Sam can mow a lawn in 6 hours. Leo can mow it in 30 hours. How many… Easy

The question

Question: Sam can mow a lawn in 6 hours.

Leo can mow it in 30 hours.

How many hours will it take them to mow it together?

1 Find what we know

What do we know?

Sam does the whole job alone in 6 hours.

Leo does it alone in 30 hours.

What do we want?

How long it takes them together.

2 Name the unknown

We don't know how long it takes together.

So we call it x hours.

Careful!

Many people add the times: 36 hours.

But two working together are faster than one.

So together it takes less than 6 hours.

The whole lawn is 1 whole job.

We split it into parts.

Let's put it in a table: the time alone, and the part done.

Sam does the whole job in 6 hours.

So in 1 hour, Sam does 1 part out of 6.

That is 1/6.

Leo does 1 part out of 30 in 1 hour: 1/30.

In x hours, Sam does x times as much: x/6.

And Leo does x/30.

3 Write the equation

In words: Sam's part plus Leo's part makes the whole job.

So the equation is x/6 + x/30 = 1.

Now we solve it.

4 Remove the fractions

The bottom numbers are 6 and 30.

30 can be divided by all of them.

We multiply every part by 30.

We do it on both sides, so they stay equal.

30 divided by 6 is 5.

30 divided by 30 is 1.

Write what is left.

Do the multiplications.

No more fractions!

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

5x + x = 6x.

Now we have 6x = 30.

6 Remove the 6

6x means 6 times x.

We want just x.

To remove times 6, we divide by 6.

We do it on both sides, so they stay equal.

6 divided by 6 is 1.

So just x is left.

30 divided by 6 is 5.

So x = 5!

7 Answer the question

So together it takes 5 hours.

8 Check

Let's check in the story.

In 5 hours, Sam does 5/6.

That is 5/6 of the lawn.

In 5 hours, Leo does 5/30.

That is 1/6 of the lawn.

Together: 5/6 plus 1/6 is 6/6.

That is 1 whole lawn, like the story says.

Does it make sense?

Together is faster than Sam alone, who needs 6 hours.

So the answer is right!

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

▶ Watch it solved

Leo can build a wall in 12 hours. Max can build it in 6 hours. How… Medium

The question

Question: Leo can build a wall in 12 hours.

Max can build it in 6 hours.

How many hours will it take them to build it together?

1 Find what we know

What do we know?

Leo does the whole job alone in 12 hours.

Max does it alone in 6 hours.

What do we want?

How long it takes them together.

2 Name the unknown

We don't know how long it takes together.

So we call it x hours.

Careful!

Many people add the times: 18 hours.

But two working together are faster than one.

So together it takes less than 6 hours.

The whole wall is 1 whole job.

We split it into parts.

Let's put it in a table: the time alone, and the part done.

Leo does the whole job in 12 hours.

So in 1 hour, Leo does 1 part out of 12.

That is 1/12.

Max does 1 part out of 6 in 1 hour: 1/6.

In x hours, Leo does x times as much: x/12.

And Max does x/6.

3 Write the equation

In words: Leo's part plus Max's part makes the whole job.

So the equation is x/12 + x/6 = 1.

Now we solve it.

4 Remove the fractions

The bottom numbers are 12 and 6.

12 can be divided by all of them.

We multiply every part by 12.

We do it on both sides, so they stay equal.

12 divided by 12 is 1.

12 divided by 6 is 2.

Write what is left.

Do the multiplications.

No more fractions!

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

x + 2x = 3x.

Now we have 3x = 12.

6 Remove the 3

3x means 3 times x.

We want just x.

To remove times 3, we divide by 3.

We do it on both sides, so they stay equal.

3 divided by 3 is 1.

So just x is left.

12 divided by 3 is 4.

So x = 4!

7 Answer the question

So together it takes 4 hours.

8 Check

Let's check in the story.

In 4 hours, Leo does 4/12.

That is 1/3 of the wall.

In 4 hours, Max does 4/6.

That is 2/3 of the wall.

Together: 1/3 plus 2/3 is 3/3.

That is 1 whole wall, like the story says.

Does it make sense?

Together is faster than Max alone, who needs 6 hours.

So the answer is right!

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

▶ Watch it solved

Ana can paint a fence in 4 hours. Ben can paint it in 6 hours. How… Harder

The question

Question: Ana can paint a fence in 4 hours.

Ben can paint it in 6 hours.

How many hours will it take them to paint it together?

1 Find what we know

What do we know?

Ana does the whole job alone in 4 hours.

Ben does it alone in 6 hours.

What do we want?

How long it takes them together.

2 Name the unknown

We don't know how long it takes together.

So we call it x hours.

Careful!

Many people add the times: 10 hours.

But two working together are faster than one.

So together it takes less than 4 hours.

The whole fence is 1 whole job.

We split it into parts.

Let's put it in a table: the time alone, and the part done.

Ana does the whole job in 4 hours.

So in 1 hour, Ana does 1 part out of 4.

That is 1/4.

Ben does 1 part out of 6 in 1 hour: 1/6.

In x hours, Ana does x times as much: x/4.

And Ben does x/6.

3 Write the equation

In words: Ana's part plus Ben's part makes the whole job.

So the equation is x/4 + x/6 = 1.

Now we solve it.

4 Remove the fractions

The bottom numbers are 4 and 6.

12 can be divided by all of them.

We multiply every part by 12.

We do it on both sides, so they stay equal.

12 divided by 4 is 3.

12 divided by 6 is 2.

Write what is left.

Do the multiplications.

No more fractions!

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

3x + 2x = 5x.

Now we have 5x = 12.

6 Remove the 5

5x means 5 times x.

We want just x.

To remove times 5, we divide by 5.

We do it on both sides, so they stay equal.

5 divided by 5 is 1.

So just x is left.

So x = 12/5!

7 Answer the question

12/5 is 2.4.

So together it takes 2.4 hours.

8 Check

Let's check in the story.

In 2.4 hours, Ana does 2.4 divided by 4: 0.6 of the fence.

In 2.4 hours, Ben does 2.4 divided by 6: 0.4 of the fence.

Together: 1 whole fence, like the story says.

Does it make sense?

Together is faster than Ana alone, who needs 4 hours.

So the answer is right!

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

▶ Watch it solved