Step by step
The question
Question: Paul can tile a floor in 7 days.
Working together, Paul and Rosa can tile it in 6 days.
How long would it take Rosa alone?
1 Find what we know
What do we know?
Paul does the whole job alone in 7 days.
Together they take 6 days.
What do we want?
How long Rosa takes alone.
2 Name the unknown
We don't know how long Rosa takes alone.
So we call it x days.
Careful!
Many people take away the times: 1 day.
But Rosa alone is slower than the two together.
So it takes more than 6 days.
The whole floor is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Paul does the whole job in 7 days.
So in 1 day, Paul does 1 part out of 7.
That is 1/7.
Rosa takes x days alone.
So in 1 day, Rosa does 1 over x.
They work 6 days together.
In 6 days, Paul does 6 times as much: 6/7.
And Rosa does 6 over x.
3 Write the equation
In words: Paul's part plus Rosa's part makes the whole job.
So the equation is 6/7 + 6/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 7 and x.
7x can be divided by all of them.
We multiply every part by 7x.
We do it on both sides, so they stay equal.
7x divided by 7 is x.
7x divided by x is 7.
7x times 1 is 7x.
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 6x from the left side.
To remove 6x, we take away 6x.
We do it on both sides, so they stay equal.
6x and − 6x make zero.
Cross them out.
7x − 6x = x.
Now we have 42 = x.
Let's swap the two sides, so x is on the left.
So x = 42!
6 Answer the question
So Rosa alone takes 42 days.
7 Check
Let's check in the story.
In 6 days, Paul does 6/7, and Rosa does 6/42.
36/42 plus 6/42 is 42/42.
That is 1 whole floor, like the story says.
Does it make sense?
Rosa 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.