Step by step
The question
Question: Ana can paint a fence in 12 hours.
Ben can paint it in 4 hours.
How many hours will it take them to paint it together?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find what we know
What do we know?
Ana does the whole job alone in 12 hours.
Ben does it alone in 4 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: 16 hours.
But two working together are faster than one.
So together it takes less 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 12 hours.
So in 1 hour, Ana does 1 part out of 12.
That is 1/12.
Ben does 1 part out of 4 in 1 hour: 1/4.
In x hours, Ana does x times as much.
x times 1/12 is x/12: the x goes on top.
And Ben does x/4.
Let's draw the whole fence as one bar.
Ana does a part: x/12.
Ben does the rest: x/4.
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/12 + x/4 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 12 and 4.
To remove them all at once, we multiply by one number that every bottom goes into.
We count up in 12s and 4s.
In 12s: 12.
In 4s: 4, 8, 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 12 is 1.
The bottom is gone.
12 divided by 4 is 3.
We write the small 3.
The bottom is gone.
We write it again, without the crossed-out parts.
3 times x is 3x.
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.
x + 3x = 4x.
Now we have 4x = 12.
Is x alone now?
Not yet: 4x means 4 times x.
How do we remove it?
6 Remove the 4
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
They cancel each other.
So just x is left.
12 divided by 4 is 3.
So x = 3.
7 Answer the question
So together it takes 3 hours.
8 Check
Let's check in the story.
In 3 hours, Ana does 3/12 of the fence.
3 goes into 12: 3 × 4 = 12.
So 3 goes into both.
So that is 1/4.
In 3 hours, Ben does 3/4 of the fence.
In 3 hours, Ben does 3 divided by 4.
That is 0.75 of the fence.
1/4 plus 3/4 is 4/4.
That is the whole fence: 1 whole job.
Does it make sense?
Together is faster than Ben alone.
Alone, Ben needs 4 hours.
So the answer is right!
9 Pick the choice
Our answer is 3 hours.
That is choice B.
10 Why not the others
A is 1/3.
Wrong: 1/12 + 1/4 = 1/3 is the part they paint in 1 hour, not the time.
C is 8.
Wrong: the mean of 12 and 4: together they are faster than either one.
D is 16.
Wrong: 12 + 4: added the times; we add the work per hour.
That is the Careful mistake from before.
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.