Step by step
The question
Question: Omar can write the code in 24 days.
Working together, Omar and Kim can write it in 8 days.
How long would it take Kim alone?
1 Find what we know
What do we know?
Omar does the whole job alone in 24 days.
Together they take 8 days.
What do we want?
How long Kim takes alone.
2 Name the unknown
We don't know how long Kim takes alone.
So we call it x days.
Careful!
Many people take away the times: 16 days.
But Kim alone is slower than the two together.
So it takes more than 8 days.
The whole code is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Omar does the whole job in 24 days.
So in 1 day, Omar does 1 part out of 24.
That is 1/24.
Kim takes x days alone.
So in 1 day, Kim does 1 over x.
They work 8 days together.
In 8 days, Omar does 8 times as much: 8/24.
And Kim does 8 over x.
3 Write the equation
In words: Omar's part plus Kim's part makes the whole job.
So the equation is 8/24 + 8/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 24 and x.
24x can be divided by all of them.
We multiply every part by 24x.
We do it on both sides, so they stay equal.
24x divided by 24 is x.
24x divided by x is 24.
24x times 1 is 24x.
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 8x from the left side.
To remove 8x, we take away 8x.
We do it on both sides, so they stay equal.
8x and − 8x make zero.
Cross them out.
24x − 8x = 16x.
Now we have 192 = 16x.
Let's swap the two sides, so x is on the left.
6 Remove the 16
16x means 16 times x.
We want just x.
To remove times 16, we divide by 16.
We do it on both sides, so they stay equal.
16 divided by 16 is 1.
So just x is left.
192 divided by 16 is 12.
So x = 12!
7 Answer the question
So Kim alone takes 12 days.
8 Check
Let's check in the story.
In 8 days, Omar does 8/24, and Kim does 8/12.
8/24 plus 16/24 is 24/24.
That is 1 whole code, like the story says.
Does it make sense?
Kim 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.