Step by step
The question
Question: Tom can clean a house in 3 hours.
Working together, Tom and Sara can clean it in 2 hours.
How long would it take Sara alone?
1 Find what we know
What do we know?
Tom does the whole job alone in 3 hours.
Together they take 2 hours.
What do we want?
How long Sara takes alone.
2 Name the unknown
We don't know how long Sara takes alone.
So we call it x hours.
Careful!
Many people take away the times: 1 hour.
But Sara alone is slower than the two together.
So it takes more than 2 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.
Tom does the whole job in 3 hours.
So in 1 hour, Tom does 1 part out of 3.
That is 1/3.
Sara takes x hours alone.
So in 1 hour, Sara does 1 over x.
They work 2 hours together.
In 2 hours, Tom does 2 times as much: 2/3.
And Sara does 2 over x.
3 Write the equation
In words: Tom's part plus Sara's part makes the whole job.
So the equation is 2/3 + 2/x = 1.
Now we solve it.
First: a bottom can never be 0.
x at the bottom can't be 0.
So x ≠ 0.
4 Remove the fractions
The bottoms are 3 and x.
3x can be divided by all of them.
We multiply every part by 3x.
We do it on both sides, so they stay equal.
3x divided by 3 is x.
3x divided by x is 3.
3x times 1 is 3x.
Write what is left.
No more fractions!
5 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 2x from the left side.
To remove 2x, we take away 2x.
We do it on both sides, so they stay equal.
2x and − 2x make zero.
Cross them out.
3x − 2x = x.
Now we have 6 = x.
Let's swap the two sides, so x is on the left.
So x = 6!
6 Answer the question
So Sara alone takes 6 hours.
7 Check
Let's check in the story.
In 2 hours, Tom does 2/3, and Sara does 2/6.
4/6 plus 2/6 is 6/6.
That is 1 whole house, like the story says.
Does it make sense?
Sara alone is slower than the two together.
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.