Step by step
The question
Question: Max can wash the dishes in 10 minutes.
Working together, Max and Ana can wash them in 6 minutes.
How long would it take Ana alone?
1 Find what we know
What do we know?
Max does the whole job alone in 10 minutes.
Together they take 6 minutes.
What do we want?
How long Ana takes alone.
2 Name the unknown
We don't know how long Ana takes alone.
So we call it x minutes.
Careful!
Many people take away the times: 4 minutes.
But Ana alone is slower than the two together.
So it takes more than 6 minutes.
The whole dishes is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Max does the whole job in 10 minutes.
So in 1 minute, Max does 1 part out of 10.
That is 1/10.
Ana takes x minutes alone.
So in 1 minute, Ana does 1 over x.
They work 6 minutes together.
In 6 minutes, Max does 6 times as much: 6/10.
And Ana does 6 over x.
3 Write the equation
In words: Max's part plus Ana's part makes the whole job.
So the equation is 6/10 + 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 10 and x.
10x can be divided by all of them.
We multiply every part by 10x.
We do it on both sides, so they stay equal.
10x divided by 10 is x.
10x divided by x is 10.
10x times 1 is 10x.
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.
10x − 6x = 4x.
Now we have 60 = 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.
60 divided by 4 is 15.
So x = 15!
7 Answer the question
So Ana alone takes 15 minutes.
8 Check
Let's check in the story.
In 6 minutes, Max does 6/10, and Ana does 6/15.
18/30 plus 12/30 is 30/30.
That is 1 whole dishes, like the story says.
Does it make sense?
Ana 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.