Step by step
The question
Question: Omar can wash a car in 20 minutes.
Ali can wash it in 30 minutes.
How many minutes will it take them to wash it together?
1 Find what we know
What do we know?
Omar does the whole job alone in 20 minutes.
Ali does it alone in 30 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: 50 minutes.
But two working together are faster than one.
So together it takes less than 20 minutes.
The whole car is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Omar does the whole job in 20 minutes.
So in 1 minute, Omar does 1 part out of 20.
That is 1/20.
Ali does 1 part out of 30 in 1 minute: 1/30.
In x minutes, Omar does x times as much: x/20.
And Ali does x/30.
3 Write the equation
In words: Omar's part plus Ali's part makes the whole job.
So the equation is x/20 + x/30 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 20 and 30.
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 20 is 3.
60 divided by 30 is 2.
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.
3x + 2x = 5x.
Now we have 5x = 60.
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.
60 divided by 5 is 12.
So x = 12!
7 Answer the question
So together it takes 12 minutes.
8 Check
Let's check in the story.
In 12 minutes, Omar does 12/20.
That is 3/5 of the car.
In 12 minutes, Ali does 12/30.
That is 2/5 of the car.
Together: 3/5 plus 2/5 is 5/5.
That is 1 whole car, like the story says.
Does it make sense?
Together is faster than Omar alone, who needs 20 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.