Step by step
The question
Question: Grace can bake the cakes in 4 hours.
Working together, Grace and Ian can bake them in 3 hours.
How long would it take Ian alone?
1 Find what we know
What do we know?
Grace does the whole job alone in 4 hours.
Together they take 3 hours.
What do we want?
How long Ian takes alone.
2 Name the unknown
We don't know how long Ian takes alone.
So we call it x hours.
Careful!
Many people take away the times: 1 hour.
But Ian alone is slower than the two together.
So it takes more than 3 hours.
The whole cakes is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Grace does the whole job in 4 hours.
So in 1 hour, Grace does 1 part out of 4.
That is 1/4.
Ian takes x hours alone.
So in 1 hour, Ian does 1 over x.
They work 3 hours together.
In 3 hours, Grace does 3 times as much: 3/4.
And Ian does 3 over x.
3 Write the equation
In words: Grace's part plus Ian's part makes the whole job.
So the equation is 3/4 + 3/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 4 and x.
4x can be divided by all of them.
We multiply every part by 4x.
We do it on both sides, so they stay equal.
4x divided by 4 is x.
4x divided by x is 4.
4x times 1 is 4x.
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 3x from the left side.
To remove 3x, we take away 3x.
We do it on both sides, so they stay equal.
3x and − 3x make zero.
Cross them out.
4x − 3x = x.
Now we have 12 = x.
Let's swap the two sides, so x is on the left.
So x = 12!
6 Answer the question
So Ian alone takes 12 hours.
7 Check
Let's check in the story.
In 3 hours, Grace does 3/4, and Ian does 3/12.
9/12 plus 3/12 is 12/12.
That is 1 whole cakes, like the story says.
Does it make sense?
Ian 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.