Playing with Percentages
Chapter 10: Playing with Percentages
So, for example, 17 percent is ,
a fraction with 100 in the denominator. You use the expression one half —
— but when you convert that to a fraction with a denominator of 100, it becomes
, or 50 percent.
What about larger percentages? You can write 300 percent as:
That doesn’t look very pretty, but you can reduce it to 3. Yes, 3 is still a fraction —
— but it’s not really eligible to be a percentage until the denominator returns to 100.
You can also have fractional percentages. If you have 100 items and then take of one of those items, you have of a 100th (because one whole item out of the hundred would be one one-hundreth), or
That’s
percent or 0.5 percent.
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A percentage is a ratio, too
A percent is a fraction. A fraction is a ratio. Therefore, a percent is also a ratio. Check out Chapter 8 for more information about ratios.
Here’s a simple example. If you stock 100 cans of motor oil, and 40 of them are SAE 5W-20, then 40 out of 100 cans are 5W-20. That’s
of the cans, or 40 percent of them. You have a 40:60 ratio of 5W-20 to other kinds of motor oil in stock.
When you have a few friends over for a hot evening of talking math, you can refer to a percentage as a dimensionless proportionality. A quantity is dimen- sionless when it doesn’t have a physical unit. The parts of a percent form a proportionality because they have a constant ratio.
Percentages Are Good Converts
When the going gets tough, the tough convert from one form to another. You convert percentages into other numbers and other numbers into percentages when doing so is convenient. Or when you’re just in the mood.
We cover these conversions in the following sections:
✓ Percentage to decimal
✓ Decimal to percent
✓ Percentage to fraction
✓ Fraction to percent
If the conversions in the following sections all seem circular, that’s because they are. Fractions to decimals to percents and back again. Percents to frac- tions to decimals and back again. Decimals to fractions to percents and back again. They’re all interrelated.
Converting percentages to decimals
You can do some percentage conversions in your head, but when you need to get serious, convert then to decimal numbers by using a calculator, a spreadsheet, or good old pencil and paper.
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Chapter 10: Playing with Percentages
The rule for going from percentages to decimals is simple: Divide the per- centage by 100. The result is a decimal number because the denominator in a percent is always 100 (as we discuss in “A percentage is a fraction, but the denominator never changes” earlier in the chapter). So (for example) 67 per- cent becomes
or 0.67.
If you need 20 percent of a 1 liter solution, you need
of it or 0.2 liters. You don’t even have to plug the numbers into a calculator.
To divide by 100, you can just move the decimal point two places to the left of its original position. The number 20 really is 20.0. When you move the deci- mal to the left, the result is 0.200 liters, or 0.2 liters — the same answer you got with the actual division.
What do you do when you don’t see a decimal point? Just remember that the decimal point is always assumed to follow the ones position in the number.
And, if the percentage itself has a decimal point, such as 37.6 percent, just shift the decimal two places to the left. The answer is .376.
Turning decimals into percentages
Converting from a decimal number to a percentage is just the opposite of converting from a percentage to a decimal (see the preceding section).
When it comes to decimal conversions, you’re faster than any calculator or spreadsheet.
To make this conversion, multiply the decimal by 100 and add a percent sign. The result is a percentage. For instance, if you want to make 0.67 a per- centage, multiply 0.67 by 100 to get 67 and then slap a percent sign on it. As a shortcut, you can simply shift the decimal point two places to the right, which is the same as multiplying by 100.
If you machine 0.42689 inches of stock off a 1-inch aluminum bar, what per- centage do you remove? To find out, just multiply by 100: 0.42689 × 100 = 42.689 percent. You took off 42.689 percent of it.
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Going from percentages to fractions
This conversion has a great taste and it’s less filling. The rule is to drop the percent sign and put the number in a fraction over 100. The result is (naturally) a fraction. Reduce as necessary. This conversion comes in handy when work- ing with a fraction is easier than working with a decimal. (Head to the earlier section “Converting percentages to decimals” for details on that calculation.) For, example to convert 26 percent to a fraction, just follow these steps:
1. Drop the percent sign and use the numerical portion of the term.
The 26 percent is just 26.
2. Set the percentage number as a numerator over a denominator.
The denominator is always 100
3. Simplify your fraction.
The answer is
Transforming fractions to percentages
This conversion is the reverse of the conversion of percentages to fractions in the preceding section. The rule for converting a fraction to a percentage is to convert the fraction into a decimal and multiply by 100. The result is a per- centage. For example, if you have
of something, you divide 33 by 100 to get 0.33 and then multiply by 100 to arrive at 33 percent.
What if you have 5⁄8 of something? First, divide 5 by 8 to get the decimal 0.625.
Multiply by 100. The result is 62.5 percent.
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Chapter 10: Playing with Percentages
Calculating Percentage Increases and Decreases
The rules for calculating percentage increases and decreases are very simple even without the available shortcuts. The following sections present the lightning round of increasing and decreasing percentages.
Percentage increases: You get 10 percent more!
To increase a number by a percentage, multiply the original number by the percentage and then add the result to that original base amount. For example, if you normally supply customers with a $50.00 item and must increase its price by 15 percent, multiply $50.00 by 15 percent (50 × 0.15) to get $7.50. Add the result to the base amount to reach the new price: $50.00 + $7.50 = $57.50.
Of course, the same procedure holds true in calculating the tip in a restaurant.
Here’s a tip about tips. To calculate a tip quickly, take the bill and figure 10 percent in your head. For a $50.00 dinner, that’s $5.00. Then figure 50 per- cent of that figure (in this case, you get $2.50). Fifty percent of 10 percent is another 5 percent, and together they make 15 percent ($7.50). So $7.50 is a 15 percent tip for a $50.00 dinner. You also may find a tip calculator on your cellphone or smartphone.
Percentage decreases:
You save 10 percent!
To decrease a number by a percentage, multiply that number by the percent- age and then subtract the result from the base amount. For example, if you have a coupon for 20 percent off, and you want to buy a $50 item for your business, multiply $50 by 20 percent (0.20) to get $10. That’s the amount of your discount. Subtract the result from the base amount ($50 – $10 = $40) to get your final price.
In the language of office supplies and fashion, “10 percent off” means the mer- chandise’s regular selling price is discounted by 10 percent.
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An advertising slogan such as “Discounts up to 20 percent or more” is totally bogus. The language doesn’t mean anything. If the discounts are “up to 20 per- cent,” your math lets you calculate any discount from 1 percent to 20 percent.
But what about “or more?” If the discounts go above 20 percent, why didn’t the merchant say so?
The 100 percent increase: You must be 100 percent satisfied!
To calculate a 100 percent increase in something, simply double the base amount. To calculate a 200 percent increase in something, simply triple the base amount.
“Wait, what?” you may be saying. “200 percent is twice something, not triple it!” That’s correct, but in this case, you’re taking a 200-percent increase — you have to account for the original amount as well. Because you’re doubling the base and then adding it in again, a 200-percent increase therefore triples the amount.
Dividing a Pie Using Percentages
When you have a set of percentages, everything is easy. You divide some- thing into parts by multiplying by a percentage.
“You divide by multiplying” has kind of a Zen ring to it, doesn’t it? Why does it work? A percentage is a fraction, so multiplying a large quantity by a percent- age has the effect of multiplying it by a fraction (which divides it into portions).
The logic of percentages works with both quantities and amounts. It can apply to marbles, pills, cans of motor oil, pounds of rice, and so on.
Your food service sells apple pies, and each slice is 12.5 percent of a whole pie. How do you divide the pie in the easiest way? Why, you just follow these easy pie-dividing steps:
1. Turn the number into a fraction.
Follow the conversion rule to turn the percentage (12.5 percent) into
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Chapter 10: Playing with Percentages
2. Reduce the first fraction by dividing both parts by a common factor (if possible).
In this example, you can divide the numerator and denominator by 12.5 to get
Cut the pie into eight equal slices.
Now here’s an example of dividing by multiplying by two percentages. Your food service features two new portions of decadent pie, the Cholesterol Colossus (37.5 percent of a whole pie) and the Microscopic Minislice (6.25 percent of a whole pie). Your boss tells you that you can evenly slice up one pie with these portions. How do you divide the pie in the easiest way? Follow these steps:
1. Turn the numbers into fractions.
Follow the conversion rule to turn the percentages into
2. Reduce the first fraction by dividing both parts by a common factor (if possible).
In this example, you can divide the numerator and denominator by 6.25 to get
3. Repeat Step 2 to reduce the second fraction.
You can divide both parts by 12.5 to get
4. Figure how many giant slices you can get from the pie and how much (if any) of the pie is left over.
You can get two big slices (totaling
of the pie) but not 3 slices (because that would be
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which is more than the whole pie). This amount
is the same as
of the pie, leaving
of it.
5. Determine how many of the skinny minislices you can get from the rest of the pie.
The remaining
pie is equal to
pie, so four of your
slices finish the pie off nicely.
6. Divide the pie by cutting it into the number of slices you determine.
In this example, that’s two
pie slices and four
pie slices.
About any sous chef or pastry chef knows about half, quarter, and eighth slices, so these measurements should be no problem. The ending fractions are good in the kitchen, while the starting percentages are good for the cost accountants.
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Chapter 10: Playing with Percentages
Example: The World of Pralines
You’ve lived in New Orleans all your life, and you have your grandmother’s famous recipe for pralines. You sell about 90 dozen a day in a storefront in the French Quarter and online.
You visit your relatives in California. You want to make 3 dozen pralines, but you’ve long since forgotten the original recipe ingredient amounts; however, you do know your commercial amounts. Use percentages to scale the recipe to make the batch you need.
The recipe for 90 dozen (1,080) pralines is
✓ 30 cups granulated sugar
✓ 30 cups light brown sugar, packed ✓ 5.5 quarts half-and-half
✓ 7.5 teaspoons salt
✓ 2 pounds butter
✓ 5 ounces vanilla
✓ 7.5 pounds chopped pecans