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### Ratio

• Ratio: A comparison of two quantities by division
• Rate: A ratio that compares quantities in different rates
• Unit rate: A rate with a denominator of 1
• To convert units, cross-cancel out units until you’re left with the correct unit

### Ratio

Write in simplest form
[(12 ÷12)/(24 ÷12)]
[1/2]
Write in simplest form
[(42 ÷6)/(48 ÷6)]
[7/8]
Tommy has 7 blue marbles, 3 green marbles, and 10 red marbles in a bag. Find the ratio of green to red marbles.
• green:red
3:10
Out of 32 students in a classroom, 17 are boys. Find the ratio of boys to girls.
• boys:girls
• Number of girls = 32 - 17 = 15
17:15
Out of 10 people in an office, 3 are male. Find the ratio of male to female.
• male:female
• Number of female = 10 - 3 = 7
3:7
Find the unit rate
\$ 18.00 for 12 boxes
• [(\$ 18 ÷12)/(12  boxes ÷12)] = [\$ 1.5/box]
[1.5/box]
Find the unit rate
A car goes 200 mi on 5 gallons of gas
• [(200  mi ÷5)/(5  gallons ÷5)] = [(40  mi)/gallon]
[(40  mi)/gallon]
Find the unit rate
A box falls 130 ft in 10 seconds
• [(130  ft ÷10)/(10  seconds ÷10)] = [(13  ft)/second]
[(13  ft)/second]
A car is moving at 10 mi/h. How many feet wil it move in 10 minutes?
• 1 mi = 5280 ft
1 hour = 60 min
• [(10  mi)/(1  hour)] ·[(5280  ft)/(1  mi)] ·[(1  hour)/(60  min)] = [(880  ft)/min]
• [(880   ft)/min] ·[(10  min)/(10  min)] = [(8800  ft)/(10  min)]
8,800 ft
A sprinkler uses 2 gal/min. How many quarts will it use in 30 seconds?
• 1 gal = 4 qts
• 1 min = 60 seconds
• [(2  gal)/(1  min)] ·[(4  qts)/(1  gal)] ·[(1  min)/(60  seconds)] = [(8  qts)/(60  seconds)]
• [(8  qts ÷2)/(60  seconds ÷2)] = [(4  qts)/(30  seconds)]
4 quarts

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Ratio

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Ratio 0:05
• Definition of Ratio
• Examples of Ratio
• Rate 2:19
• Definition of Rate
• Unit Rate
• Example: \$10 / 20 pieces
• Converting Rates 6:46
• Example: Converting Rates
• Extra Example 1: Write in Simplest Form 16:22
• Extra Example 2: Find the Ratio 20:53
• Extra Example 3: Find the Unit Rate 22:56
• Extra Example 4: Convert the Unit 26:34

### Transcription: Ratio

Welcome back to Educator.com; for the next lesson, we are going to go over ratio.0000

A ratio is when you compare two things.0007

You are making a comparison between two quantities; it is also same as division.0011

If you look here, there is three ways to express a ratio.0020

There is three ways to write a ratio.0023

If A is 1 and B is something else, you can say A to B.0028

You can write it out, A to B.0033

All this, whether you write it like this, like this, or like this, they are all read as A to B.0036

But you can write it like this, like this using a colon or as a fraction A to B.0045

You are still comparing A and B.0053

For example, if I said what is the ratio, you are comparing boys to girls.0056

Because I said boys first, boys is going to be written as A, the first one.0064

Then girls has to be the second one; boys to girls.0071

If I ask for the ratio of boys to girls, then I can't give you the number of girls to boys.0077

You can't do this. You have to write out the ratio in the order that was asked for; boys to girls.0086

If I say there are 5 boys and there are 3 girls, then the ratio of boys to girls would be 5 to 3.0095

You can also write it as 5 to 3 like that.0108

If I ask you what is the ratio of girls to boys, then you would have to0114

give me this number first, the number of girls to the number of boys.0117

You always have to write out the ratio in the order that it was asked for.0124

A to B, A:B, and A/B as a fraction; this is called ratio.0132

A rate is a ratio; you are still comparing A to B.0141

But you are given different rates; for example, if I say miles per hour.0148

Miles per hour would be... you have the number of miles and you have however many number of hours.0158

You are comparing, you are making a ratio between the number of miles and the number of hours.0169

A rate would be a ratio, same thing, A to B, but using different rates.0175

If I say \$5 for 5 candies, then that is a ratio.0183

You are making the comparison between the amount over the number of candy.0194

If you make a comparison between two things, it is called a ratio.0204

When those two things have some kind of unit, then it is called a rate.0209

A unit rate is a rate with a denominator of 1.0219

That means that if I say I traveled 2 miles in 2 hours.0225

Here is my ratio, 2 miles every 2 hours; this is my ratio.0239

A unit rate would mean to make this, the denominator, the bottom number, a 1.0247

That means I need to change this to become 1 hour.0255

That would be a unit rate.0259

That means in order to turn this 2 into a 1, I have to divide the 2.0261

I am going to divide this by 2 which means I have to divide the top number by 2.0267

This would be 1 mile per hour because it is 1 hour.0274

1 mile per hour would be the unit rate.0282

This alone would just be a rate.0286

But when you make the denominator a 1, a unit of 1, then this is a unit rate.0290

This here is a unit rate because the denominator is 1.0298

Here is an example; \$10 per 20 pieces; that is like the candy example.0306

If it is \$10 for every 20 pieces,0310

in order to give me a unit rate, I want to find out how much it is per piece.0323

One piece, I am turning this denominator into a 1.0328

That means in order to turn this denominator into 1, I have to divide it by this number, divide it by itself.0333

That means I have to divide this top number.0340

Because this is money, I want to change it to a decimal.0346

I know that 10 divided by 20 or 10 divided by 20 is going to give me 0.5.0350

0.5 in money is the same thing as 50 cents.0368

If I add a 0 here, that becomes 50 cents.0373

Not 5 cents, be careful; this is 50 cents.0375

The unit rate would be 50 cents per piece.0380

I can put 1 in front of it if I want.0391

But if I just say per piece, then I am talking about 1 piece.0393

You can leave it like this; this would be your unit rate.0400

When you are converting rates, rate remember it is a ratio of two different rates.0408

You have different units on the top and the bottom; that is a rate.0416

To convert rates means you are going to go from whatever rates they give you,0422

whatever units they give you, and you are changing it to something else, changing it to different rates.0429

You are converting them.0433

For example, if I have miles per hour, let's say I want miles per hour.0435

I am going to put miles on top; I am going to put... 1 mile per hour.0444

This is the rate that I am starting off with.0451

I want to convert it to feet per minute.0455

This is miles; mi is miles; min is for minutes.0467

I am going to convert this number here, this ratio, this rate, to this rate, this ratio.0471

Remember rates are ratios; I am going to convert this to this.0482

That means miles I need to change to feet and hours I am going to change to minutes.0486

Miles and feet, they are both measurements of distance.0493

Mile and feet, they are both measuring the distance of something.0498

Hour and minutes, they are both measuring time.0503

I can convert miles to feet and hours to minutes.0507

In order for you to be able to convert rates, this to this, you have to know the equivalent units.0512

How many feet equals a mile?--1 mile equals 5280 feet.0525

This and this are the same; 1 mile is equal to 5280 feet.0540

Same thing for hours and minutes; I know that 60 minutes equals 1 hour.0545

If it helps, you need to just write this on the side.0556

First thing I do, I am going to start here and I am going to end here.0565

I am going to change all these into these.0571

I am going to start from 1 mile to 1 hour.0574

I am going to multiply it to different units because I can cross cancel things out.0582

If I say that 5280 feet is the same thing as 1 mile, if they equal each other,0594

then I can say 5280 feet over 1 mile is going to equal 1 because this number and this are the same thing.0603

We said they are equal; this over this is equal to 1.0623

Anything over itself is 1.0629

If I said, for example, 5/5, isn't that 1?--because it is the same number over itself.0631

Same thing here; this equals this.0638

If I say 5280 feet over 5280 feet, isn't that equal to 1?0642

This does equal 5280 feet.0649

If I write it like this, you have to understand that this is the same thing as 1.0653

If I multiply this by 1, I am not changing this.0659

I can multiply this by 1 if I want because it doesn't change.0665

Instead of multiplying it by 1, I want to multiply it by this.0670

This is the same thing as 1.0674

I am going to multiply all this to this.0682

I want the miles to go away because the miles is going to have to change to feet.0689

I need to the miles to go away.0696

In order for me to cross cancel the miles, I have to have one on the top and one on the bottom.0697

This miles is going to go down here.0702

On the top, it is going to go 5280 feet.0705

That way this and this will cancel.0718

Again this whole thing is just equal to 1.0724

I can just multiply it to this if I want.0730

It is not going to change my answer because I am just multiplying it by 1.0734

Same thing for hours.0740

I also know that since this 60 minutes is equal to 1 hour,0743

if I put 60 minutes and I divide it by 1 hour,0749

since this whole thing equals this whole thing, this is also equal to 1, isn't it?0755

They equal each other.0760

Whenever the top and the bottom equal each other, that always equals 1.0762

I want to multiply this whole thing to this whole thing because again this is equal to 1.0769

I want the hours to go away.0777

That means if this is already in the bottom, then I need to write this on the top.0779

This is going to go 1 hour over 60 minutes.0783

I just flip this; this went to the top; this went to the bottom.0790

Because again if this is the same thing as this, then isn't this over this the same thing?0795

I am writing it on the top and the bottom, depending on where I have to cancel it.0803

If this is already in the bottom, then I need to cancel this.0808

That is going to go like that.0811

If I look on the top, what units am I left with?--feet.0816

For my answer, if I multiply all this out, then I am going to be left with feet which is what I want.0823

On the bottom, what am I left with?--minutes.0828

That is what I want left on the bottom.0834

I know that all I have to do is now solve this out.0836

I cancelled out everything that I need to cancel out.0840

If I just multiply this out and then multiply that out, solve for it, I will get my answer.0844

Here my top is going to be 1 times 5280 times 1 which is 5280 feet over 60 minutes.0852

1 mile per hour is the same thing as 5280 feet over 60 minutes.0873

This would be my answer; I can simply this if I want.0885

This is a ratio, is a rate.0890

But if I want to change it to a unit rate, I can divide this by... let me use a different color.0893

I can divide this by 60 and then divide this by 60.0901

5280 divided by 60 is going to give me... I am going to cross out these 0s.0910

Remember you can cross out the 0s if you want.0921

It is going to give me... 8, 4, 8; it is going to give me 88.0926

Again to multiply this, you are going to do 6 times 8.0949

Write it out, 48; my remainder is 4.0952

I am going to bring down this 8; then 6 times 8 is 48.0954

My answer is 88 feet per minute; this would be my unit rate.0966

Let's do a few more examples; the first example, write in simplest form.0981

Here these are just ratios; it is comparing this number to this number.0991

They look like fractions.0997

But you can also think of these numbers, the top number and the bottom number as ratios.0998

It is like division.1005

To write this in simplest form, 12/36, I can look for a common factor.1008

The greatest common factor is 12 because 12 goes into 12 here and 12 goes into 36.1019

If you don't see that 12 is the biggest factor,1030

you can just look for any factor because 12 and 36 have a lot of common factors.1032

If you want, you can just divide the 2 first and then just keep making the numbers smaller.1038

You can divide this by 4; divide it by 3.1043

Since I know that 12 is my biggest factor, I am going to divide this by 12 and then divide this by 12.1050

Whenever I am going to simplify, then I need to divide both1057

the top number and the bottom number by the same number, the same factor.1060

This is 1 over... 36 divided by 12 is 3.1066

This is saying that the ratio of 12 to 36 is the same as 1 to 3.1074

They are the same ratio; they are equivalent; they are the same.1081

This next one, I know because this ends in a 0 and this ends in a 5, that they are both divisible by 5.1088

I am going to take 30 divided by 5; 35 divided by 5.1100

30 divided by 5 is 6; 35 divided by 5 is 7.1108

This is simplest form.1118

That means the ratio of 6 to 7 is the same as 30 to 35.1119

Same thing here; let's divide this by...1125

Again if you just see any common factor, you can just keep dividing until you get simplest form.1128

Or if you find the greatest factor, that would be the fastest way.1137

But let's say that we wanted to just divide this by 2 because I just noticed that they are both even.1141

That is not the greatest factor; but let's just do that first.1147

I am going to divide both the top and the bottom by 2.1152

This is going to be 8/12.1155

This is still not simplest form because they are both even still.1161

4. or maybe 2 if you just notice that they are both even numbers.1167

But from these two, the greatest factor is 4; let's just divide them by 4.1174

8 divided by 4 is 2; over 3; that would be simplest form.1180

For this, let's see, this is not an even number so I know that 2 is not going to go into them.1187

If I add these two together, 5 plus 1, that is 6.1195

6 is a multiple of 3.1200

This bottom one, 1 plus 9, 9 is a multiple of 3.1204

So I know that 3 can go into both of these.1209

Divide this one by 3; divide this one by 3; 51 divided 3.1213

If you don't know what that is, you can always just divide it.1222

51 divided by 3; 1; subtract the number; bring this down.1226

That will be 17 over... 18 divided by 3; 3 times what equals 18?1236

That is 6; that would be simplest form.1246

Next example, Tommy has 4 blue marbles, 3 green marbles, and 7 red marbles in a bag.1254

Find the ratio of red to blue marbles; 4 blue, 3 green, 7 red.1264

We want to find the ratio of red to blue.1274

The ratio is going to be red to blue.1278

How many red do we have?--7 to 4 blue.1286

Make sure you have to write it as 7 to 4 and not 4 to 7.1292

Because they ask for red first before the blue, you have to write out the red first.1298

It is 7 to 4; you can say 7 to 4.1304

Or you can say 7 to 4 like that as a fraction.1309

Next, out of 27 students in classroom, 15 are boys.1317

Find the ratio of boys to girls; ratio is boys to girls.1323

They don't give us a number of girls.1333

They just tell us that there is 15 boys.1334

But I know that if there is 27 students total and 15 are boys, then the rest of the students have to be girls.1339

I have to subtract 27 students minus 15; I am going to get 12.1350

That means I know that 15 are boys and 12 are girls.1358

The ratio of boys to girls would be 15 to 12 or 15/12.1364

Find the unit rate.1379

Remember unit rate is when you have a ratio and the bottom number, the denominator, has to be 1.1380

Here the ratio is 250 for every 2 dozen.1389

I want to find how much it is going to be for 1 dozen or how much per dozen.1403

Think of it as per.1411

Every time you see unit rate, you are going to think of per.1413

Per whatever the unit is on the bottom; how much per dozen?1418

That means I need to turn this into a 1.1424

I divide this by 2 then to turn 2 divided by 2 into 1.1428

Then I have to multiply the top by 2.1433

250 divided by 2.... remember to bring out the decimal.1436

It is going to be 1; 2; bring down the 5.1444

2; 4; 1; bring down the 0; 5.1455

Going to be \$1.25 per dozen; there is my unit rate.1460

A car goes 300 miles... mi means miles... on 10 gallons of gas.1479

300 miles on 10 gallons of gas; find the unit rate.1488

That means I want to turn this into 1.1498

It is going to be how many miles per gallon.1501

Then again divide this by 10; divide the top number by 10.1504

300 divided by 10... every time you divide by a number with the 0 at the end of it,1512

and they both have 0s at the end, you can just cross out one of the 0s.1519

If I cross out this 0 and cross out this 0, then I am going to be left with 30.1524

30 miles per gallon; this is the unit rate.1529

A skydiver falls 240 feet in 5 seconds; 240 feet every 5 seconds.1541

How many feet per second?--1 second; divided by 5; divided by 5.1556

240... let's do it over here; 240 divided by 5 is going to be 41564

because that is going to be 20; 4, 0; that will be 8.1571

48 feet per second; here is my unit rate.1581

The fourth example, we are going to convert the units.1595

This is going to be the most difficult part of this lesson.1599

But just make sure you are going to...1603

Just try to cancel out the units so that you end up with the units that you want for your answer.1606

A car is moving at 8 miles per hour.1616

I am going to write that as a fraction; 8 miles per hour.1620

I want to convert this to feet on the top with what units on the bottom?--minutes.1627

I know it is 10 minutes.1638

But then I just want to focus on converting these units first--miles per hour to feet per minute.1639

I am going to start off here; I am going to write that over 1 hour.1657

Again I am going to multiply; I want to turn the miles into feet.1667

On the side, let's find out... 5280 feet equals 1 mile; this equals this.1673

That means if I put this over that as a numerator and denominator, it is going to equal 1.1685

Why don't I just do that right now; I am going to do times...1698

From the feet and the miles, which one do I want to go as my numerator?1701

I want my miles to go on the denominator because I want them to cross cancel out.1707

From these two, I am going to put this on the bottom.1714

I am going to put 1 mile on the bottom and then 5280 feet as my numerator.1717

That way my miles cancel out.1727

Hours I am going to change to minutes; 1 hour equals 60 minutes.1733

Write that out first so you can see it; it is a lot easier.1744

Then from this and here, one is going to go on as my numerator.1747

One is going to go on my denominator.1752

Which one do I want to go on the top?1755

The hours because here the hours is on the bottom.1758

I want it to cancel out so I have to put it on the top.1760

This is going to go on the top; this is going to go on the bottom.1763

1 hour over 60 minutes; cross cancel that out.1767

Here I want to now solve this out because if I look at the top, what units are left on the top?1779

Feet is left which is what I want.1789

This is what I want my answer in; I am on the right track.1791

On the bottom, what do I want left?--minutes.1797

That is where I am at; so I am good.1800

Now I know I just have to multiply this out and solve these numbers out.1803

Before we start multiplying this times this and get a big number and then1809

have to divide by a big number, let's try to cross cancel some stuff out.1813

Anytime you are multiplying numbers and you have numbers on top and you have numbers on the bottom,1820

you can start cross cancelling things out if they have common factors.1825

First thing I see is I see a 0 here and I see a 0 here.1829

I can cross cancel those out.1834

This is going to change to 6; this is going to change to 528.1840

Cross out that 0; cross out that 0.1846

8 and 6, I know that they have a common denominator of 2.1852

I can cross this out, divide this by 2; I get 3; that changes to a 3.1858

I am going to change that because that common factor was a 2 so that changes to a 4.1865

That means I divided this by 2 and I divided this by 2.1871

This became 4; this became 3.1875

Here, does 528, is it divisible by 3?1879

If you add this, it becomes 5 plus 2 is 7; plus 8 is 15.1886

Is 15 a multiple of 3?--it is.1892

Therefore I know that this is divisible by 3.1896

If you are wondering what I just did, I used the divisibility rule.1899

The divisibility rule of 3 is you add up all the digits.1905

You are going to do 5 plus 2 plus 8 which gives you 15.1911

You are going to see if that number is a multiple of 3.1918

Does 3 go into that number?1922

5 plus 2 plus 8 is 15; 3 does go into 15.1925

I know that 3 will go into this number; 528 divided by 3.1929

This is 1; this becomes 3; subtract it; you get 2; bring this 2 down.1940

3 goes into 22 seven times; that is 21; 1; bring down the 8.1948

3 times 6 is 18; this goes away; this became 176.1957

Now all I have to do to find my answer is just...1979

Since the bottom number is 1 times 1 and these all canceled out, then it is just 1 times 1.1983

That is just 1 minute; I just multiplied all the numbers; I get this left.1991

On my top, my numerator, it is just 4 times 176.1997

That is 24; 7 times 4 is 28; add 2; this becomes feet.2005

This is in a unit rate; 704 feet per minute.2037

I want to know how many feet it will move in 10 minutes.2045

What does that mean?--this is my unit rate; I am converting the units.2051

This would be the correct answer.2059

But then it is asking me how many feet it will move in 10 minutes, not per minute.2061

If they asked how many feet it will move per minute or in one minute,2069

this would be my answer, 704 feet per minute, for one minute.2074

But since they are asking for 10 minutes, I need to change my denominator to a 10.2079

They are not asking for a unit rate.2086

They want to know how many feet for 10 minutes.2088

I need to change this 1 to a 10.2093

In order to do that, I have to multiply by 10.2094

Same thing here; I need to multiply by 10.2098

If I need to multiply this by 10, I just have to add a 0 at the end of it.2102

That is pretty easy.2106

It just becomes 7 thousand, 0, 4, add the 0, feet per 10 minutes.2106

My answer, how many feet?--it is 7040 feet.2118

I know that problem seemed a little bit complicated.2128

But all I had to do was convert the units at miles per hour to feet per minute which is what I did.2131

If you want, you didn't have to cross cancel all this stuff out.2143

That is why it looks so complicated, because we ended up cross cancelling numbers out.2146

But if you want, forget about the cross cancelling.2150

Just multiply all the numbers straight across; get this number.2153

Multiply all the bottom numbers straight across and get this number.2158

Then simplify if you want; you can do it that way.2161

Once we get this, this is per minute; denominator is 1.2165

This is our unit rate.2170

But then because they are asking for 10 minutes,2173

I need to change this denominator to 10 by multiplying by 10.2174

We multiply the top by 10; you get 7040 feet per 10 minutes.2179

Let's try one more problem; the sprinklers used 2 gallons per minute.2187

How many quarts will it use in 30 seconds?--again we have to convert units.2194

This is 2 gallons per minutes.2202

I want to convert this to quarts... this is quarts... per seconds.2209

I am going to put just 30 seconds here.2219

Question mark, how many quarters per 30 seconds?2223

I am going to start off with this again; 2 gallons per minute.2226

Since I need to convert gallons to quarts, I know that 1 gallon is equal to 4 quarts.2237

Remember if this is equal to this, I can change this to a fraction, 4 quarts over 1 gallon.2254

That is going to equal 1.2262

1 gallon over 4 quarts, that is also going to be the same.2263

I can multiply this by... what do I want to get rid of?2269

I want to get rid of the gallons first by using this.2275

The gallons is going to go on the bottom.2280

This one is going to go on the bottom; that way this will cancel like this.2282

This one will go on the top like that.2287

See how one goes on the bottom and one goes on the top?2293

Or one goes on the top and one goes on the bottom?2295

Just depends on what you have to cancel.2299

Then I need to convert minutes to seconds.2303

The minutes to seconds is going to be 1 minute is equal to 60 seconds.2310

I need to write the minute one on the top so that it will cancel.2322

This is going to go on the top; this is top, bottom.2326

1 minute over 60 seconds; minutes will cancel.2330

What units do I have left on the top?2341

I have quarts which is what I want.2343

And I have seconds on the bottom which is what I want.2346

Now I just have to solve it out.2350

If I want, I can cross cancel out this 2 and this 60.2354

2 goes into 2; this changes to a 1.2359

2 goes into 60; cut it in half; that is 30.2362

You can cross cancel out again.2369

But then otherwise you can just write it out; 4 quarts over 30 seconds.2371

The reason why I decided to leave it...2384

You could have cross cancelled it out; that is fine.2386

Here they ask for 30 seconds.2390

They want to know how many quarts will it use in 30 seconds.2393

Here it will be 4 quarts every 30 seconds.2399

I know that my answer will be 4; 4 quarts.2404

It is 4 quarts per 30 seconds; that is my answer.2410

That is it for this lesson; thank you for watching Educator.com.2417