Worked examples
The question
Question: Ana can paint a fence in 6 hours.
Working together, Ana and Ben can paint it in 4 hours.
How long would it take Ben alone?
1 Find what we know
What do we know?
Ana does the whole job alone in 6 hours.
Together they take 4 hours.
What do we want?
How long Ben takes alone.
2 Name the unknown
We don't know how long Ben takes alone.
So we call it x hours.
x = hours for Ben alone
Careful!
Many people take away the times: 2 hours.
But Ben alone is slower than the two together.
So it takes more than 4 hours.
Careful: not 6 − 4 = 2 hours ✗
The whole fence is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Ana does the whole job in 6 hours.
So in 1 hour, Ana does 1 part out of 6.
That is 1/6.
Ben takes x hours alone.
So in 1 hour, Ben does 1 over x.
They work 4 hours together.
In 4 hours, Ana does 4 times as much: 4/6.
And Ben does 4 over x.
3 Write the equation
In words: Ana's part plus Ben's part makes the whole job.
46 + 4x = 1
So the equation is 4/6 + 4/x = 1.
Now we solve it.
46 + 4x = 1
First: a bottom can never be 0.
x at the bottom can't be 0.
So x ≠ 0.
x ≠ 0
4 Remove the fractions
The bottoms are 6 and x.
6x can be divided by all of them.
We multiply every part by 6x.
We do it on both sides, so they stay equal.
6x × 46 + 6x × 4x = 6x × 1
6x divided by 6 is x.
6x divided by x is 6.
6x times 1 is 6x.
Write what is left.
No more fractions!
4x + 24 = 6x
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 + 24 − 4x = 6x − 4x
4x and − 4x make zero.
Cross them out.
6x − 4x = 2x.
6x − 4x = 2x
Now we have 24 = 2x.
24 = 2x
Let's swap the two sides, so x is on the left.
2x = 24
6 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2x2 = 242
2 divided by 2 is 1.
So just x is left.
24 divided by 2 is 12.
242 = 12
So x = 12!
x = 12
7 Answer the question
So Ben alone takes 12 hours.
8 Check
Let's check in the story.
In 4 hours, Ana does 4/6, and Ben does 4/12.
in 4 hours: Ana 46, Ben 412
8/12 plus 4/12 is 12/12.
That is 1 whole fence, like the story says.
812 + 412 = 1212 = 1
Does it make sense?
Ben alone is slower than the two together.
So the answer is right!
makes sense: 12 hours > 4 hours: slower alone than together
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.
▶ Watch it solved
The question
Question: Ben can paint a room in 15 hours.
Working together, Ben and Zoe can paint it in 10 hours.
How many hours would it take Zoe alone?
1 Find what we know
What do we know?
Ben does the whole job alone in 15 hours.
Together they take 10 hours.
What do we want?
How long Zoe takes alone.
2 Name the unknown
We don't know how long Zoe takes alone.
So we call it x hours.
x = hours for Zoe alone
Careful!
Many people take away the times: 5 hours.
But Zoe alone is slower than the two together.
So it takes more than 10 hours.
Careful: not 15 − 10 = 5 hours ✗
The whole room is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Ben does the whole job in 15 hours.
So in 1 hour, Ben does 1 part out of 15.
That is 1/15.
Zoe takes x hours alone.
So in 1 hour, Zoe does 1 over x.
They work 10 hours together.
In 10 hours, Ben does 10 times as much: 10/15.
And Zoe does 10 over x.
3 Write the equation
In words: Ben's part plus Zoe's part makes the whole job.
1015 + 10x = 1
So the equation is 10/15 + 10/x = 1.
Now we solve it.
1015 + 10x = 1
First: a bottom can never be 0.
x at the bottom can't be 0.
So x ≠ 0.
x ≠ 0
4 Remove the fractions
The bottoms are 15 and x.
15x can be divided by all of them.
We multiply every part by 15x.
We do it on both sides, so they stay equal.
15x × 1015 + 15x × 10x = 15x × 1
15x divided by 15 is x.
15x divided by x is 15.
15x times 1 is 15x.
Write what is left.
No more fractions!
10x + 150 = 15x
5 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 10x from the left side.
To remove 10x, we take away 10x.
We do it on both sides, so they stay equal.
10x + 150 − 10x = 15x − 10x
10x and − 10x make zero.
Cross them out.
15x − 10x = 5x.
15x − 10x = 5x
Now we have 150 = 5x.
150 = 5x
Let's swap the two sides, so x is on the left.
5x = 150
6 Remove the 5
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5x5 = 1505
5 divided by 5 is 1.
So just x is left.
150 divided by 5 is 30.
1505 = 30
So x = 30!
x = 30
7 Answer the question
So Zoe alone takes 30 hours.
8 Check
Let's check in the story.
In 10 hours, Ben does 10/15, and Zoe does 10/30.
in 10 hours: Ben 1015, Zoe 1030
20/30 plus 10/30 is 30/30.
That is 1 whole room, like the story says.
2030 + 1030 = 3030 = 1
Does it make sense?
Zoe alone is slower than the two together.
So the answer is right!
makes sense: 30 hours > 10 hours: slower alone than together
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.
▶ Watch it solved
The question
Question: Sara can paint a barn in 18 hours.
Working together, Sara and Tom can paint it in 12 hours.
How long would it take Tom alone?
1 Find what we know
What do we know?
Sara does the whole job alone in 18 hours.
Together they take 12 hours.
What do we want?
How long Tom takes alone.
2 Name the unknown
We don't know how long Tom takes alone.
So we call it x hours.
x = hours for Tom alone
Careful!
Many people take away the times: 6 hours.
But Tom alone is slower than the two together.
So it takes more than 12 hours.
Careful: not 18 − 12 = 6 hours ✗
The whole barn is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Sara does the whole job in 18 hours.
So in 1 hour, Sara does 1 part out of 18.
That is 1/18.
Tom takes x hours alone.
So in 1 hour, Tom does 1 over x.
They work 12 hours together.
In 12 hours, Sara does 12 times as much: 12/18.
And Tom does 12 over x.
3 Write the equation
In words: Sara's part plus Tom's part makes the whole job.
1218 + 12x = 1
So the equation is 12/18 + 12/x = 1.
Now we solve it.
1218 + 12x = 1
First: a bottom can never be 0.
x at the bottom can't be 0.
So x ≠ 0.
x ≠ 0
4 Remove the fractions
The bottoms are 18 and x.
18x can be divided by all of them.
We multiply every part by 18x.
We do it on both sides, so they stay equal.
18x × 1218 + 18x × 12x = 18x × 1
18x divided by 18 is x.
18x divided by x is 18.
18x times 1 is 18x.
Write what is left.
No more fractions!
12x + 216 = 18x
5 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 12x from the left side.
To remove 12x, we take away 12x.
We do it on both sides, so they stay equal.
12x + 216 − 12x = 18x − 12x
12x and − 12x make zero.
Cross them out.
18x − 12x = 6x.
18x − 12x = 6x
Now we have 216 = 6x.
216 = 6x
Let's swap the two sides, so x is on the left.
6x = 216
6 Remove the 6
6x means 6 times x.
We want just x.
To remove times 6, we divide by 6.
We do it on both sides, so they stay equal.
6x6 = 2166
6 divided by 6 is 1.
So just x is left.
216 divided by 6 is 36.
2166 = 36
So x = 36!
x = 36
7 Answer the question
So Tom alone takes 36 hours.
8 Check
Let's check in the story.
In 12 hours, Sara does 12/18, and Tom does 12/36.
in 12 hours: Sara 1218, Tom 1236
24/36 plus 12/36 is 36/36.
That is 1 whole barn, like the story says.
2436 + 1236 = 3636 = 1
Does it make sense?
Tom alone is slower than the two together.
So the answer is right!
makes sense: 36 hours > 12 hours: slower alone than together
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.
▶ Watch it solved
