Step by step
The question
Question: Ana can rake a yard in 30 minutes.
Ben can rake it in 60 minutes.
How many minutes will it take them to rake it together?
1 Find what we know
What do we know?
Ana does the whole job alone in 30 minutes.
Ben does it alone in 60 minutes.
What do we want?
How long it takes them together.
2 Name the unknown
We don't know how long it takes together.
So we call it x minutes.
Careful!
Many people add the times: 90 minutes.
But two working together are faster than one.
So together it takes less than 30 minutes.
The whole yard is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Ana does the whole job in 30 minutes.
So in 1 minute, Ana does 1 part out of 30.
That is 1/30.
Ben does 1 part out of 60 in 1 minute: 1/60.
In x minutes, Ana does x times as much: x/30.
And Ben does x/60.
3 Write the equation
In words: Ana's part plus Ben's part makes the whole job.
So the equation is x/30 + x/60 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 30 and 60.
60 can be divided by all of them.
We multiply every part by 60.
We do it on both sides, so they stay equal.
60 divided by 30 is 2.
60 divided by 60 is 1.
Write what is left.
Do the multiplications.
No more fractions!
5 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
2x + x = 3x.
Now we have 3x = 60.
6 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
60 divided by 3 is 20.
So x = 20!
7 Answer the question
So together it takes 20 minutes.
8 Check
Let's check in the story.
In 20 minutes, Ana does 20/30.
That is 2/3 of the yard.
In 20 minutes, Ben does 20/60.
That is 1/3 of the yard.
Together: 2/3 plus 1/3 is 3/3.
That is 1 whole yard, like the story says.
Does it make sense?
Together is faster than Ana alone, who needs 30 minutes.
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.