Step by step
The question
Question: Sara can paint a barn in 18 hours.
Working together, Sara and Tom can paint it in 12 hours.
How long would it take Tom alone?
1 Find what we know
What do we know?
Sara does the whole job alone in 18 hours.
Together they take 12 hours.
What do we want?
How long Tom takes alone.
2 Name the unknown
We don't know how long Tom takes alone.
So we call it x hours.
Careful!
Many people take away the times: 6 hours.
But Tom alone is slower than the two together.
So it takes more than 12 hours.
The whole barn is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Sara does the whole job in 18 hours.
So in 1 hour, Sara does 1 part out of 18.
That is 1/18.
Tom takes x hours alone.
So in 1 hour, Tom does 1 over x.
They work 12 hours together.
In 12 hours, Sara does 12 times as much: 12/18.
And Tom does 12 over x.
3 Write the equation
In words: Sara's part plus Tom's part makes the whole job.
So the equation is 12/18 + 12/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 18 and x.
18x can be divided by all of them.
We multiply every part by 18x.
We do it on both sides, so they stay equal.
18x divided by 18 is x.
18x divided by x is 18.
18x times 1 is 18x.
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 12x from the left side.
To remove 12x, we take away 12x.
We do it on both sides, so they stay equal.
12x and − 12x make zero.
Cross them out.
18x − 12x = 6x.
Now we have 216 = 6x.
Let's swap the two sides, so x is on the left.
6 Remove the 6
6x means 6 times x.
We want just x.
To remove times 6, we divide by 6.
We do it on both sides, so they stay equal.
6 divided by 6 is 1.
So just x is left.
216 divided by 6 is 36.
So x = 36!
7 Answer the question
So Tom alone takes 36 hours.
8 Check
Let's check in the story.
In 12 hours, Sara does 12/18, and Tom does 12/36.
24/36 plus 12/36 is 36/36.
That is 1 whole barn, like the story says.
Does it make sense?
Tom 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.