Step by step
The question
Question: Nina can sort the mail in 30 minutes.
Working together, Nina and Ali can sort it in 20 minutes.
How long would it take Ali alone?
1 Find what we know
What do we know?
Nina does the whole job alone in 30 minutes.
Together they take 20 minutes.
What do we want?
How long Ali takes alone.
2 Name the unknown
We don't know how long Ali takes alone.
So we call it x minutes.
Careful!
Many people take away the times: 10 minutes.
But Ali alone is slower than the two together.
So it takes more than 20 minutes.
The whole mail is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Nina does the whole job in 30 minutes.
So in 1 minute, Nina does 1 part out of 30.
That is 1/30.
Ali takes x minutes alone.
So in 1 minute, Ali does 1 over x.
They work 20 minutes together.
In 20 minutes, Nina does 20 times as much: 20/30.
And Ali does 20 over x.
3 Write the equation
In words: Nina's part plus Ali's part makes the whole job.
So the equation is 20/30 + 20/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 30 and x.
30x can be divided by all of them.
We multiply every part by 30x.
We do it on both sides, so they stay equal.
30x divided by 30 is x.
30x divided by x is 30.
30x times 1 is 30x.
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 20x from the left side.
To remove 20x, we take away 20x.
We do it on both sides, so they stay equal.
20x and − 20x make zero.
Cross them out.
30x − 20x = 10x.
Now we have 600 = 10x.
Let's swap the two sides, so x is on the left.
6 Remove the 10
10x means 10 times x.
We want just x.
To remove times 10, we divide by 10.
We do it on both sides, so they stay equal.
10 divided by 10 is 1.
So just x is left.
600 divided by 10 is 60.
So x = 60!
7 Answer the question
So Ali alone takes 60 minutes.
8 Check
Let's check in the story.
In 20 minutes, Nina does 20/30, and Ali does 20/60.
40/60 plus 20/60 is 60/60.
That is 1 whole mail, like the story says.
Does it make sense?
Ali 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.