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Percentages are another way of talking about ratios, fractions, and decimals.
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Percent means “by the hundred,” and it demonstrates the relationship between a part and its whole in terms of 100. For example, if 50 out 100 people enjoy chocolate, then you can say that 50% of the people you surveyed enjoy chocolate. Percentages are often used to look at increases and decreases in values, chances of something happening (or probability), discounts and sales, and polling.
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A city’s population can decrease by 10% in a year, or a Monday can have a 35% chance of rain. An electronics store can have a 25% sale on speakers, or a company can find that 80% of its employees are in favor of decreasing the number of meetings they attend. Each of these examples show us the direction a value is going and can help guide decisions we make in real life.
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But let’s start with the basics--percentages are represented as a number combined with the % sign. It is important to understand that a percentage (such as 50%) is the same thing as:
And each of these can be converted into a percentage (and vise versa).
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Let’s test this with 25%. For ratios and percents, we always use the part-to-a-whole ratio, with the part on the left side of the colon symbol and the whole on the right. Since percentages are always in terms of 100, 25% is written as 25 for the part and 100 for the whole and is read as 25 out of 100.
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You can reduce these values just as you would with other ratios. In this case, 25 can divide into both sides of the ratio and give you 1 out of 4.
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To convert a ratio into a percentage, you manipulate the right side of the ratio to equal 100 and then do the exact same thing to the part that you did to the whole.
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For example, if the ratio is 1:4, then you can change the right side to 100 by multiplying 4 by 25. In order for the ratio to remain true, you must then do the same thing to the part and multiply 1 by 25. This gives you 25 out of 100 or 25%.
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In many cases, manipulating the whole to equal 100 will require an extra step or two. For example, we want to know the percentage a team has lost their games, and the team has lost 3 out of 12 games played.
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Notice that while 12 does not easily divide into 100, we can manipulate the ratio to a value that can, by dividing both sides of the ratio by 3 (which gives us 1:4).
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Fractions and percentages follow the same process, with the part as the numerator (or top number) and the whole as the denominator (or bottom number) in a fraction.
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Since 25% is 25 out of 100, you would write this as 25 over 100. Since 25/100 is not the fraction’s simplest form, we reduce it by dividing the top and bottom numbers by 25, which gives us ¼.
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To convert a fraction to a percentage, we manipulate the denominator to equal 100 and take the exact same action with the numerator.
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For example, with the fraction ⅕, we multiply 5 by 20 to reach 100 and then do the same to 1, which gives us 20 and ultimately 20%. As before with ratios, many denominators will require extra steps to reach 100.
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Converting between decimals and percentages is even easier. If you are converting a percentage to a decimal, then you will move the decimal point to the left two place value. For example, 25% will become 0.25 (which is 25/100 as a fraction)
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If you are converting a decimal to a percentage, then you will move the decimal point to the right two place values. For example, 0.67 will become 67% (which is 67/100 as a fraction).
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This is also true for more complex percentages and decimals. 4.58% will become 0.0458 by moving the decimal point to the left twice. And 0.6085 will become 60.85% by moving the decimal point to the right twice.