Step by step
The question
Question: Ana can type a report in 8 hours.
Working together, Ana and Max can type it in 6 hours.
How long would it take Max alone?
1 Find what we know
What do we know?
Ana does the whole job alone in 8 hours.
Together they take 6 hours.
What do we want?
How long Max takes alone.
2 Name the unknown
We don't know how long Max takes alone.
So we call it x hours.
Careful!
Many people take away the times: 2 hours.
But Max alone is slower than the two together.
So it takes more than 6 hours.
The whole report is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Ana does the whole job in 8 hours.
So in 1 hour, Ana does 1 part out of 8.
That is 1/8.
Max takes x hours alone.
So in 1 hour, Max does 1 over x.
They work 6 hours together.
In 6 hours, Ana does 6 times as much: 6/8.
And Max does 6 over x.
3 Write the equation
In words: Ana's part plus Max's part makes the whole job.
So the equation is 6/8 + 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 8 and x.
8x can be divided by all of them.
We multiply every part by 8x.
We do it on both sides, so they stay equal.
8x divided by 8 is x.
8x divided by x is 8.
8x times 1 is 8x.
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.
8x − 6x = 2x.
Now we have 48 = 2x.
Let's swap the two sides, so x is on the left.
6 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
48 divided by 2 is 24.
So x = 24!
7 Answer the question
So Max alone takes 24 hours.
8 Check
Let's check in the story.
In 6 hours, Ana does 6/8, and Max does 6/24.
18/24 plus 6/24 is 24/24.
That is 1 whole report, like the story says.
Does it make sense?
Max 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.