Algebra-041Ratios, Rates, and Proportions

The concept of a ratio is similar to that of a fraction, but instead of taking a part and a whole in the same group, a ratio can compare two independent groups. Using ratios to compare different things can get you things like proportions and rates, which are explored in these resources.

Rates and Proportions

This is an interactive activity that demonstrates how rates and proportions work.

PhET Interactive Simulations: Ratio and Proportions

Ratios

This video explains 3 different ways for determining if two ratios are proportional (using simplified ratios, cross products, and equivalent fractions).

Transcript

0:00 Welcome to math with Mr. J. in this video I’m going to cover how to determine if two ratios are proportional.

0:10 I’ll go through three ways we can do this. I’ll cover how to use simplifying, cross multiplication, and equivalent fractions.

0:20 Now remember, a proportion is formed by equal or equivalent ratios. They represent the same comparison and relationship.

0:31 We'll start by using simplifying to determine if two ratios are proportional. Let's jump into number one, where we have 10 to 15 and 8 to 12.

0:43 Now when it comes to checking if two ratios are proportional using simplifying, all we need to do is simplify both ratios.

0:53 If they simplify to the same ratio, they are proportional. If they are different, they are not proportional. Let's set these ratios up in fractional form and simplify.

1:06 We'll start with 10 to 15. Now the greatest common factor between 10 and 15 is 5, so let's divide both of these terms by 5.

1:22 So 10 divided by 5 and 15 divided by 5. 10 divided by 5 is 2, 15 divided by 5 is 3. The greatest common factor between 2 and 3 is 1, so we are in simplest form. 10 to 15 simplifies to 2 to 3.

1:45 Now, we need to simplify 8 to 12. The greatest common factor between 8 and 12 is 4, so let's divide both of these terms by 4. 8 divided by 4 and 12 divided by 4.

2:04 8 divided by 4 is 2. 12 divided by 4 is 3. So both of our ratios 10 to 15 and 8 to 12 simplify to 2 to 3.

2:18 Since they simplify to the same ratio, they are proportional, so we can set them up as equal to each other and write them as a proportion. So 10 to 15 is equal or proportional to 8 to 12.

2:41 Let's move on to number 2, where we have 21 to 12 and 14 to 6. Let’s set these up in fractional form and simplify.

2:53 We'll start with 21 to 12. The greatest common factor between 21 and 12 is 3, so let's divide both of these terms by 3.

3:07 21 divided by 3 and 12 divided by 3. 21 divided by 3 is 7. 12 divided by 3 is 4. The greatest common factor between 7 and 4 is 1, so we are in simplest form. 21 to 12 simplifies to 7 to 4.

3:30 Now we need to simplify 14 to 6. The greatest common factor between 14 and 6 is 2. Let's divide our terms by 2. So 14 divided by 2 and 6 divided by 2.

3:49 14 divided by 2 is 7. 6 divided by 2 is 3. The greatest common factor between 7 and 3 is 1, so we are in simplest form. 14 to 6 simplifies to 7 to 3.

4:07 So for number 2, our ratios of 21 to 12 and 14 to 6 did not simplify to the same ratio. That means that these ratios are not proportional. We can write this as 21 to 12 is not equal or not proportional to 14 to 6.

4:36 So there you have it. There's how you determine if ratios are proportional by simplifying. Simplify each ratio, if the ratios simplify to the same ratio, they are proportional. If they don't simplify to the same ratio, they are not proportional.

4:54 Next, we'll move on to using cross multiplication. Here are our examples for cross multiplication. Let's jump into number one where we have 8 to 3 and 11 to 7.

5:09 Now when it comes to checking if two ratios form a proportion using cross multiplication, all we need to do is cross multiply and see if our cross products, the numbers we get once we multiply, are equal or not.

5:27 If they are equal, then we have a proportion. If they are not equal, then we do not have a proportion.

5:34 Let's start by setting our ratios up in fractional form and setting them equal to each other. Now they may not turn out to be equal, but for now set them up as a proportion.

5:47 We have 8 to 3 and 11 to 7. Now we can cross multiply, so we multiply diagonally. We'll start with 8 times 7, so 8 times 7, and then bring our equal sign straight down.

6:12 And now we can do 3 times 11. 8 times 7 is 56, 3 times 11 is 33. 56 is not equal to 33, so our cross products were not equal.

6:35 That means we do not have a proportion. 8 to 3 and 11 to 7 are not proportional. They do not form a proportion. So we can write this out as 8 to 3 is not equal, or not proportional, to 11 to 7.

6:58 Now one more thing I do want to mention about cross multiplying these ratios. Don't get too held up on what numbers to multiply first or which diagonal you should put on which side of the equal sign.

7:11 It will all work out the same either way. So if you do 3 times 11 or 11 times 3 first and then 8 times 7 or 7 times 8, that will give you the same answer. Just make sure to multiply diagonally and have the equal sign in between your cross products.

7:32 Let's move on to number two, where we have 6 to 10 and 9 to 15. We’ll start by setting these up as a proportion, so 6 to 10 and then 9 to 15.

7:49 Now we're ready to cross multiply. We'll start with 6 times 15, so 6 times 15. And now we have 10 times 9, so 10 times 9.

8:10 6 times 15, that equals 90. 10 times 9, that also equals 90. So both of our cross products are 90.

8:21 That means our ratios of 6 to 10 and 9 to 15 are proportional, so we can write this as 6 to 10 is equal or proportional to 9 to 15. So those ratios form a proportion.

8:43 So there you have it. There's how you determine if ratios are proportional using cross multiplication. Cross multiply, if the cross products are equal, the ratios are proportional. If the cross products are not equal, the ratios are not proportional.

8:59 Lastly, we will move on to using equivalent fractions. Here are our examples for equivalent fractions. Let's jump into number one where we have 6 to 8 and 7 to 12.

9:15 Now when it comes to checking if two ratios are proportional using equivalent fractions, we first need to write the ratios in fractional form.

9:25 Then, we find the least common multiple between the bottom terms of our ratios and rename the ratios as equivalent ratios with that new bottom term.

9:37 This is just like renaming fractions with a common denominator, so the same exact steps, we are just working with ratios.

9:47 We'll start by rewriting 6 to 8 and 7 to 12 in fractional form, so 6 to 8 and then 7 to 12. Now we need to find the least common multiple between those bottom terms 8 and 12.

10:08 Now when it comes to multiples, think multiply or skip count. For number one I’ll write out some multiples of 8 and 12 as a refresher, and we can look for that least common multiple.

10:20 We'll start with 8. So I’ll come to the side here, and again, think multiply or skip count. 8 times 1 is 8. 8 times 2 is 16. 8 times 3 is 24, and 8 times 4 is 32.

10:38 Now multiples go on forever. My suggestion: Write 4 or 5 for each term. Look for a least common multiple. If you still don't have one, you can always extend your lists. So let's write four multiples of 12 as well and find the least common multiple between our bottom terms.

10:59 So 12 times 1 is 12, 12 times 2 is 24, 12 times 3 is 36, 12 times 4 is 48. So we can look to see if we have any in common, and we want the least common multiple, so the smallest in value that we have in common.

11:16 That's going to be 24. Now that we have that least common multiple, we are going to rename our original ratios with that 24 as the bottom term, the denominator.

11:30 So again, we're going to rename our original ratios as equivalent ratios, or equivalent fractions, with 24 being the bottom number, the denominator.

11:41 So let's come back over here and go underneath our original ratios and rename. So let's start with 6 to 8. How do we get 8 to equal 24? Well 8 times 3 is 24.

11:57 Whatever we do to the bottom we must do to the top in order to keep this equivalent, so multiply 6 by 3 as well. 6 times 3 is 18. 18 to 24 is equivalent to 6 to 8. So we're not changing the value of that ratio at all.

12:18 Now let's go to 7 to 12. How do we get 12 to equal 24? Well 12 times 2 is 24. Whatever we do to the bottom we must do to the top in order to keep this equivalent.

12:33 7 times 2 is 14. So 14 to 24 is equivalent to that 7 to 12. Now 18 to 24 is not equal to 14 to 24.

12:49 That means our original ratios of 6 to 8 and 7 to 12 are not proportional, so we can write that as 6 to 8 is not equal or proportional to 7 to 12.

13:10 Let's move on to number two, where we have 2 to 10 and 5 to 25. We'll start by rewriting these ratios in fractional form, so 2 to 10 and 5 to 25.

13:26 Now we need to think of the least common multiple between 10 and 25. We're going to do this one without writing out the lists of multiples.

13:36 The least common multiple between 10 and 25 is 50, so we need to rename these ratios using that 50 as our bottom term, the denominator.

13:47 So we're going to write equivalent ratios or equivalent fractions. So let's come underneath our ratios and use that 50 for our bottom term, the denominator.

13:59 We'll start with 2 to 10. So how do we get 10 to equal 50? Well 10 times 5 equals 50. Whatever we do to the bottom we must do to the top in order to keep this equivalent. 2 times 5 is 10. 10 to 50 is equivalent to 2 to 10.

14:19 Now we need to do 5 to 25. How do we get 25 to equal 50? Well 25 times 2 is 50. Do the same thing to the top in order to keep this equivalent. 5 times 2 is 10. 10 to 50 is equivalent to 5 to 25.

14:40 Now we have that same bottom term, that same denominator, for our equivalent ratios. Our equivalent fractions, 10 to 50 equals 10 to 50. That means that our original ratios, 2 to 10 and 5 to 25, are proportional, they form a proportion.

15:01 So we can write that as 2 to 10 is equal or proportional to 5 to 25. So there you have it. There's how to determine if ratios are proportional using equivalent fractions.

15:19 Rewrite the ratios in fractional form then find the least common multiple between the bottom terms of the ratios. That number will be the bottom term, or denominator, of the renamed equivalent ratios or renamed equivalent fractions.

15:36 If these equivalent ratios or equivalent fractions are the same, then the ratios are proportional. If they are different, they are not proportional.

15:46 As a quick recap, we can use simplifying, cross multiplication, or equivalent fractions to determine if two ratios form a proportion.

15:57 I hope that helped. Thanks so much for watching. Until next time. Peace!

Attribution

Math with Mr. J.(2022). "Determining if Two Ratios are Proportional (3 Ways)." Youtube.com. https://www.youtube.com/watch?v=TWUK7uMgEXE

Percents

This video explains the meaning of a percent, as well as how to convert between percents, fractions, and decimals.

Transcript

0:00 Write 18% as a decimal and as a fraction in simplest form. So let's do it as a decimal first. So 18% is the same thing as 18 per 100, or 18 per cent.

0:18 I'm actually separating out the percent, it's only going to be one word, but I'm writing it as, literally, per cent. Cent means the same thing as 100. So this literally means 18 per 100.

0:29 Actually, I said I would do the decimal first, but we can start putting it into a fraction first. 18 per 100 as a fraction literally means 18/100.

0:39 We're literally doing the fraction first. This literally means 18 per 100, or 18 hundredths. From here we can go straight to a decimal or we could do this fraction in simplest form.

0:52 Let's do the decimal first, just because that's actually what I said I would do first. Let's do that first. This is the same thing as 18 hundredths. And we know how to write that in decimal form. It's 0.18.

1:12 You could view this as 1 tenth and 8 hundredths, which is the same thing, or 10 hundredths and 8 hundredths, which is 18 hundredths. So this is written in decimal form.

1:22 And if we write it as a simplified fraction, we need to see if there is a common factor for 18 and 100. And they're both even numbers, so we know they're both divisible by 2, so let's divide both the numerator and the denominator by 2.

1:35 So we have 18 divided by 2 over 100 divided by 2. And we're going to get 18 divided by 2 is 9. 100 divided by 2 is 50. And I don't think these guys share any common factors.

1:51 50 is not divisible by 3. 9 is only divisible by 3 and 1 and 9. So this is the fraction in simplest form. So we have 18% is the same thing as 0.18, which is the same thing, in simplest form, as 9/50.

2:08 Now, I went through a lot of pain here to show you that this really just comes from the word, from percent, from per 100.

2:16 But if you ever were to see this in a problem, the fast way to do this is to immediately say, OK, if I have 18%, you should immediately say, anything in front of the percent-- that's that anything, whatever this anything is-- it should be equal to that anything.

2:32 In this case it's 18/100. And another way to think about it, you could view this as 18.0%. I just added a trailing zero there, just so that you see the decimal, really.

2:50 But if you want to express this as a decimal without the percent, you just move the decimal to the left two spaces. So if we move the decimal to the left two spaces, one, two, this becomes 0.18.

3:11 Or you could immediately say that 18% as a fraction is 18/100. When you put it in simplified form, it's 9/50.

3:18 But you should also see that 18/100, and we have seen this, is the exact same thing as 18 hundredths, or 0.18. Hopefully, this made some connections for you and didn't confuse you.

Attribution

Khan Academy. (2023). Khanacademy.org. https://www.khanacademy.org/math/cc-sixth-grade-math/x0267d782:cc-6th-rates-and-percentages/x0267d782:equivalent-representations-of-percent-problems/v/representing-a-number-as-a-decimal-percent-and-fraction

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