Step by step
The question
Question: Eva can fill the shelves in 15 minutes.
Working together, Eva and Dan can fill them in 10 minutes.
How long would it take Dan alone?
1 Find what we know
What do we know?
Eva does the whole job alone in 15 minutes.
Together they take 10 minutes.
What do we want?
How long Dan takes alone.
2 Name the unknown
We don't know how long Dan takes alone.
So we call it x minutes.
Careful!
Many people take away the times: 5 minutes.
But Dan alone is slower than the two together.
So it takes more than 10 minutes.
The whole shelves is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Eva does the whole job in 15 minutes.
So in 1 minute, Eva does 1 part out of 15.
That is 1/15.
Dan takes x minutes alone.
So in 1 minute, Dan does 1 over x.
They work 10 minutes together.
In 10 minutes, Eva does 10 times as much: 10/15.
And Dan does 10 over x.
3 Write the equation
In words: Eva's part plus Dan's part makes the whole job.
So the equation is 10/15 + 10/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 15 and x.
15x can be divided by all of them.
We multiply every part by 15x.
We do it on both sides, so they stay equal.
15x divided by 15 is x.
15x divided by x is 15.
15x times 1 is 15x.
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 10x from the left side.
To remove 10x, we take away 10x.
We do it on both sides, so they stay equal.
10x and − 10x make zero.
Cross them out.
15x − 10x = 5x.
Now we have 150 = 5x.
Let's swap the two sides, so x is on the left.
6 Remove the 5
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
So just x is left.
150 divided by 5 is 30.
So x = 30!
7 Answer the question
So Dan alone takes 30 minutes.
8 Check
Let's check in the story.
In 10 minutes, Eva does 10/15, and Dan does 10/30.
20/30 plus 10/30 is 30/30.
That is 1 whole shelves, like the story says.
Does it make sense?
Dan 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.