Step by step
The question
Question: Raj can wash the cars in 12 hours.
Working together, Raj and Ella can wash them in 4 hours.
How long would it take Ella alone?
1 Find what we know
What do we know?
Raj does the whole job alone in 12 hours.
Together they take 4 hours.
What do we want?
How long Ella takes alone.
2 Name the unknown
We don't know how long Ella takes alone.
So we call it x hours.
Careful!
Many people take away the times: 8 hours.
But Ella alone is slower than the two together.
So it takes more than 4 hours.
The whole cars is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Raj does the whole job in 12 hours.
So in 1 hour, Raj does 1 part out of 12.
That is 1/12.
Ella takes x hours alone.
So in 1 hour, Ella does 1 over x.
They work 4 hours together.
In 4 hours, Raj does 4 times as much: 4/12.
And Ella does 4 over x.
3 Write the equation
In words: Raj's part plus Ella's part makes the whole job.
So the equation is 4/12 + 4/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 12 and x.
12x can be divided by all of them.
We multiply every part by 12x.
We do it on both sides, so they stay equal.
12x divided by 12 is x.
12x divided by x is 12.
12x times 1 is 12x.
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 4x from the left side.
To remove 4x, we take away 4x.
We do it on both sides, so they stay equal.
4x and − 4x make zero.
Cross them out.
12x − 4x = 8x.
Now we have 48 = 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.
48 divided by 8 is 6.
So x = 6!
7 Answer the question
So Ella alone takes 6 hours.
8 Check
Let's check in the story.
In 4 hours, Raj does 4/12, and Ella does 4/6.
4/12 plus 8/12 is 12/12.
That is 1 whole cars, like the story says.
Does it make sense?
Ella 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.