Step by step
The question
Question: Lily can plant a garden in 9 hours.
Working together, Lily and Noah can plant it in 6 hours.
How long would it take Noah alone?
1 Find what we know
What do we know?
Lily does the whole job alone in 9 hours.
Together they take 6 hours.
What do we want?
How long Noah takes alone.
2 Name the unknown
We don't know how long Noah takes alone.
So we call it x hours.
Careful!
Many people take away the times: 3 hours.
But Noah alone is slower than the two together.
So it takes more than 6 hours.
The whole garden is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Lily does the whole job in 9 hours.
So in 1 hour, Lily does 1 part out of 9.
That is 1/9.
Noah takes x hours alone.
So in 1 hour, Noah does 1 over x.
They work 6 hours together.
In 6 hours, Lily does 6 times as much: 6/9.
And Noah does 6 over x.
3 Write the equation
In words: Lily's part plus Noah's part makes the whole job.
So the equation is 6/9 + 6/x = 1.
Now we solve it.
First: a bottom can never be 0.
x at the bottom can't be 0.
So x ≠ 0.
4 Remove the fractions
The bottoms are 9 and x.
9x can be divided by all of them.
We multiply every part by 9x.
We do it on both sides, so they stay equal.
9x divided by 9 is x.
9x divided by x is 9.
9x times 1 is 9x.
Write what is left.
No more fractions!
5 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 6x from the left side.
To remove 6x, we take away 6x.
We do it on both sides, so they stay equal.
6x and − 6x make zero.
Cross them out.
9x − 6x = 3x.
Now we have 54 = 3x.
Let's swap the two sides, so x is on the left.
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.
54 divided by 3 is 18.
So x = 18!
7 Answer the question
So Noah alone takes 18 hours.
8 Check
Let's check in the story.
In 6 hours, Lily does 6/9, and Noah does 6/18.
12/18 plus 6/18 is 18/18.
That is 1 whole garden, like the story says.
Does it make sense?
Noah alone is slower than the two together.
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.