Step by step
The question
Question: Nadia can paint a room in 2 hours.
Sara can paint it in 2 hours too.
How many hours will it take them to paint it together?
1 Find what we know
What do we know?
Nadia does the whole job alone in 2 hours.
Sara does it alone in 2 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: 4 hours.
But two working together are faster than one.
So together it takes less than 2 hours.
The whole room is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Nadia does the whole job in 2 hours.
So in 1 hour, Nadia does 1 part out of 2.
That is 1/2.
Sara does 1 part out of 2 in 1 hour: 1/2.
In x hours, Nadia does x times as much: x/2.
And Sara does x/2.
3 Write the equation
In words: Nadia's part plus Sara's part makes the whole job.
So the equation is x/2 + x/2 = 1.
Now we solve it.
4 Remove the fractions
The bottom number is 2.
Times 2 will remove it.
We multiply every part by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
2 divided by 2 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.
x + x = 2x.
Now we have 2x = 2.
6 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
2 divided by 2 is 1.
So x = 1!
7 Answer the question
So together it takes 1 hour.
8 Check
Let's check in the story.
In 1 hour, Nadia does 1/2.
That is 1/2 of the room.
In 1 hour, Sara does 1/2.
That is 1/2 of the room.
Together: 1/2 plus 1/2 is 2/2.
That is 1 whole room, like the story says.
Does it make sense?
Together is faster than Nadia 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.