Step by step
The question
Question: Mia can clean a house in 4 hours.
Zoe can clean it in 12 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 4 hours.
Zoe does it alone in 12 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: 16 hours.
But two working together are faster than one.
So together it takes less than 4 hours.
The whole house 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 4 hours.
So in 1 hour, Mia does 1 part out of 4.
That is 1/4.
Zoe does 1 part out of 12 in 1 hour: 1/12.
In x hours, Mia does x times as much: x/4.
And Zoe does x/12.
3 Write the equation
In words: Mia's part plus Zoe's part makes the whole job.
So the equation is x/4 + x/12 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 4 and 12.
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 12 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.
3x + x = 4x.
Now we have 4x = 12.
6 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
12 divided by 4 is 3.
So x = 3!
7 Answer the question
So together it takes 3 hours.
8 Check
Let's check in the story.
In 3 hours, Mia does 3/4.
That is 3/4 of the house.
In 3 hours, Zoe does 3/12.
That is 1/4 of the house.
Together: 3/4 plus 1/4 is 4/4.
That is 1 whole house, like the story says.
Does it make sense?
Together is faster than Mia 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.