Step by step
The question
Question: Ana can paint a fence in 6 hours.
Ben can paint it in 12 hours.
How many hours does it take them working 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 times 1/6 is x/6: the x goes on top.
And Ben does x/12.
Let's draw the whole fence as one bar.
Ana does a part: x/6.
Ben does the rest: x/12.
Together, the two parts make the whole fence: 1.
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.
To remove them all at once, we multiply by one number that every bottom goes into.
We count up in 6s and 12s.
In 6s: 6, 12.
In 12s: 12.
12 is the first number in both lists.
So 12 works for both.
We multiply every part by 12, not only the fractions.
Each whole side is times 12, so 1 is times 12 too.
We do it on both sides, so they stay equal.
12 divided by 6 is 2.
We write the small 2.
The bottom is gone.
12 divided by 12 is 1.
The bottom is gone.
We write it again, without the crossed-out parts.
2 times x is 2x.
12 times 1 is 12.
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.
Is x alone now?
Not yet: 3x means 3 times x.
How do we remove it?
6 Remove the 3
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
They cancel each other.
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 of the fence.
What goes into 4 and 6?
We list the numbers that go into each.
Into 4: 1, 2, 4.
Into 6: 1, 2, 3, 6.
2 is the biggest one in both lists.
So that is 2/3.
In 4 hours, Ben does 4/12 of the fence.
4 goes into 12: 4 × 3 = 12.
So 4 goes into both.
So that is 1/3.
2/3 plus 1/3 is 3/3.
That is the whole fence: 1 whole job.
Does it make sense?
Together is faster than Ana alone.
Alone, Ana 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.