Step by step
The question
Question: One machine can print a book in 8 hours.
Another machine can print it in 24 hours.
How many hours will it take both machines together?
1 Find what we know
What do we know?
machine 1 does the whole job alone in 8 hours.
machine 2 does it alone in 24 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: 32 hours.
But two working together are faster than one.
So together it takes less than 8 hours.
The whole book is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
machine 1 does the whole job in 8 hours.
So in 1 hour, machine 1 does 1 part out of 8.
That is 1/8.
machine 2 does 1 part out of 24 in 1 hour: 1/24.
In x hours, machine 1 does x times as much: x/8.
And machine 2 does x/24.
3 Write the equation
In words: machine 1's part plus machine 2's part makes the whole job.
So the equation is x/8 + x/24 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 8 and 24.
24 can be divided by all of them.
We multiply every part by 24.
We do it on both sides, so they stay equal.
24 divided by 8 is 3.
24 divided by 24 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.
3x + x = 4x.
Now we have 4x = 24.
6 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
24 divided by 4 is 6.
So x = 6!
7 Answer the question
So together it takes 6 hours.
8 Check
Let's check in the story.
In 6 hours, machine 1 does 6/8.
That is 3/4 of the book.
In 6 hours, machine 2 does 6/24.
That is 1/4 of the book.
Together: 3/4 plus 1/4 is 4/4.
That is 1 whole book, like the story says.
Does it make sense?
Together is faster than machine 1 alone, who needs 8 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.