TEK 6.4 G Learning Goal: I can generate equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money.

TEK 6.5 B Learning Goal: I can solve real-world problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole, including the use of concrete and pictorial models.

TEK 6.5 C Learning Goal: I can use equivalent fractions, decimals, and percents to show equal parts of the same whole.

Vocabulary

Percent: A ratio that compares a number to 100.

Scaling Up: Multiplying the numerator and denominator of a fraction/ratio by the same factor

Scaling Down: Multiplying the numerator and denominator of a fraction/ratio by the same factor

Equivalent Ratios: Ratios that express the same relationship between two quantities

Proportion: An equation stating that two ratios are equal

Percent Proportion: A proportion with one ratio or fraction that compares part of a quantity to the whole quantity. The other ratio is the equivalent percent written as a fraction with a denominator of 100.

Interest: Money paid regularly at a particular rate in return for being able to borrow money.

Percents

Percents are ratios that compare numbers to 100. As you already know, they are written with a percent sign (%). The percent sign represents ":100." So, 70% is just another way of writing 70:100.

In most other ratios, you are comparing two unrelated things - apples to oranges, for example. Percents are unique in that the number you are comparing 100 to is always part of the 100 as well. A percent would be used to compare apples to the total amount of fruit, for example. The total amount of fruit includes apples, and also other fruit like oranges as well.

Models like this can help you visualize this:

In this picture, there are 100 squares. Each square represents a piece of fruit. The shaded squares represent all of the apples. There are 45 shaded squares, so the ratio of apples to total fruit is 45:100. ":100" can be written as %, so 45% is represented in this picture. 45% of the fruit are apples.

Even though the apple squares are shaded, they are still attached to the rest of the fruits in the same giant square because they are part of a whole. A lot of times, people would say that 45 out of 100 fruits are apples. "Out of" means that, out of all of the items here, a specific group is being focused on.

Percents to Fractions

Continuing on from the paragraph before, "out of" is also another way of saying "divided by." So, our fruit ratio can be rewritten as 45 divided by 100. Fractions are basically just division problems, so this is the same thing as writing:

So, percents can easily be turned into fractions if you just put 100 on bottom and put the number you are comparing to 100 on top. But you already knew this. If 45% is just a ratio of 45:100, then of course it can be turned into a fraction by putting the first quantity on top and the second quantity on bottom. All ratios can be rewritten as fractions, so it's no different than what we did in the ratios lesson (check that out if you haven't already.)

The only thing you need to make sure to do is simplify your fraction. 45 and 100 have a greatest common factor of 5, so we can simplify the fraction:

Percents to Decimals

Finding the decimal form of a percent is just taking the fraction form and actually doing the division. A lot of times, you will be able to do this quickly in your head. If you had a fraction of 1/2, for example, you know that dividing 1 by 2 will give you 0.5 because you have used that number a lot in your life.

However, you might not know how to convert the apple percentage we were working with before into decimal form. You can divide 9 by 20 on your calculator (or even 45 by 100, since they are equivalent). You will get the answer of 0.45. But there is also a quick way to do this that doesn't involve a calculator.

Try dividing 100 by 100. You get 1. Now divide 900 by 100. You get 9. Now divide 1400 by 100. You get 14. What you should notice is, every time you divide by 100, the decimal place moves to the left two times.

100.00 changes to 1.0000, and the decimal goes from sitting after the third number to sitting after the first.

900.00 changes to 9.0000, and the decimal goes from sitting after the third number to sitting after the first.

1400.00 changes to 14.0000, and the decimal goes from sitting after the fourth number to sitting after the second.

Each time, the decimal place gets to sit two spots to the left. What this means is that if you want to divide any number by 100, just move the decimal to the right two times. This is very helpful when it comes to percents, because percents are always out of 100.

When you are finding the decimal form, start with the unsimplified fraction (the one that is over 100), then move the decimal place of the numerator twice to the left:

So, 45% in decimal form is 0.45.

Decimals to Percents

When you change a percent to a decimal, you divide not only the percentage part of the ratio by 100 but also the 100. When you turn 45% (45:100) into a decimal, you move the decimal to the left two times on the 45 (to 0.45), but also on the 100 (to 1). The new ratio is 0.45:1. The reason we write it as just 0.45 is that any number divide by 1 is just that number.

Remember that a percent is always a comparison to 100. So, if we have a ratio of 0.45:1, we need to multiply both parts by 100 so that the 1 turns into 100.

You know that dividing a number by 100 is the same as moving the decimal place twice to the left. Multiplication is the opposite of division, so if you want to multiply by 100, you move the decimal place twice to the right. So, multiplying 0.45 by 100 moves the decimal over from being after the first number to after the third number, 45. The new ratio is 45:100, or 45%.

In short, all you need to do to turn a decimal to a percent is move the decimal place to the right twice.

Fractions to Percents

Changing fractions to percents can be a little bit trickier than changing decimals to percents. If you have a fraction that is over 100, like 6/100, you can easily change it into a percent because it is already in the form of a ratio being compared to 100 (in fraction form). All you have to do is look at it like this: 6:100, and change the ":100" into a %, so you have 6%.

But most of the time, fractions won't have 100 in the denominator, especially since you simplify them. 6/100 can be simplified to 3/50 if you divide the numerator and denominator by 2, the greatest common factor. Now, this isn't a ratio that has a comparison to 100, so it can't immediately be turned into a percent. What you have to do is figure out how to make it a comparison to 100. Remember that when you multiply both parts of a ratio by the same number, it creates an equivalent ratio. So, just find a number that multiplies with 50 to create 100, and multiply both parts by that number. A good way to figure out this number is to divide 100 by the denominator of the fraction, 3/50, or divide 100 by 50. The result is 2. So, multiply both parts of the ratio by two. The result is 6/100, which turns into 6%.

However, sometimes the denominator of your fraction doesn't fit into 100 evenly. Let's say you have the fraction 11/55. 100 divided by 55 is not a whole number. So, you have to reduce the fraction before you can scale up to where the denominator is 100. Find a number that you can divide the denominator by so that the denominator fits evenly into 100. For 55, we can divide by 11 to give us 5, which goes into 100 evenly. So, divide both sides by 11 and you get 1/5. Now, you can divide 100 by 5 to see what number you need to scale up by. 100 divided by 5 is 20, so multiply both parts by 20. 20/100 is the new fraction, which is 20%.

When you have to convert a fraction to a percent:

1. If the fraction's denominator is 100, just put the numerator in front of a percent sign (%), and that is your answer.

2. If the fraction's denominator is not 100, divide 100 by the denominator of the fraction. If the denominator of the fraction goes into 100 evenly, multiply the numerator and denominator of the fraction by the quotient. You should have a number over 100 now, and you can put the numerator in front of a percent sign (%).

3. If the denominator of the fraction does not go into 100 evenly, scale the fraction down until the denominator goes into 100 evenly. Then, divide 10 by the new denominator, and scale the fraction up by multiplying by the quotient.

Percents in the Real World

Percents are usually used when analyzing data. If you have a lot of data, and it is all similar in some way, you can classify groups within the data using percents. In our fruit example from before, the data is each fruit that we have. All of the data points are similar because they are all fruits, but there are also differences between them. We can separate the data based on these differences - all the apples can be in a group, all the oranges can be in a group, all the lemons can be in a group, etc. We can compare the percentages of the different types of fruits and start making conclusions. Maybe oranges have a low percentage compared to the other fruits, and we can make a hypothesis that oranges must not be in season, or whoever bought the fruit doesn't really like oranges. Of course, you wouldn't know why there aren't many oranges from just looking at the percentages; you would have to do some additional research. But comparing percentages in data is a good first step for scientists trying to solve a problem.

In the fruit example, we said earlier that we had 100 pieces of fruit. However, most of the time, there isn't going to be 100 data points. Let's say a team of scientists collected a sample of air from the Earth's atmosphere. They put their data into this table:

The nitrogen and oxygen in the atmosphere are both molecules, so it makes sense that they'd both be part of the same whole - the total amount of molecules in this sample.

But the whole isn't going to be 100 molecules. As you probably know, air molecules are very small, and a lot of them can fit into a very small space. Just one liter (the size of a disposable water bottle) can fit over 20,000,000,000,000,000,000,000 molecules of air - maybe even more depending on the conditions.

Lucky for us, these particular scientists have very advanced equipment, and they can trap a sample as small as 800 molecules. That's still a lot more than 100. The percentages in the table are actually simplified versions of the ratio of the amount of a specific type of molecule to the total amount of molecules. They are not necessarily the most simplified version of the ratio, but they are simplified enough so that a number is being compared to 100.

The actual number of nitrogen molecules in the 800 molecule sample is 624. The ratio of nitrogen molecules to total molecules is 624:800. What the scientists did to find the percentage of nitrogen is simplify this ratio until the second number is 100. 800 divided by 100 is 8, so they divided both sides by 8. 624 divided by 8 is 78. The new ratio was 78:100. Now you can see where they got the percentage from.

In 78%, each percent does not represent one individual molecule - it represents multiple - 8, to be exact, because the original ratio was scaled down by dividing by 8.

Sometimes, you will have to scale up to find a percentage. Look at this model of Ramiro's garden. Each square represents a square foot in his backyard.

There are only 20 squares. Like with the molecules, the percentage of a certain type of vegetable will be an equivalent ratio of the actual ratio of the square of that vegetable to the total square feet in the garden. You need to scale the original ratio up until the second number is 100.

Let's find the percentage of cucumbers. There are 8 squares labeled "cucumbers", so the ratio of square feet of cucumbers to total square feet in the garden is 8:20. You need to scale up until the second number of this ratio is 100. 100 divided by 20 is 5, so you know that multiplying 20 by 5 will get you to 100. If you multiply both sides of 8:20 by 5, you get 40:100, so your percentage is 40%.

Since you scaled up by multiplying by 5, each percent represents one fifth (1/5) of a square foot.

The point is, one percent doesn't always represent one point of data. It could represent many data points, or part of a data point. It just depends on the problem.

Percent Proportions

The percentage 40% could be written as 40:100, which can simplify to 2:5 if you divide both parts by the greatest common factor, 20. 40% is equal to 2:5 - every number in that phrase is part of the identity of this percentage. 40 tells us the percentage in the form of a number compared to 100, 2 gives us the part, and 5 gives us the whole.

The part is the numerator in the simplified fraction form, the whole is the denominator in the simplified fraction form, and the percent is the numerator in the unsimplified fraction form that has 100 as the denominator. Since the ratio of part:whole and percentage:100 are equivalent, we can write a proportion:

This proportion that compares the part to the whole and the percentage to 100 is called a percent proportion.

To describe this relationship, we often say that the part is a percent of the whole. You might say that 2 is 40% of 5, for example. The part corresponds with the percent because they are both in the numerator of their ratio - that's why they have an "is a" relationship. The whole and 100 correspond with each other because they are in the denominator.

Sometimes, you will see problems that give you the percent and the part but not the whole, or give you the percent and the whole but not the part, or give you the part and the whole but not the percent.

The ratio of the part to the whole, and the percent ratio (the percent number to 100) both should be equivalent ratios (like 40:100 and 2:5 are equivalent.) Because of this, we can set up a proportion between these two ratios, and solve for whatever the missing number is.

Problems that don't give you the Whole

Some problems will give you the percent and the part but not the whole. Like we said before, set up a proportion with what you know and then solve for the whole. Here is an example:

14 is 10% of what number?

Remember that this is read as "the part is a percent of the whole," so 14 (the part) is 10% of the whole. Set up a proportion using the ratio of part/whole and percentage/100:

To solve for the whole, cross multiply. You end up with 1400 = 10 × whole. If you are a bit familiar with algebra, you know that you can divide both sides by 10, and that will give you 140 = whole. Or you can just think that 10 times some number equals 1400. If you divide 1400 by 10, that tells you what number multiplies with 10 to get 1400. That number is 140, which is your answer.

Problems that don't give you the Part

Other problems won't give you the part. Here is an example of how questions like this may be asked:

What is 23% of 200?

Remember that the part is a percent of the whole, so when you solve for the part, it will be 23% of 200. If you have trouble seeing this, just look at the numbers we have here. 23% is obviously a percent, so you know we have a percent. The word "of" before 200 shows that something is a portion, or a part, of 200. So, 200 must be the whole, and you know we have a percent and a whole.

Same as before, just set up the proportion:

Now solve by dividing both sides by 100, or find what number you multiply 100 by to get 4600.

Problems that don't give you the Percent

The last type of problem is those that give you the part and the whole but not the percent. They are typically asked like this:

85 is what percent of 340?

The question directly tells you that you need to find the percent because it asks "what percent." Like we said before, the word "of" shows that something else is a part or portion of the following number - so "of 340" shows that whatever is before it is a piece of 340. This means that 340 is the whole, and 85 is the part. The part will always be first and the whole will always be second when the problem is asked in a simple way like this, following the order of the part:whole ratio.

You might be thinking that we actually have two numbers missing in our proportion, because a percent is a ratio of two numbers, and we are not given either of them. However, you know that a percent is always out of 100, so you know that the denominator of the percent is 100. Set up the proportion:

Now, divide both sides by 340, or find what number you have to multiply with 340 to get 8500. The answer should be 25%.

Percents and Money

When we talked about percents in the real world, most of our discussion was on experiments or data being collected. But another very important application of percents is money, especially for bank accounts and shopping for items on sale.

Bank Accounts

Banks offer you interest if you make a savings account. When you making a savings account and put money into it, you are basically loaning that money to the bank. So, you earn interest, a percentage of the money in your account, every so often in return for doing that.

Every time a bank gives you your interest, they add the percentage of your current balance to your balance, making the balance higher. If a savings account at a particular bank offers you 4% interest per year, and you start off with $500, after the first year they will add 4% of $500 to your bank account. Does the phrase "4% of $500" sound familiar? This is a problem where they give you a percent and a whole and you have to find out what the part is. Set up the percent proportion:

And solve the proportion:

The part is 20, so $20 is added to $500. Now you have $520 in your bank account. Next year, the bank will add 4% of $520, so they will give you even more money.

Sometimes, banks have a formula for their interest. An example of a formula they could have is:

I = prt

Where I represents the total interest given to the customer, p represents the initial, or original, amount, r represents the interest rate, and t represents the time in years (or months, depending on how often your bank awards interest.) If you have a formula like this, the percent you use needs to be in decimal form. When you multiply a number by a decimal, you are taking the percent of that number that is equivalent to that decimal.

For example, if you wanted to take 80% of 10, you could multiply 10 by the decimal form of 80%, which is 0.80. This works because you are setting up a proportion. The ratio 80/100 was scaled down to 0.80/1 when you turned it into a decimal. So, we are setting 0.80/1 equal to part/10.

You don't have to do this math every time - just know that multiplying by a decimal is the same as finding the percentage that the decimal represents.

Shopping

At stores, you have probably noticed how sales are in percents - 15% off, 20% off, 50% off. A sale is kind of the opposite of interest - you are taking the percentage of the price and then subtracting that, rather than adding the percentage like we did with bank accounts.

If a pair of $20 jeans is 10% off, you would find 10% of $20 and then subtract that from the original price. Again, you have the percent and the whole but not the part. Set up the percent proportion and solve:

10% of $20 is $2. So, $2 is subtracted from the price and the jeans now only cost $18.

Sometimes problems will ask about sales tax. Sales tax is like interest in that it is added to the amount. Sales tax is calculated after any sales an item may have on it. You find the percent of the cost of the item after money has been taken off because of the sale, and you add that percentage to the cost.

If the jeans from above were still 10% off, but now they have a sales tax of 5%, you will take 5% of $18 and add that to the cost.

5% of $18 is $0.90, so this is added to $18 for a final price of $18.90.

For practicing vocabulary with flashcards, go to:

https://quizlet.com/_6u6fo1

Use these quizzes to make sure you fully understand what you have just learned. Click "view score" to see explanations for any wrong answers. Click "submit another response" to try again.