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Solving Percent Problems

Related Links

  • To solve percent problems, translate the sentence into an equation
    • “of” means times
    • “what” means unknown variable

Solving Percent Problems

25 is 50% of what number?
  • 50% = .50
  • 25 = .50x
  • [25/.50] = x
11 is 20% of what number?
  • 20% = .20
  • 11 = .20x
  • [11/.20] = x
18 is 40% of what number?
  • 40% = .40
  • 18 = .40x
  • [18/.40] = x
What percent of 70 is 35?
  • x ·70 = 35
  • x = [35/70]
  • x = 0.5
What percent of 16 is 4?
  • x ·16 = 4
  • x = [4/16]
  • x = 0.25
15 is what percent of 1500?
  • 15 = x ·1500
  • [15/1500] = x
  • 0.01 = x
30 percent of 60 is what number?
  • 30% = .30
  • .30(60) = x
5% of what number is 15?
  • .5x = 15
  • x = [15/.5]
x = 30
100% of 154 is what number?
  • 100% = 1
  • 1 ×154 = x
16 is 25% of what number?
  • .25x = 16
  • x = [16/.25]

*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.


Solving Percent Problems

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
  • Solving Percent Problems 0:06
    • Translate the Sentence into an Equation
  • Extra Example 1: Solving Percent Problems 0:56
  • Extra Example 2: Solving Percent Problems 14:49
  • Extra Example 3: Solving Percent Problems 23:44

Transcription: Solving Percent Problems

Welcome back to

For the next lesson, we are going to go over solving percent problems.0002

To solve a problem that involves percents, we want to first translate whatever the sentence is into an equation.0008

Whenever you have a number, you are going to write that down in your equation.0018

If you have a percent, you need to change it to a decimal.0022

You see the word of; that means times; you are going to be multiplying.0027

When you see the word what or what number, that means you are going to have a variable.0032

That is what you are going to be looking for.0037

That is what you are going to be solving for.0039

This is almost the same as what we just did the last lesson.0041

But now we are going to be looking at variables and solving equations.0047

We are going to have to do a little bit deeper into these problems.0053

The first set of examples; for example one, 15.0060

Remember if we have a number, we are just going to write it straight down into our equation.0068

Is; the word is remember means equals; 25 percent; this is write down.0074

This we are also going to write straight down.0083

But because it is a percent, in our equation, we need to make it into a decimal.0085

Percent to decimal, remember we have to move two decimal spaces to the left0091

because think of percent as a bigger version of a decimal; decimals are small numbers.0105

Whenever you are converting from a percent to a decimal, you have to get smaller.0112

The way for you to get smaller is to move the decimal point over to the left.0118

The decimal point is over on this side.0127

If you don't see a decimal point, it is always at the end of the number.0130

We are going to move it one, two spaces.0134

The decimal point of 25 percent is going to be 0.25.0139

Of course obviously we have to drop the percent sign.0145

25 percent is 0.25; that is what I am going to write in my equation.0148

0.25; of means times; times what number; this is my variable X.0154

Be careful here because whenever you write a dot for times, that looks like the decimal point.0168

The best way to represent multiplying is to either write it in parentheses.0174

If you are going to show that you are going to be multiplying two numbers, write them in parentheses.0179

Or if it is a number with a variable, a letter, then you can just put them together.0184

0.25X, that would mean 0.25 times X.0191

Here is our equation now; this is what I am solving for.0198

15 is 25 percent of what number?--this is the number that I am looking for.0202

When I solve for X, remember that since this is 0.25 times X, I need to do the inverse operation.0207

0.25 times X, the inverse operation is divide.0219

In order to solve for X, I have to divide 0.25.0224

0.25 over itself is going to be 1.0232

Whatever I do to one side, I have to do to the other side of the equal sign.0236

I have to divide this side also by 0.25; this, it looks like a fraction.0241

But fractions are division problems; this is the same thing as 15 divided by 0.25.0247

Another way to explain why we have to divide, let's say I have 8 equals 4X.0255

If I have 4 times a number equals 8, what do I know about X?0268

Isn't 4 times 2, 8?--so I know X has to be 2.0275

In the same way, I have to solve for X and I can just divide this number by this number.0285

Divide this by 4; divide this by 4; 8 divided by 4 is 2.0292

That gives me 2 equals X.0298

Here 15 divided by 0.25, I need to actually solve that out in order to find X.0305

Let's review over how to divide numbers with decimals.0313

If I am going to divide these two numbers, remember that the top number is what goes inside when you divide.0321

15 on the inside; 0.25 on the outside.0330

The decimal point for this number is at the end because we don't see one.0337

It is always at the end.0342

If I need to add 0s to this number, I can because0349

I can always add 0s to the end of a number behind a decimal.0352

If it was before the decimal, I can't because then that will just become 150 instead of 15.0358

As long as it is behind the decimal and it is at the end of a number,0364

you can add as many 0s as you want.0370

I can add two 0s; I can add three; I can add ten; however many I need.0372

This number, we want to change to a whole number.0379

I need to move the decimal point over two spaces to the right to make the decimal point at the end.0384

If I move two decimal places for this number, then I have to move this two decimal places over here.0392

Then I am going to take that decimal point up.0399

25 goes into 15 zero times; 25 goes into 150 how many times?0403

Think about quarters; 25 cents or 25 is like a quarter.0412

How many of those fit into 100 or a dollar?--four.0421

Four quarters is a dollar; think of 150 cents.0425

25 cents goes into 150 cents or 1 dollar 50, how many of them?0428

That would be six.0434

If you want to just check that, this is 12, 13, 14, 15.0437

That becomes 0 when you subtract it; I can bring down this 0.0449

25 goes into 0 zero times; I have to fill in this space right here.0454

That is 0; subtract it; that is nothing.0461

I don't have to bring down another 0 because my remainder is 0.0464

My answer then, if I do 15 divided by 0.25, is this number up here, 60.0471

This number, if there is a 0 in front of the number like that, then that doesn't mean anything either.0479

I can just drop the 0; that would just be 60.0485

This 0 I cannot drop.0489

This 0 has to stay there because if I drop it, my number is going to change to 6.0491

We know that 6 is not the same as 60.0498

This 0, it is not after the decimal point so we can't drop that 0.0501

My answer just becomes 60.0507

If you need to review over this, you can either go back to that lesson, dividing decimals.0513

Or we are going to do a few more problems that involve dividing decimals.0520

The next one, again we are going to change this into an equation.0527

1; is is equals; 4 percent.0535

4 percent, again we have to change it to a decimal.0540

Be careful; 4 percent is not 0.4.0543

Again 4 percent to decimal; the decimal point is at the end right here.0549

I go one, two; then put the decimal point there.0556

I have an empty space that I have to fill.0561

I have to fill that with a 0; it is going to be 0.04.0563

Again at the end here, one, two, decimal point; 0.04.0568

Of means times; blank, that is what we are looking for; that is my variable.0580

I am going to put X there.0589

Remember if I put number with variable, that means times.0591

This represents... I don't have to put a dot here.0595

I don't have to use parentheses when it is number with variable.0597

Again how do I solve for X?--look at this example again.0603

If we are going to do 8 equals 4 times a number, I can take this number, divide it by this number.0607

8/4; that is going to give me X.0614

Then I have to do this number divided by this number.0617

Remember that this over this becomes 1.0626

This whole side, my right side, just becomes 1X.0630

1X is the same thing as X.0635

That is probably a little bit hard to understand, 1X being the same as X.0639

But it is like me saying I have an apple.0645

If I say I have an apple, you know I only have 1 apple.0650

Even though I didn't say I have 1 apple, you just know because how many A's do you see?0655

You see one; an A is the same thing as 1A.0661

An apple is the same thing as 1 apple.0667

Just think of that as having 1X; again we have to divide that; 1.0672

Be careful, the top number is going to go inside.0683

0.04, the bottom number, is going to go on the outside.0689

Again I have to move this decimal point over one, two spaces to the right.0693

That means I have to take this decimal point; it is at the end.0697

Go one, two spaces; I have to fill these in with 0s.0702

There is my new spot for my decimal point; I bring it up.0709

4 goes into 10 how many times?--4 times 2 is 8.0717

4 times 3 is 12; 12 is too big; it only fits into 10 twice.0724

2 times 4 is 8; subtract it; I get 2.0732

I am going to bring down this 0.0738

4 goes into 20 how many times?--five times.0741

4 times 5 is 20; subtract it; I get a remainder of 0.0746

I can stop there; my answer becomes 25.0753

I don't have to put that decimal because it is a whole number and it is at the end.0758

My answer X is 25; right here, this is 25.0764

Again 1X is the same thing as X; what did this become?0773

This became 25; if 25 is X, then I can just say that X is 25.0778

It is the same exact thing.0786

The next one, 20 equals 100 percent; to decimal.0792

Again start at the end; you are going to go one, two; right there.0806

It is going to be after the 1; 1.0.0810

Remember if the 0s are at the end of a number behind the decimal point, then I can just drop it.0814

Isn't this the same thing as 1?--I can just write 1.0819

100 percent as a decimal is 1; times; of means times.0824

What number, that is my variable; 1 times X; 20 equals 1X.0832

Remember 1X is the same thing as X because if I have 1 apple,0841

that is the same thing as just saying I have an apple.0846

If you want, you can go ahead and divide the 1 just like we did the other problems.0850

20 divided by 1 is 20.0856

Whenever you have a number divided by 1, it is always itself.0860

20 equals X; or I can flip this around.0864

If 20 equals X, then isn't that the same thing as X being 20?0870

Either way, that is correct; we just want to know what the number is.0877

The number is 20; or you can say 20 is the number.0881

It is the same exact thing.0886

Let's do a few more examples; these are a little bit different.0889

What percent of 50 is 10?0895

Now the variable, what we are solving for, is a percent.0900

Be careful here; what percent, I am going to make that X.0907

Times, times; 50, 50; is, equals; 10, 10; X times 50 equals 10.0913

Remember you can change this if you want to 50X just like we did the other problems0929

because a number times a variable, you just put it together with the number in front.0935

50X equals 10; it is the same thing.0939

How do we solve for X then?--how do we get what X is?0944

Remember my example?--let's say I have 3 times X equals 6.0949

You can do this in your head and know that 3 times 2 equals 6.0959

Another way for you to solve it is to do 6 divided by this number; this divided by this.0964

Same thing; we can just do 10 divided by 50.0972

It is not 50 divided by 10.0975

It is this number divided by this number, the one that is multiplied to the variable.0978

I can show you this way; 50/50, that is 1.0985

10/50, that is what you have to do; 10 divided by 50.0993

Again fractions are the same thing as division; 10 divided by 50.0997

A shortcut way of doing this is if you are dividing two numbers1009

with 0s at the end of it, you can just cross out the 0.1014

If there is one 0 up here and one 0 down here, you can just cross out1020

one 0 from each of the numbers as long as there is 0s in both numbers.1022

But we are just going to go ahead and just divide it this way.1027

50 divided by 10; it is not going to go into this number.1032

This number is too big to go into this number.1037

I am going to have to use my decimal point.1040

Do I move it at all?--no, because there is no decimal point here.1042

I can just bring it up, bring it straight up.1047

I can add 0s at the end behind the decimal at the end of a number.1050

Now I can just look at this, 1-0-0, 100.1057

50 goes into 100... 50 plus 50 is 100; or 50 times 2 is 100.1061

Think of 50 cents; 50 cents goes into 100 cents how many times?1072

100 cents is the same thing as a dollar.1080

50 cents goes into a dollar twice; this becomes that.1082

Subtract it; you get 0; no remainder; that is my answer.1087

I don't have to bring down anymore 0s because my remainder became 0.1092

When I divided this, my answer became 0.2; X equals 0.2.1099

Here is the thing though; they are asking for percent.1114

Even though this is my answer, this is my answer as a decimal.1119

They want it in percent; they are asking what percent.1124

They are not asking what decimal; what percent?1127

I have to change this number to a percent; decimal to percent.1129

Remember decimal is a small number; percents are larger.1141

I have to go from a small to a larger.1146

That means I have to move the decimal point over two spaces; but which way?1148

If I go to the left, I am going to get smaller.1154

But if I go to the right, then I start getting whole numbers.1158

I make the number bigger.1162

0.2, to make it into a percent, I need to make it bigger.1165

I need to go to the right; one, two.1169

I have to fill this space with something.1173

0.2 as a percent will be 2-0 and then percent.1177

The decimal point is right here; it is at the end.1185

If it is at the end, remember you can just... it doesn't have to be there.1187

You can make it invisible.1190

Then we have to write the percent sign because we changed it to a percent.1193

My answer is then 20 percent; that is my answer.1196

20 percent of 50 is 10.1204

Another one; again what percent, make that X, your variable.1208

Times 75; is is equals; 7.5.1214

Again we have to do this number divided by this number; 7.5 divided by 75.1224

Again if you want to see it, I can show you this way.1235

because this you have to get rid of by dividing it.1239

This 75/75 is 1; X times 1 is just X.1244

X equals; then I have to actually divide that to find the answer.1251

Here I don't have to move the decimal point anywhere because it is at the end.1258

This decimal point will just come straight up.1263

75 goes into 75 how many times?--once.1267

1 times 75 is 75; subtract it; I get 0.1274

I don't have to go any further; 0.1 is my answer; X equals 0.1.1280

But again remember it is asking for percent.1285

Be careful that you don't forget to change it to a percent.1288

I am going to put that here to represent decimal.1294

To change it percent, I am going to go one, two, point.1301

1; fill this space with a 0; put the percent sign.1306

0.1 in decimal becomes 10 percent; X equals 10 percent; there is my answer.1314

The last one for this; again what percent X times... of is times... 4 equals 4?1325

This one we can just do in our head; what times 4 equals 4?1340

Isn't this 1?--1 times 4 equals 4; 4 times 1, it equals itself.1345

I don't even have to solve this; I can just make X equal to 1.1352

If the problem is fairly easy, you can just do it in your head, then go ahead.1356

There is no need for you to do all the work unless your teacher wants you to show the work.1360

Then X equals 1; since X equals 1... I didn't mean to box this.1367

That is not my final answer so I don't want to box it.1381

Since X equals 1, I need to change it to percent.1385

How do you change a 1, a whole number, into a percent?1391

Again where is the decimal point?--I don't see it.1395

If I don't see it, it is invisible, it is at the end like that.1399

Go one, two, point here; I have two spaces to fill.1404

This becomes 1-0-0 percent; this X equals 100 percent.1411

The next example, we are just going to do a few more, just different types though.1428

The other examples, they were the same kind, all the problems on that page.1433

15 equals what percent, X, times 150; let me rewrite this equation out.1439

Since this is 150 times X, let me just write it here.1452

15 equals... remember whenever I do a variable times a number,1458

I want to write it together but with the number in front; 150X, like that.1463

Then to solve for X, remember I have to do this number divided by this number.1471

I am going to divide this side by 150.1478

Whatever I do to one side, I have to do to the other side.1480

That way this becomes 1X or 1 times X; 1X.1486

That is the same thing as X.1492

15 divided by 150; no decimal point here; I don't have to move it.1495

Instead I need to draw that in; bring it up; add 0s.1509

150, we know it doesn't go into 15.1518

If you want, you can put a 0 up here; if not, then that is fine.1523

Just remember that it is now the next three or just these three.1525

150 goes into 150 one time; that becomes 150.1530

If you subtract it, it becomes 0; I drew an extra 0 there.1536

But you don't even have to bring it down because it is just going to be 0s.1542

Remember 0s at the end of a number behind the decimal point means nothing.1545

0.1 is my answer; X is going to equal 0.1.1551

You can also say 0.1 is going to equal 1X or X; same thing.1558

But I can switch it like this; it is asking for percent.1564

I need to take this decimal point and go one, two; fill in this space.1572

It is going to be 1-0 percent; 0.1 is the same thing as 10 percent.1579

The next one, 30 percent, this is written out as percent.1592

But it still means the same thing as percent like that.1599

30 percent, I have to change that to a decimal.1602

30 percent becomes... at the end, you go one, two, right there.1606

0.30 or 0.3 because remember it is 0s at the end behind the decimal.1613

It doesn't mean anything.1619

0.3 or 0.30; of, times; 12; it is number times number.1621

I can't write it next to each other like how I do numbers with letters.1629

I have to put it in parentheses.1633

Is is equals; what number, this is my variable.1637

All I have to do to figure out what X is is multiply those two numbers.1645

0.30, you know what?--we know that the 0 means nothing.1654

Let's just make it easier and just not even have that 0.1660

One digit number is easier to multiply.1665

I am going to put the 12 on the top; 0.3 on the bottom.1669

Multiply it; 2 times 3 is 6; 3 times 1 is 3.1673

From here, I only have one number behind the decimal point.1680

Start at the end; go place the decimal in there.1685

What I just did, when you multiply decimals, you have to count...1692

This decimal point is at the end, right here.1695

You count to see how many numbers are behind the decimal point.1698

Here I only have one; then I start at the end.1702

I only go one inwards; that is where the decimal place goes.1705

X is 3.6; to finish this equation, 3.6 equals X.1711

3.6 is the number; or I can say the number is 3.6.1723

The next one, 5 percent, change that to a decimal.1734

Start here; go one, two; it is point... fill in that space.1741

It is 0.05; it is not 0.5; 0.05.1745

0.05 times my unknown which is X; X equals 4.1752

Again I have to solve for X which means I have to divide.1764

4 divided by 0.05; 4 divided by 0.05.1768

I have a decimal point here; I have to move it one, two.1779

There is my decimal point; I am going to move it one, two.1784

Bring it up; fill in these spaces with 0s; 5 goes into 4 zero times.1788

5 goes into 40... 5 times we know 8 is 40.1797

Write 40 down here; subtract it; 0.1804

I am not finished with the number yet because I still have space up here.1808

Bring down the 0; 5 goes into 0 zero times.1814

That is why for this, I have to keep going.1818

Even though this became a 0, I have to bring down the 0 and solve it again1820

because there was an empty space before my decimal point.1826

In that case, you have to continue.1831

If it is after the 0 like in this problem right here... I'm sorry.1833

If it is after the decimal point, then I can stop once I get 0 as my remainder.1838

But for this, if there is a space here before the decimal point,1844

then I have to go again until I fill in those spaces.1849

Here this is 80; X is 80; they are not asking for percent.1852

5 percent of 80 is 4.1861

The last one, 100 percent of 3448 is what number?1866

They are asking for 100 percent of this number.1874

100 percent is all of it, is the whole thing.1877

100 percent of this number is just this number.1883

If you want to solve it out how we solved out the rest of them,1888

100 percent as a decimal, again move the decimal point over one, two spaces.1891

That becomes 1 or 1.0 which is the same thing as 1.1898

Times; times 3448; I am going to change this to parentheses.1905

3448 equals what number?--X; 1 times this number is just that number.1912

I can say 3448 is the number or the number is 3448.1926

That is it for this lesson; thank you for watching