Step by step
The question
Question: Leo can fix the fence in 20 hours.
Working together, Leo and Mia can fix it in 12 hours.
How many hours would it take Mia alone?
1 Find what we know
What do we know?
Leo does the whole job alone in 20 hours.
Together they take 12 hours.
What do we want?
How long Mia takes alone.
2 Name the unknown
We don't know how long Mia takes alone.
So we call it x hours.
Careful!
Many people take away the times: 8 hours.
But Mia alone is slower than the two together.
So it takes more than 12 hours.
The whole fence is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Leo does the whole job in 20 hours.
So in 1 hour, Leo does 1 part out of 20.
That is 1/20.
Mia takes x hours alone.
So in 1 hour, Mia does 1 over x.
They work 12 hours together.
In 12 hours, Leo does 12 times as much: 12/20.
And Mia does 12 over x.
3 Write the equation
In words: Leo's part plus Mia's part makes the whole job.
So the equation is 12/20 + 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 20 and x.
20x can be divided by all of them.
We multiply every part by 20x.
We do it on both sides, so they stay equal.
20x divided by 20 is x.
20x divided by x is 20.
20x times 1 is 20x.
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.
20x − 12x = 8x.
Now we have 240 = 8x.
Let's swap the two sides, so x is on the left.
6 Remove the 8
8x means 8 times x.
We want just x.
To remove times 8, we divide by 8.
We do it on both sides, so they stay equal.
8 divided by 8 is 1.
So just x is left.
240 divided by 8 is 30.
So x = 30!
7 Answer the question
So Mia alone takes 30 hours.
8 Check
Let's check in the story.
In 12 hours, Leo does 12/20, and Mia does 12/30.
36/60 plus 24/60 is 60/60.
That is 1 whole fence, like the story says.
Does it make sense?
Mia 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.