Step by step
The question
Question: Jo can sort the mail in 40 minutes.
Kim can sort it in 120 minutes.
How many minutes will it take them to sort it together?
1 Find what we know
What do we know?
Jo does the whole job alone in 40 minutes.
Kim does it alone in 120 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: 160 minutes.
But two working together are faster than one.
So together it takes less than 40 minutes.
The whole mail is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Jo does the whole job in 40 minutes.
So in 1 minute, Jo does 1 part out of 40.
That is 1/40.
Kim does 1 part out of 120 in 1 minute: 1/120.
In x minutes, Jo does x times as much: x/40.
And Kim does x/120.
3 Write the equation
In words: Jo's part plus Kim's part makes the whole job.
So the equation is x/40 + x/120 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 40 and 120.
120 can be divided by all of them.
We multiply every part by 120.
We do it on both sides, so they stay equal.
120 divided by 40 is 3.
120 divided by 120 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 = 120.
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.
120 divided by 4 is 30.
So x = 30!
7 Answer the question
So together it takes 30 minutes.
8 Check
Let's check in the story.
In 30 minutes, Jo does 30/40.
That is 3/4 of the mail.
In 30 minutes, Kim does 30/120.
That is 1/4 of the mail.
Together: 3/4 plus 1/4 is 4/4.
That is 1 whole mail, like the story says.
Does it make sense?
Together is faster than Jo alone, who needs 40 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.