Step by step
The question
Question: Sam can pack the boxes in 20 minutes.
Working together, Sam and Ava can pack them in 12 minutes.
How long would it take Ava alone?
1 Find what we know
What do we know?
Sam does the whole job alone in 20 minutes.
Together they take 12 minutes.
What do we want?
How long Ava takes alone.
2 Name the unknown
We don't know how long Ava takes alone.
So we call it x minutes.
Careful!
Many people take away the times: 8 minutes.
But Ava alone is slower than the two together.
So it takes more than 12 minutes.
The whole boxes is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Sam does the whole job in 20 minutes.
So in 1 minute, Sam does 1 part out of 20.
That is 1/20.
Ava takes x minutes alone.
So in 1 minute, Ava does 1 over x.
They work 12 minutes together.
In 12 minutes, Sam does 12 times as much: 12/20.
And Ava does 12 over x.
3 Write the equation
In words: Sam's part plus Ava's part makes the whole job.
So the equation is 12/20 + 12/x = 1.
Now we solve it.
First: a bottom can never be 0.
x at the bottom can't be 0.
So x ≠ 0.
4 Remove the fractions
The bottoms are 20 and x.
20x can be divided by all of them.
We multiply every part by 20x.
We do it on both sides, so they stay equal.
20x divided by 20 is x.
20x divided by x is 20.
20x times 1 is 20x.
Write what is left.
No more fractions!
5 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 12x from the left side.
To remove 12x, we take away 12x.
We do it on both sides, so they stay equal.
12x and − 12x make zero.
Cross them out.
20x − 12x = 8x.
Now we have 240 = 8x.
Let's swap the two sides, so x is on the left.
6 Remove the 8
8x means 8 times x.
We want just x.
To remove times 8, we divide by 8.
We do it on both sides, so they stay equal.
8 divided by 8 is 1.
So just x is left.
240 divided by 8 is 30.
So x = 30!
7 Answer the question
So Ava alone takes 30 minutes.
8 Check
Let's check in the story.
In 12 minutes, Sam does 12/20, and Ava does 12/30.
36/60 plus 24/60 is 60/60.
That is 1 whole boxes, like the story says.
Does it make sense?
Ava alone is slower than the two together.
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.