Step by step
The question
Question: Ben can paint a room in 15 hours.
Working together, Ben and Zoe can paint it in 10 hours.
How many hours would it take Zoe alone?
1 Find what we know
What do we know?
Ben does the whole job alone in 15 hours.
Together they take 10 hours.
What do we want?
How long Zoe takes alone.
2 Name the unknown
We don't know how long Zoe takes alone.
So we call it x hours.
Careful!
Many people take away the times: 5 hours.
But Zoe alone is slower than the two together.
So it takes more than 10 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.
Ben does the whole job in 15 hours.
So in 1 hour, Ben does 1 part out of 15.
That is 1/15.
Zoe takes x hours alone.
So in 1 hour, Zoe does 1 over x.
They work 10 hours together.
In 10 hours, Ben does 10 times as much: 10/15.
And Zoe does 10 over x.
3 Write the equation
In words: Ben's part plus Zoe's part makes the whole job.
So the equation is 10/15 + 10/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 15 and x.
15x can be divided by all of them.
We multiply every part by 15x.
We do it on both sides, so they stay equal.
15x divided by 15 is x.
15x divided by x is 15.
15x times 1 is 15x.
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 10x from the left side.
To remove 10x, we take away 10x.
We do it on both sides, so they stay equal.
10x and − 10x make zero.
Cross them out.
15x − 10x = 5x.
Now we have 150 = 5x.
Let's swap the two sides, so x is on the left.
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.
150 divided by 5 is 30.
So x = 30!
7 Answer the question
So Zoe alone takes 30 hours.
8 Check
Let's check in the story.
In 10 hours, Ben does 10/15, and Zoe does 10/30.
20/30 plus 10/30 is 30/30.
That is 1 whole room, like the story says.
Does it make sense?
Zoe 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.