Step by step
The question
Question: Ava can load a truck in 45 minutes.
Working together, Ava and Sam can load it in 30 minutes.
How long would it take Sam alone?
1 Find what we know
What do we know?
Ava does the whole job alone in 45 minutes.
Together they take 30 minutes.
What do we want?
How long Sam takes alone.
2 Name the unknown
We don't know how long Sam takes alone.
So we call it x minutes.
Careful!
Many people take away the times: 15 minutes.
But Sam alone is slower than the two together.
So it takes more than 30 minutes.
The whole truck is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Ava does the whole job in 45 minutes.
So in 1 minute, Ava does 1 part out of 45.
That is 1/45.
Sam takes x minutes alone.
So in 1 minute, Sam does 1 over x.
They work 30 minutes together.
In 30 minutes, Ava does 30 times as much: 30/45.
And Sam does 30 over x.
3 Write the equation
In words: Ava's part plus Sam's part makes the whole job.
So the equation is 30/45 + 30/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 45 and x.
45x can be divided by all of them.
We multiply every part by 45x.
We do it on both sides, so they stay equal.
45x divided by 45 is x.
45x divided by x is 45.
45x times 1 is 45x.
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 30x from the left side.
To remove 30x, we take away 30x.
We do it on both sides, so they stay equal.
30x and − 30x make zero.
Cross them out.
45x − 30x = 15x.
Now we have 1350 = 15x.
Let's swap the two sides, so x is on the left.
6 Remove the 15
15x means 15 times x.
We want just x.
To remove times 15, we divide by 15.
We do it on both sides, so they stay equal.
15 divided by 15 is 1.
So just x is left.
1350 divided by 15 is 90.
So x = 90!
7 Answer the question
So Sam alone takes 90 minutes.
8 Check
Let's check in the story.
In 30 minutes, Ava does 30/45, and Sam does 30/90.
60/90 plus 30/90 is 90/90.
That is 1 whole truck, like the story says.
Does it make sense?
Sam 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.