n. There's an "of" but it doesn't have a number with it. Therefore x must be the number needed to solve the problem. That means that the only other number, 12, must be y. You also could have identified y by its proximity to "is".
In formula form we have: 28 = 12%(x) or 28 = .12(x). Then x is 233 1/3 or 233.33...
Example 13: 45 is what percent of 300? This time there is no value with "percent" making n the value you need to find. The word "of" tells us that x is 300 and "is" reveals that y is 45. Then our formula is 45 = n(300). The swolution is n = .15 or 15%. This same problem could be stated "What percent of 300 is 45?" Can you see how y, n, and x are still the same?
Example 14: You currently make $8 and hour and you're getting a 5% raise effective next week. How much is your raise? Hmm. This is not as obvious as the prior examples, but it can still fit into the formula. There's a % symbol so n must be 5. There's only one other number. Is it x or y? There isn't any "of" to help us out but there is an "is". There's no number with the "is" so that must be y. That means the 8 must be x. Another way to identify x is that your raise must be based on your current wage. Thus, the raise is 5% of your current wage of $8. Even though "of" did not appear in the problem, it is grammatically implied: "what is 5% of $8". In formula form we have y = 5%(8) or y = .05(8) and y = .40 or 40 cents.
Example 15: A pair of pants that originally cost $32 are on sale for $10 off. What percent discount is this? There's no value with "percent" so n must be what we need to find and the other two numbers must be x and y. We just need to figure out which is which. We have "is" followed by "this", Grammatically "this" refers to the "$10 off". Thus the question is "what percent discount is $10." Therefore y must be 10. That leaves 32 for x. If it had been worded like the first three examples, this problem would have been" What percent of 32 is 10?" Our formula is 10 = n%(32). Then n = .3125 or 31.25%.
The wording of the first three examples was somewhat "forced" so that they fit the standard mold. More realistic problems are less clear like that last two. When faced with a problem that does not directly fit the form "y is n percent of x" , many students find it easier to translate it in words before trying to assign values to variables.
5.3 A Word About Rounding
Many problems, percents and others, do not have nice round answers. In these cases you will usually have to round off your answer. The number of digits needed vary by context but statistics usually need, at most, two difuts beyond the decimal point and probabilities usually need, at most, four digits. Keep in mind that four digits is equivalent to two digits beyond the decimal point when answers are in percent form.
When possible, it's best to avoid any rounding before the final answer is obtained. You can often accomplish this by storing any partial answers in your calculator's memory. If you must round off a partial answer, a good guideline is to keep four digits beyond the decimal point.
Keep in mind that rounding and truncating are not the same thing. Truncating means that you simply chop off the"extra" digits. Rounding means that you adjust the last remaining digit to be closest to the correct number. If the first eliminated digit is less than 5, round down. If the first eliminated digit is 5 or greater, round up. Here are some examples of rounding to two digits:
• 3.456712 rounds to 3.46
• 3.454712 rounds to 3.45
• 3.455712 rounds to 3.46
Here are some examples of rounding to four digits:
• 3.456712 rounds to 3.4567
• 3.456792 rounds to 3.4568
• 3.456752 rounds to 3.4568