Step by step
The question
Question: Joe can rake the leaves in 6 hours.
Working together, Joe and Amy can rake them in 2 hours.
How long would it take Amy alone?
1 Find what we know
What do we know?
Joe does the whole job alone in 6 hours.
Together they take 2 hours.
What do we want?
How long Amy takes alone.
2 Name the unknown
We don't know how long Amy takes alone.
So we call it x hours.
Careful!
Many people take away the times: 4 hours.
But Amy alone is slower than the two together.
So it takes more than 2 hours.
The whole leaves is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Joe does the whole job in 6 hours.
So in 1 hour, Joe does 1 part out of 6.
That is 1/6.
Amy takes x hours alone.
So in 1 hour, Amy does 1 over x.
They work 2 hours together.
In 2 hours, Joe does 2 times as much: 2/6.
And Amy does 2 over x.
3 Write the equation
In words: Joe's part plus Amy's part makes the whole job.
So the equation is 2/6 + 2/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 6 and x.
6x can be divided by all of them.
We multiply every part by 6x.
We do it on both sides, so they stay equal.
6x divided by 6 is x.
6x divided by x is 6.
6x times 1 is 6x.
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 2x from the left side.
To remove 2x, we take away 2x.
We do it on both sides, so they stay equal.
2x and − 2x make zero.
Cross them out.
6x − 2x = 4x.
Now we have 12 = 4x.
Let's swap the two sides, so x is on the left.
6 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
12 divided by 4 is 3.
So x = 3!
7 Answer the question
So Amy alone takes 3 hours.
8 Check
Let's check in the story.
In 2 hours, Joe does 2/6, and Amy does 2/3.
2/6 plus 4/6 is 6/6.
That is 1 whole leaves, like the story says.
Does it make sense?
Amy 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.