Step by step
The question
Question: One pipe can fill a pool in 9 hours.
Another pipe can fill it in 18 hours.
How long will it take both pipes together?
1 Find what we know
What do we know?
pipe 1 does the whole job alone in 9 hours.
pipe 2 does it alone in 18 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: 27 hours.
But two working together are faster than one.
So together it takes less than 9 hours.
The whole pool is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
pipe 1 does the whole job in 9 hours.
So in 1 hour, pipe 1 does 1 part out of 9.
That is 1/9.
pipe 2 does 1 part out of 18 in 1 hour: 1/18.
In x hours, pipe 1 does x times as much: x/9.
And pipe 2 does x/18.
3 Write the equation
In words: pipe 1's part plus pipe 2's part makes the whole job.
So the equation is x/9 + x/18 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 9 and 18.
18 can be divided by all of them.
We multiply every part by 18.
We do it on both sides, so they stay equal.
18 divided by 9 is 2.
18 divided by 18 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 = 18.
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.
18 divided by 3 is 6.
So x = 6!
7 Answer the question
So together it takes 6 hours.
8 Check
Let's check in the story.
In 6 hours, pipe 1 does 6/9.
That is 2/3 of the pool.
In 6 hours, pipe 2 does 6/18.
That is 1/3 of the pool.
Together: 2/3 plus 1/3 is 3/3.
That is 1 whole pool, like the story says.
Does it make sense?
Together is faster than pipe 1 alone, who needs 9 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.