Step by step
The question
Question: Ana can paint a fence in 6 hours.
Working together, Ana and Ben can paint it in 4 hours.
How long would it take Ben alone?
1 Find what we know
What do we know?
Ana does the whole job alone in 6 hours.
Together they take 4 hours.
What do we want?
How long Ben takes alone.
2 Name the unknown
We don't know how long Ben takes alone.
So we call it x hours.
Careful!
Many people take away the times: 2 hours.
But Ben alone is slower than the two together.
So it takes more than 4 hours.
The whole fence is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Ana does the whole job in 6 hours.
So in 1 hour, Ana does 1 part out of 6.
That is 1/6.
Ben takes x hours alone.
So in 1 hour, Ben does 1 over x.
They work 4 hours together.
In 4 hours, Ana does 4 times as much: 4/6.
And Ben does 4 over x.
3 Write the equation
In words: Ana's part plus Ben's part makes the whole job.
So the equation is 4/6 + 4/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 6 and x.
6x can be divided by all of them.
We multiply every part by 6x.
We do it on both sides, so they stay equal.
6x divided by 6 is x.
6x divided by x is 6.
6x times 1 is 6x.
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 4x from the left side.
To remove 4x, we take away 4x.
We do it on both sides, so they stay equal.
4x and − 4x make zero.
Cross them out.
6x − 4x = 2x.
Now we have 24 = 2x.
Let's swap the two sides, so x is on the left.
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.
24 divided by 2 is 12.
So x = 12!
7 Answer the question
So Ben alone takes 12 hours.
8 Check
Let's check in the story.
In 4 hours, Ana does 4/6, and Ben does 4/12.
8/12 plus 4/12 is 12/12.
That is 1 whole fence, like the story says.
Does it make sense?
Ben 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.