6.3+Percents

Section 6.3 - Percents
We use percents for a lot of things - sales tax, tips, special end-of-season sales, change in the value of a currency, etc.
 * Per Cento**

The concept of percent or percentage was probably born during the centuries of the Roman Empire. People would meet in the market to buy and sell goods. The seller paid a tax on the value of the goods they sold. The Roman currency is fairly complicated, but let's use dollars as an example. If the Roman seller made 100 dollars in a day, he or she would have to give the government 3 dollars, or 3 dollars per hundred. Hundred was Cento, so 3 per cento, which becomes 3%.

The percent sign has also changed somewhat over time, but if you look at it, you might recognize that it represents a fraction - something divided by something, or X over X. In reality, the bottom number is always 100. ** A percentage is a fraction with a denominator of 100. ** 45% would be 45 over 100. 465% would be 465 over 100. 100% is 100 over 100, which is equal to 1. ** 100% = 1 **


 * Converting between Decimals and Percents**

If a percentage is a number divided by 100, the hundredths spot is the key place value for percents. 0.35 is the same as 35 over 100, which is 35%. This is the key understanding: ** To change from a decimal number to a percent, or the other way around, you will move the decimal point two places. ** The trick may be remembering which way.



It is not always needed, but usually, when you do math with percents, you will convert to a decimal first. Either way, it is important to remember that 75% = 0.75 and not just 75.

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 * Percent Equation**

Often, percentage problems are story problems, written with words. 80% of the cattle are already rounded up, so far there are 440. 80% is already a math term. Of means times. The total number of cattle is unknown, so X, and that is supposed to equal 440. 80%X = 440 However, the % sign and X right next to each other cause confusion, and we have never really learned to divide by a percent in that form. So, change 80% to 0.80 and then we have 0.8X = 440, which is a much more familiar "solve for X" type problem.

There is a general format for what we will call the Percent Equation: The % of the Whole is the part. or (%)(whole) = (part). There are three variables here. If you know any two of those, you can solve for the third.

84% of the 5,964 people who were arrested in Salt Lake last year made bail within a week. How many people made bail within a week? 0.84(5,964) = X 835 of those arrested saw a judge within 2 days. About what percent is that? X(5,964) = 835 Almost 23% of those arrested the year before were found innocent. If 43 were found innocent, how many were arrested the year before? 0.23(X) = 43

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 * Percent Proportion**

The proportions that were used earlier in this chapter to find similar rates or ratios can also be used to find percentages. The format for the percent proportion is: percent over 100 = part over whole.



Because the denominator of the percentage will always be 100, this form has the exact same variables as the percent equation. Again, using cross multiplication, you will be able to solve for any of the three variables if you know the other two.

84% of the 5,964 people who were arrested in Salt Lake last year made bail within a week. How many people made bail within a week?

835 of those arrested saw a judge within 2 days. About what percent is that?

Almost 23% of those arrested the year before were found innocent. If 43 were found innocent, how many were arrested the year before?


 * Applications**

__Discounts__ - When you go shopping, some things are on sale. The discount is often a percentage of the normal price. The formula looks like this: Sales Price = (Original Price) - (Original Price)(% off) Just plug in the information that you have. The question may include any two of the three variables: Original price, % off, or sales price.

__Sales tax__ -

If you buy an item, you must also pay an additional cost that is a percentage of the price of the item. (Item Cost) + (Item Cost) (% tax) = (Total Price)

__Tips__ -

Tips are just like sales tax. (Cost) + (Cost )(% tip) - (total cost)

__Simple Interest__ -

Simple interest is not the same as compound interest. With compound interest, you pay interest on the interest. It's what makes the rich richer and the poor poorer. Simple interest, on the other hand, is much like the sales tax or tips above, except that you pay the tax each year (12 months). The formula is I=PRT. (Interest) = (Principal borrowed) (% Rate) (Time in years)

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 * Increase and Decrease Problems**





Last year there were 110 faculty parking spaces by the Library. This year there are 124. What percent increase is that?

There are now 820 student parking places, but that is quite a few more than 7 years ago, when there were only 450. What percent increase is that?


 * Chapter Six Practice Problems**

**Where to from here?**
6.1 Ratios 6.2 Rates 6.3 Percents Chapter Six Summary Chapter Six Homework