Step by step
The question
Question: Dan can mow a field in 10 hours.
Raj can mow it in 15 hours.
How many hours will it take them to mow it together?
1 Find what we know
What do we know?
Dan does the whole job alone in 10 hours.
Raj does it alone in 15 hours.
What do we want?
How long it takes them together.
2 Name the unknown
We don't know how long it takes together.
So we call it x hours.
Careful!
Many people add the times: 25 hours.
But two working together are faster than one.
So together it takes less than 10 hours.
The whole field is 1 whole job.
We split it into parts.
Let's put it in a table: the time alone, and the part done.
Dan does the whole job in 10 hours.
So in 1 hour, Dan does 1 part out of 10.
That is 1/10.
Raj does 1 part out of 15 in 1 hour: 1/15.
In x hours, Dan does x times as much: x/10.
And Raj does x/15.
3 Write the equation
In words: Dan's part plus Raj's part makes the whole job.
So the equation is x/10 + x/15 = 1.
Now we solve it.
4 Remove the fractions
The bottom numbers are 10 and 15.
30 can be divided by all of them.
We multiply every part by 30.
We do it on both sides, so they stay equal.
30 divided by 10 is 3.
30 divided by 15 is 2.
Write what is left.
Do the multiplications.
No more fractions!
5 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
3x + 2x = 5x.
Now we have 5x = 30.
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.
5 divided by 5 is 1.
So just x is left.
30 divided by 5 is 6.
So x = 6!
7 Answer the question
So together it takes 6 hours.
8 Check
Let's check in the story.
In 6 hours, Dan does 6/10.
That is 3/5 of the field.
In 6 hours, Raj does 6/15.
That is 2/5 of the field.
Together: 3/5 plus 2/5 is 5/5.
That is 1 whole field, like the story says.
Does it make sense?
Together is faster than Dan alone, who needs 10 hours.
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.