Percentages, Ratios, and Proportions

This section covers:

Note:  For more problems with percents and ratios, see the Algebra Word Problems section.

Percentages and Percent Changes

Percentages are something you are probably quite familiar with because of your shopping habits, right? How many times have you been to the store when everything is 20% off? Do you notice how many people around you (adults, usually!) have no idea how to figure out what the sale price is? The easiest example of percentages is 50% off, which means that the item is half price.

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Percentages really aren’t that difficult if you truly understand what they are. The word “percentage” comes from the word “per cent”, which means “per hundred” in Latin.  Remember that “per” usually means “over”. So “per cent” literally means “over 100” or “divided by 100“. And remember what “of” typically means? I’ll write it again, since it’s so important:

OF  =  TIMES

When we say “20% off of something”, let’s translate it to “20 over (or divided by) 100 — then times the original price”, and that will be the amount we subtract from the original price.

Remember that we cannot use a percentage in math;  we need to turn it into a decimal. To turn a percentage into a decimal, we move the decimal 2 places to the left (because we need to divide by 100), and if we need to turn a decimal back into a percentage, we move the decimal 2 places to the right (because we need to multiply by 100).

I like to think of it this way: When we’re taking away the %, we are afraid of it, so we move 2 decimal places away from it (or to the left). When we need to turn a number into a %, we like it, so we move 2 decimals towards it (or to the right).

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Let’s get back to our percentage example. If there’s a dress we like for say $50, and it’s 20% off (“off” means take-away or minus!), we’ll do the math to figure out the sales price. This is called a percent change problem. Amount of sale: $$\displaystyle 20\%\,\,\text{of }\,\50=.2\times \50=\10.\,\,\,\,\50-\10=\40$$. The dress would be$40.

(See how we had to turn the 20% into a decimal by taking away the % sign and moving 2 decimals to the left, or away from it, since we didn’t like it?)

We could have also multiplied the original price by $$\displaystyle 80\%\,(100\%-20\%)$$, or $$\displaystyle \frac{{80}}{{100}}$$, since that’s what we’ll be paying if we get 20% off (100% full price minus 20% discount equals 80% discounted price):

Price of discounted dress: $$\displaystyle 80\%\,\,\text{of }\,\50=.8\times \50=\40$$. This method has fewer steps.

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This shopping example is a percent decrease problem; the following is the formula for that. Make sure you relate this formula back to the example above.

$$\displaystyle \text{Newer}\,\,\text{lower}\,\,\text{price =}\,\,\text{original}\,\,\text{price}\,\,-\,\,\left( {\text{original}\,\,\text{price}\,\,\times \,\,\left. {\frac{{\text{percentage}\,\,\text{off}}}{{100}}} \right)} \right.$$

$$\displaystyle \50-\left( {\50\,\,\times \,\,\left. {\frac{{20}}{{100}}} \right)} \right.\,\,=\,\,\50-\10=\40$$

Notice that we worked the math in the parentheses first (we will get to this in more detail later).

Now let’s talk about a percent increase problem, which is also a percent change problem. A great example of a percent increase is the tax you pay on this dress. Tax is a percentage (usually) that you add on to what you pay so we can continue driving on the streets free and going to public school free.

If we need to add on 8.25% sales tax to the $40 that we are going to spend on the dress, we’ll have to know the percent increase formula, but let’s first figure it out without the formula. Tax is the amount we have to add that is based on a percentage of the price that we’re paying for the dress. The tax would be 8.25% or .0825 (remember – we don’t like the %, so we take it away and move away from it?) times the price of the dress and then add it back to the price of the dress. Total price with tax: $$\displaystyle \50+(8.25\%\times 50)=\50+(.0825\times 50)=\50+\4.125=\54.125=\54.13$$. Note that we rounded up to two decimal places, since we’re dealing with money. Note also that we did the math inside the parentheses first. < The total price of the dress would be$54.13.

Here’s the formula:

$$\displaystyle \text{Price}\,\,\text{with}\,\,\text{tax}=\,\,\text{original}\,\,\text{price}+\,\,\left( {\text{original}\,\,\text{price}\,\,\times \,\,\left. {\frac{{\text{tax}\,\,\text{percentage}}}{{100}}} \right)} \right.$$

$$\displaystyle \50+\left( {\50\,\,\times \,\,\left. {\frac{{8.25}}{{100}}} \right)} \right.\,\,=\,\,\50+\left( {\50\times \left. {.0825} \right)} \right.\,\,=\,\,\50+\4.125=\54.125=\54.13$$

Another way we can figure percent increase is to just multiply the original amount by 1 (to make sure we include it) and also multiply it by the tax rate and add them together (this is actually using something called distributing, which we’ll talk about in Algebra):

$$\displaystyle \text{Price}\,\,\text{with}\,\,\text{tax}\,\,\text{=}\,\,\text{original}\,\,\text{price}\,\,\times \,\,\left( {1+\left. {\frac{{\text{tax}\,\,\text{percentage}}}{{100}}} \right)} \right.$$

$$\displaystyle \50\,\times \,\left( {1+\left. {\frac{{8.25}}{{100}}} \right)} \right.=\50\,\times \,\left( {1+\left. {.0825} \right)} \right.=\50\times 1.0825=\54.125=\54.13$$

If we need to figure out the actual percent decrease or increase (percent change), we can use the following formula:

$$\displaystyle \text{Percent Increase}=\frac{{\text{New Price}-\text{Old Price}}}{{\text{Old Price}}}\,\times 100$$

$$\displaystyle \text{Percent Decrease}\,=\frac{{\text{Old Price}-\text{New Price}}}{{\text{Old Price}}}\,\,\times \,100$$

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For example, say we want to work backwards to get the percentage of sales tax that we pay (percent increase). If we know that the original (old) price is $50, and the price we pay (new price) is$54.13, we could get the % we pay in tax this way (note that since we rounded to get the 54.13, our answer is off a little):

$$\displaystyle \text{Percent Increase (Tax)}\,\,=\frac{{54.13-50}}{{50}}\,\,\times \,100\,=\,8.26%$$

Sometimes we have to work a little backwards in the problem to get the right answer. For example, we may have a problem that says something like this:

Your favorite pair of shoes are on sale for 30% off. The sale price is $62.30. What was the original price? To do this problem, we have to think about the fact that if the shoes are on sale for 30%, we need to pay 70% for them. Also remember that “of = times”. We can set it up this way: $$\displaystyle \,\,.7\,\,\times \,\,?=\62.30\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,? = \frac{{62.30}}{{.7}} = \89$$ The original price of the shoes would have been$89 before tax.

In the Algebra sections, we will address solving the following types of percentage problems, but I’ll briefly address them here if you need to do them now. If you don’t totally follow how to get the answers, don’t worry about it, since we’ll cover “word problems” later!

 What is 20% of 100? $$20\%\,\,\text{of}\,\,100=.2\times 100=20$$ 100 is what percentage of 200? $$\begin{array}{c}100=\,\,?\%\times 200\\\frac{{100}}{{200}}=\,\,?\%\\?=50\end{array}$$ 200 is 50% of what number? $$\begin{array}{c}200=50\%\,\,\times \,\,?\\200=.5\,\,\times \,\,?\\200\,\,\text{is half of what}?\\?=400\end{array}$$

One other way to address percentages is the “$$\displaystyle \frac{{\text{is}}}{{\text{of}}}$$” trick, which we’ll address below.

Ratios and Proportions

Ratios are just a comparison of two numbers. They look a little scary since they involve fractions, but they really aren’t bad at all. Again, they are typically used when you are comparing two things — like cost of one pair of shoes to another pair, or maybe even the number of shirts you have compared to the number of jeans you have.

Let’s use that as an example. Let’s say you have about 5 shirts for every one pair of jeans you have, and you figure this same ratiois pretty typical among your friends. You can write your ratio in a fraction like $$\displaystyle \frac{5}{1}$$, or you can use a colon in between the two numbers, like 5 : 1 (spoken as “5 to 1”). The fractions over 1 is actually a rate (this word is related to the word ratio!), for example, just like when you think of miles per hour. Our rate is shirts per one pair of jeans – 5 shirts for every pair of jeans.

Also note that this particular ratio is a unit rate, since the second number (denominator in the fraction) is 1.

Let’s say you know your friend Alicia has 7 pairs of jeans and you’re wondering how many shirts she has, based on the ratio or rate of 5 shirts to one pair of jeans. We can do this with math quite easily by setting up the following proportion, which is an equation (setting two things equal to one another) with a ratio on each side:

$$\displaystyle \frac{{\text{shirts}}}{{\text{jeans}}}=\frac{5}{1}=\frac{?}{7}$$

How do we figure out how many shirts Alicia has? One way is just to think about reducing or expanding fractions. Let’s expand the fraction $$\displaystyle \frac{5}{1}$$ to another fraction that has 7 on the bottom:

$$\displaystyle \frac{{\text{shirts}}}{{\text{jeans}}}\,=\,\frac{5}{1}\,=\,\frac{5}{1}\,\times \,1\,=\frac{5}{1}\,\,\times \,\,\frac{7}{7}\,=\frac{{35}}{7}$$

Alicia would have about 35 shirts.

Now I’m going to also show you a concept called cross-multiplying, which is very, very useful, even when we get into Algebra, Geometry and up through Calculus! This is a much easier way to do these types of problems.

Remember the “butterfly up” concept when we’re comparing fractions, and remember how the fractions are equal when the “butterfly up” products are equal?

We’re going to use this concept to set the fractions or ratios equal so we know how many shirts Alicia has:

We know that 5 × 7 = 35, so we need to know what multiplied by 1 will give us 3535!! Alicia has 35 shirts!!! See how easy that was? Now if we didn’t have the 1 as a factor to get to 35, we’d have to divide 35 by the number under the 5 to get the answer. This is because dividing “undoes” multiplying.

One of my students also suggested to use the “WON” method for proportions. To do this, you set up a table with WON at the top.  “W” stands for Words, “O” stands for Original or Old, and “N” stands for New (in this example, for Alicia). Put the words and numbers in the table, and then cross multiply like we did earlier. Again, we get that Alicia has 35 shirts, based on my proportion of 5 shirts to every pair of jeans, and the fact that she has 7 pairs of jeans.

Let’s try a cooking example with proportions, since sometimes the recipe might give you the amounts in tablespoons, for example, and you only have a measuring spoon with teaspoons. We know from the Fractions section that 1 tablespoon = 3 teaspoons, and let’s say the recipe calls for 2 tablespoons. This seems pretty easy to do without the proportion, but let’s set it up anyway, so you can see how easy it is to use proportions:

$$\require{cancel} \displaystyle \frac{{\text{teaspoons}}}{{\text{tablespoons}}}\,\,\,\,\,={}^{6}{{\xcancel{{\frac{3}{1}\,\,\,=\,\,\,\frac{?}{2}}}}^{6}}$$

We know that 3 × 2 = 6, so we need to know what multiplied by 1 will give us 6. We would need 6 teaspoons for our 2 tablespoons.

Now let’s go on to a more complicated example that relates back to converting numbers back and forth between the metric system and our customary system here. (For more discussion on the metric system, see the Metric System section).

Let’s say we have 13 meters of something and we want to know how many feet this is. We can either look up how many feet are in 1 meter, or how many meters are in 1 foot – it really doesn’t matter – but we need a conversion number.

We find that 1 meter equals approximately 3.28 feet. Let’s set all this up in a proportion. Remember to keep the same unit either on the tops of the proportion, or on the sides – it works both ways:

$$\displaystyle \frac{{\text{meters}}}{{\text{meters}}}\,\,=\,\,\frac{{\text{feet}}}{{\text{feet}}}\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\frac{{\text{meters}}}{{\text{feet}}}\,\,=\,\,\frac{{\text{meters}}}{{\text{feet}}}\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\frac{{\text{feet}}}{{\text{meters}}}\,\,=\,\,\frac{{\text{feet}}}{{\text{meters}}}$$

Let’s solve both two different ways to get the number of feet in 13 meters. Notice that we can turn proportions sideways, move the “=” sideways too, and solve – this is sort of how we got from the first equation to the second above.

 Proportion Cross Multiplying Explanation $$\displaystyle \frac{{\text{meters}}}{{\text{meters}}}\,\,\text{=}\,\,\frac{{\text{feet}}}{{\text{feet}}}$$ $$\require{cancel} \displaystyle {}^{{42.64}}{{\xcancel{{\frac{1}{{13}}\,\,\,=\,\,\,\frac{{3.28}}{?}}}}^{{42.64}}}$$ We know 1 meter is 3.28 feet, so we put these numbers across. We put 13 under the 1, since it’s also a meter. Then we cross multiply to get  $$?\,\times 1=42.64$$ feet so there are 42.64 feet in 13 meters. $$\displaystyle \frac{{\text{meters}}}{{\text{feet}}}\,\,=\,\,\frac{{\text{meters}}}{{\text{feet}}}$$ $$\displaystyle {}^{{42.64}}{{\xcancel{{\frac{1}{{3.28}}\,\,\,=\,\,\,\frac{{13}}{?}}}}^{{42.64}}}$$ We know 1 meter is 3.28 feet, so we put those on the left. We put 13 across from the 1, since it’s also a meter. Then we cross multiply to get  $$?\,\times 1=42.64$$ feet so there are 42.64 feet in 13 meters.

Here’s an example where we have to do some dividing with our cross multiplying. Try to really understand why we have to divide by 2 to get the answer (it “undoes” the multiplying):

$$\displaystyle \frac{5}{2}\,=\,\frac{?}{9}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5\,\,\times \,\,9\,=2\,\times \,?\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,?\,=\,\frac{{5\,\times \,9}}{2}\,=\,\frac{{45}}{2}\,=\,\,22\frac{1}{2}$$

Unit Multipliers

We can also use what we call unit multipliers to change numbers from one unit to another.  The idea is to multiply fractions to get rid of the units we don’t want. You probably will use this technique some day when you take Chemistry; it may be called Dimensional Analysis.

Let’s say we want to use two unit multipliers to convert 58 inches to yards.

Since we have inches and we want to end up with yards, we’ll multiply by ratios (fractions) that relate the units to each other. We can do this because we are really multiplying by “1”, since the top and bottom amounts will be the same (just the units will be different). Let’s first set this up with the units we have to see what we’ll need to have on the top and the bottom. I put 1’s under the first and last items to make them look like fractions:

$$\displaystyle \frac{{58\text{ inches}}}{1}\,\,\times \,\,\frac{?}{?}\,\,\times \,\,\frac{?}{?}\,\,=\,\,\frac{{?\text{ yards}}}{1}$$

We need to get rid of the inches unit on the top and somehow get the yards unit on the top; since the problem calls for 2 unit multipliers, we’ll include feet to do this:

$$\require{cancel} \displaystyle \frac{{58\text{ }\cancel{{\text{inches}}}}}{1}\,\times \,\frac{{?\text{ }\cancel{{\text{feet}}}}}{{?\text{ }\cancel{{\text{inches}}}}}\,\times \,\frac{{?\text{ }\,\text{yards}}}{{?\text{ }\cancel{{\text{feet}}}}}\,=\,\frac{{\text{? }\,\text{yards}}}{\text{1}}$$

Now just fill in how many inches are in a foot, and how many feet are in a yard, and we can get the answer with real numbers:

$$\displaystyle \frac{{58\text{ }\cancel{{\text{inches}}}}}{1}\,\times \,\frac{{1\text{ }\cancel{{\text{foot}}}}}{{12\text{ }\cancel{{\text{inches}}}}}\,\times \,\frac{{1\text{ yard}}}{{3\text{ }\cancel{{\text{feet}}}}}\,=\,\frac{{58\times 1\times 1\text{ yards}}}{{1\times 12\times 3}}\,=\,\frac{{58}}{{36}}\text{ }\,\text{yards}\,=\,\frac{{29}}{{18}}\text{ }\,\text{yards}$$

Here’s another example where we use two unit multipliers since we are dealing with square units:

Use two unit multipliers to convert 100 square kilometers to square meters.

$$\displaystyle \frac{{100\text{ }\cancel{{\text{kilometers}}}\times \cancel{{\text{kilometers}}}}}{1}\,\times \,\frac{{1000\text{ meters}}}{{1\text{ }\cancel{{\text{kilometer}}}}}\,\times \,\frac{{1000\text{ meters}}}{{1\text{ }\cancel{{\text{kilometer}}}}}\,=\,100,000,000\,\, \text{meter}{{\text{s}}^{2}}$$

Using Percentages with Ratios

Now let’s revisit percentages and show how proportions can help with them too! One trick to use is the $$\displaystyle \frac{{\text{is}}}{{\text{of}}}$$ and $$\displaystyle \frac{{\text{part}}}{{\text{whole}}}$$ tricks. You can remember these since the word that comes first in the alphabet (“is” and “part”) are on the top of the fractions.

You can typically solve percentage problems by using the following formula:

$$\displaystyle \frac{{\text{is}}}{{\text{of}}}=\frac{\text{ }\!\!\%\!\!\text{ }}{{100}}$$

What this means is that the number around the “is” in an equation is on top of the proportion, and the number that comes after the “of” in an equation is on bottom of the proportion, and the percentage is over the 100.

You can also think of this as the following, but you have to remember that sometimes the part may be actually be bigger than the whole (if the percentage is greater than 100):

$$\displaystyle \frac{{\text{part}}}{{\text{whole}}}=\frac{\text{ }\!\!\%\!\!\text{ }}{{100}}$$

Here are some examples, using the same problems that we did above in the Percentages section.  (Later, in the Algebra section, we’ll learn how to translate math word problems like these word-for-word from English to math.)

• What is 20% of 100?  Since we know that the 20 of the % part, we put that over the 100. The 100 comes after the “of”, so we put that on the bottom. Also, we’re looking for the “part” of the “whole” here.

$$\displaystyle \frac{{\text{is}}}{{\text{of}}}=\,\frac{\%}{{100}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\frac{{\text{part}}}{{\text{whole}}}=\frac{\%}{{100}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{?}{{100}}=\frac{{20}}{{100}}\,\,\,\,\,\,\,\,\,?=20$$

• 100 is what percentage of 200?  The 100 is close to the “is” so we put that on the top. The 200 comes after the “of”, so we put that on the bottom. Also, we know the 100 is the “part” of the 200.

$$\displaystyle \frac{{\text{is}}}{{\text{of}}}=\frac{\%}{{100}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\frac{{\text{part}}}{{\text{whole}}}=\frac{\%}{{100}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{100}}{{200}}=\frac{{?\,\,\,\%}}{{100}}\,\,\,\,\,\,\,\,\,?=50$$

• 200 is 50% of what number?   The 200 is close to the “is” and we don’t know what the “of” is. The 50 is the percentage. Also, 200 is the “part”, so we need to find the “whole”.

$$\displaystyle \frac{{\text{is}}}{{\text{of}}}=\frac{%}{{100}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\frac{{\text{part}}}{{\text{whole}}}=\frac{%}{{100}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{200}}{?}=\frac{{50}}{{100}}\,\,\,\,\,\,\,\,\,?=400$$

Remember – if you’re not quite sure what you’re doing, think of the problems with easier numbers and see how you’re doing it!  This can help a lot of the time.

Learn these rules and practice, practice, practice!

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On to Negative Numbers and Absolute Value – you are ready!!

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