Step by step
The question
Question: Sam can mow a lawn in 6 hours.
Leo can mow it in 30 hours.
How many hours will it take them to mow it together?
1 Find what we know
What do we know?
Sam does the whole job alone in 6 hours.
Leo does it alone in 30 hours.
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 hours.
Careful!
Many people add the times: 36 hours.
But two working together are faster than one.
So together it takes less than 6 hours.
The whole lawn is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Sam does the whole job in 6 hours.
So in 1 hour, Sam does 1 part out of 6.
That is 1/6.
Leo does 1 part out of 30 in 1 hour: 1/30.
In x hours, Sam does x times as much: x/6.
And Leo does x/30.
3 Write the equation
In words: Sam's part plus Leo's part makes the whole job.
So the equation is x/6 + x/30 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 6 and 30.
30 can be divided by all of them.
We multiply every part by 30.
We do it on both sides, so they stay equal.
30 divided by 6 is 5.
30 divided by 30 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.
5x + x = 6x.
Now we have 6x = 30.
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.
30 divided by 6 is 5.
So x = 5!
7 Answer the question
So together it takes 5 hours.
8 Check
Let's check in the story.
In 5 hours, Sam does 5/6.
That is 5/6 of the lawn.
In 5 hours, Leo does 5/30.
That is 1/6 of the lawn.
Together: 5/6 plus 1/6 is 6/6.
That is 1 whole lawn, like the story says.
Does it make sense?
Together is faster than Sam alone, who needs 6 hours.
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.