Step by step
The question
Question: Ivy can paint the gates in 21 hours.
Working together, Ivy and Ken can paint them in 14 hours.
How long would it take Ken alone?
1 Find what we know
What do we know?
Ivy does the whole job alone in 21 hours.
Together they take 14 hours.
What do we want?
How long Ken takes alone.
2 Name the unknown
We don't know how long Ken takes alone.
So we call it x hours.
Careful!
Many people take away the times: 7 hours.
But Ken alone is slower than the two together.
So it takes more than 14 hours.
The whole gates is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Ivy does the whole job in 21 hours.
So in 1 hour, Ivy does 1 part out of 21.
That is 1/21.
Ken takes x hours alone.
So in 1 hour, Ken does 1 over x.
They work 14 hours together.
In 14 hours, Ivy does 14 times as much: 14/21.
And Ken does 14 over x.
3 Write the equation
In words: Ivy's part plus Ken's part makes the whole job.
So the equation is 14/21 + 14/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 21 and x.
21x can be divided by all of them.
We multiply every part by 21x.
We do it on both sides, so they stay equal.
21x divided by 21 is x.
21x divided by x is 21.
21x times 1 is 21x.
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 14x from the left side.
To remove 14x, we take away 14x.
We do it on both sides, so they stay equal.
14x and − 14x make zero.
Cross them out.
21x − 14x = 7x.
Now we have 294 = 7x.
Let's swap the two sides, so x is on the left.
6 Remove the 7
7x means 7 times x.
We want just x.
To remove times 7, we divide by 7.
We do it on both sides, so they stay equal.
7 divided by 7 is 1.
So just x is left.
294 divided by 7 is 42.
So x = 42!
7 Answer the question
So Ken alone takes 42 hours.
8 Check
Let's check in the story.
In 14 hours, Ivy does 14/21, and Ken does 14/42.
28/42 plus 14/42 is 42/42.
That is 1 whole gates, like the story says.
Does it make sense?
Ken 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.