Step by step
The question
Question: Ana can paint a fence in 6 hours.
Ben can paint it in 12 hours.
How many hours will it take them to paint it together?
1 Find what we know
What do we know?
Ana does the whole job alone in 6 hours.
Ben does it alone in 12 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: 18 hours.
But two working together are faster than one.
So together it takes less than 6 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 does 1 part out of 12 in 1 hour: 1/12.
In x hours, Ana does x times as much: x/6.
And Ben does x/12.
3 Write the equation
In words: Ana's part plus Ben's part makes the whole job.
So the equation is x/6 + x/12 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 6 and 12.
12 can be divided by all of them.
We multiply every part by 12.
We do it on both sides, so they stay equal.
12 divided by 6 is 2.
12 divided by 12 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.
2x + x = 3x.
Now we have 3x = 12.
6 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
12 divided by 3 is 4.
So x = 4!
7 Answer the question
So together it takes 4 hours.
8 Check
Let's check in the story.
In 4 hours, Ana does 4/6.
That is 2/3 of the fence.
In 4 hours, Ben does 4/12.
That is 1/3 of the fence.
Together: 2/3 plus 1/3 is 3/3.
That is 1 whole fence, like the story says.
Does it make sense?
Together is faster than Ana alone, who needs 6 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.