Step by step
The question
Question: Lina can wash the dishes in 15 minutes.
Kai can wash them in 30 minutes.
How many minutes will it take them to wash them together?
1 Find what we know
What do we know?
Lina does the whole job alone in 15 minutes.
Kai 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: 45 minutes.
But two working together are faster than one.
So together it takes less than 15 minutes.
The whole dishes is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Lina does the whole job in 15 minutes.
So in 1 minute, Lina does 1 part out of 15.
That is 1/15.
Kai does 1 part out of 30 in 1 minute: 1/30.
In x minutes, Lina does x times as much: x/15.
And Kai does x/30.
3 Write the equation
In words: Lina's part plus Kai's part makes the whole job.
So the equation is x/15 + x/30 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 15 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 15 is 2.
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.
2x + x = 3x.
Now we have 3x = 30.
6 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
30 divided by 3 is 10.
So x = 10!
7 Answer the question
So together it takes 10 minutes.
8 Check
Let's check in the story.
In 10 minutes, Lina does 10/15.
That is 2/3 of the dishes.
In 10 minutes, Kai does 10/30.
That is 1/3 of the dishes.
Together: 2/3 plus 1/3 is 3/3.
That is 1 whole dishes, like the story says.
Does it make sense?
Together is faster than Lina alone, who needs 15 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.