### Overview

Percents are special types of ratios that compare the number to 100. Similar to any ratio, a percent can be expressed as a fraction. Since percents compare to 100, they can also be expressed in decimal form.

### Fractions to Percents

Percent or the percent sign (%) means “per 100,” so 5% means the same as 5/100, 31% means 31/100 and so on. Many common fractions can be expressed as percents by converting the fraction in terms of 1/100, so ½ is 50/100 or 50%, ¾ is 75/100 or 75%, 4/5 is 80/100 or 80%. Percentages, like fractions, can be greater than 1, so that something that is 200% is two times as much, just as 200 is twice as much as 100.

### Decimals to Percents

Any decimal can be expressed as a percentage, and any percentage as a decimal. (This can be a handy shortcut if the calculator does not have a percent key). For example, 20% is 0.20, and 7% is 0.07. Similarly, 0.003 is .03%, and 3.12 is 312%. A percentage such as 120% can be expressed in decimal form as 1.20.

### Percents as Proportions

Percents can be expressed by the formula part/whole =percent/100, so that they can be compared as 2 equivalent ratios, the definition of proportion. Crossproducts can be used to then solve the equation. Suppose the problem were to find 30 is 20% of what number. 30/x = 20/100 is the equation for the proportion, or 20x = 30·100, or 3000. Dividing both sides by 20 to isolate the variable, x = 3000/20, or 150. Similarly, find 50% of 50. In this case, 50/100 = x/50 gives the proportion, so 100x =2500, or x= 2500/100, or x = 25. Also, 30 is what percentage of 150? This is also a proportion, as 30/150 = x/100, or 150x =3000, or x=3000/150, x = 20%.

### Applications of Percents

Percents are used in many different ways. For example, if a ring is 24-karat gold, it is nearly 100% pure gold. It is too soft for most jewelry, so most gold is in the 10-karat (41.7% pure gold) to 18-karat (75% pure gold) range. Suppose a ring weighs about 10 g. If it were 10 karat, 4.17 g would be gold and the rest of the ring would be other metals, but if it were 18 karat, 7.5 g would be gold. Percents are often used in sale prices, as items are often on sale for 30% off or 40% off.

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