Step by step
The question
Question: One pipe can fill a tank in 6 hours.
Another pipe can fill it in 9 hours.
How many hours will it take both pipes together?
1 Find what we know
What do we know?
pipe 1 does the whole job alone in 6 hours.
pipe 2 does it alone in 9 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: 15 hours.
But two working together are faster than one.
So together it takes less than 6 hours.
The whole tank 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 6 hours.
So in 1 hour, pipe 1 does 1 part out of 6.
That is 1/6.
pipe 2 does 1 part out of 9 in 1 hour: 1/9.
In x hours, pipe 1 does x times as much: x/6.
And pipe 2 does x/9.
3 Write the equation
In words: pipe 1's part plus pipe 2's part makes the whole job.
So the equation is x/6 + x/9 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 6 and 9.
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 6 is 3.
18 divided by 9 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.
3x + 2x = 5x.
Now we have 5x = 18.
6 Remove the 5
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
So just x is left.
So x = 18/5!
7 Answer the question
18/5 is 3.6.
So together it takes 3.6 hours.
8 Check
Let's check in the story.
In 3.6 hours, pipe 1 does 3.6 divided by 6: 0.6 of the tank.
In 3.6 hours, pipe 2 does 3.6 divided by 9: 0.4 of the tank.
Together: 1 whole tank, like the story says.
Does it make sense?
Together is faster than pipe 1 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.