Step by step
The question
Question: Mia can clean a classroom in 2 hours.
Zoe can clean it in 3 hours.
How many hours will it take them to clean it together?
1 Find what we know
What do we know?
Mia does the whole job alone in 2 hours.
Zoe does it alone in 3 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: 5 hours.
But two working together are faster than one.
So together it takes less than 2 hours.
The whole classroom is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Mia does the whole job in 2 hours.
So in 1 hour, Mia does 1 part out of 2.
That is 1/2.
Zoe does 1 part out of 3 in 1 hour: 1/3.
In x hours, Mia does x times as much: x/2.
And Zoe does x/3.
3 Write the equation
In words: Mia's part plus Zoe's part makes the whole job.
So the equation is x/2 + x/3 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 2 and 3.
6 can be divided by all of them.
We multiply every part by 6.
We do it on both sides, so they stay equal.
6 divided by 2 is 3.
6 divided by 3 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 = 6.
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 = 6/5!
7 Answer the question
6/5 is 1.2.
So together it takes 1.2 hours.
8 Check
Let's check in the story.
In 1.2 hours, Mia does 1.2 divided by 2: 0.6 of the classroom.
In 1.2 hours, Zoe does 1.2 divided by 3: 0.4 of the classroom.
Together: 1 whole classroom, like the story says.
Does it make sense?
Together is faster than Mia alone, who needs 2 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.