Step by step
The question
Question: Omar can stack the shelves in 30 minutes.
Ali can stack them in 45 minutes.
How many minutes will it take them to stack them together?
1 Find what we know
What do we know?
Omar does the whole job alone in 30 minutes.
Ali does it alone in 45 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: 75 minutes.
But two working together are faster than one.
So together it takes less than 30 minutes.
The whole shelves 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 30 minutes.
So in 1 minute, Omar does 1 part out of 30.
That is 1/30.
Ali does 1 part out of 45 in 1 minute: 1/45.
In x minutes, Omar does x times as much: x/30.
And Ali does x/45.
3 Write the equation
In words: Omar's part plus Ali's part makes the whole job.
So the equation is x/30 + x/45 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 30 and 45.
90 can be divided by all of them.
We multiply every part by 90.
We do it on both sides, so they stay equal.
90 divided by 30 is 3.
90 divided by 45 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 = 90.
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.
90 divided by 5 is 18.
So x = 18!
7 Answer the question
So together it takes 18 minutes.
8 Check
Let's check in the story.
In 18 minutes, Omar does 18/30.
That is 3/5 of the shelves.
In 18 minutes, Ali does 18/45.
That is 2/5 of the shelves.
Together: 3/5 plus 2/5 is 5/5.
That is 1 whole shelves, like the story says.
Does it make sense?
Together is faster than Omar 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.