Step by step
The question
Question: Leo can build a wall in 12 hours.
Max can build it in 6 hours.
How many hours will it take them to build it together?
1 Find what we know
What do we know?
Leo does the whole job alone in 12 hours.
Max does it alone in 6 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 wall is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Leo does the whole job in 12 hours.
So in 1 hour, Leo does 1 part out of 12.
That is 1/12.
Max does 1 part out of 6 in 1 hour: 1/6.
In x hours, Leo does x times as much: x/12.
And Max does x/6.
3 Write the equation
In words: Leo's part plus Max's part makes the whole job.
So the equation is x/12 + x/6 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 12 and 6.
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 12 is 1.
12 divided by 6 is 2.
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.
x + 2x = 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, Leo does 4/12.
That is 1/3 of the wall.
In 4 hours, Max does 4/6.
That is 2/3 of the wall.
Together: 1/3 plus 2/3 is 3/3.
That is 1 whole wall, like the story says.
Does it make sense?
Together is faster than Max 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.