# cancelling common factors

#### RobertB1985

##### New member
Hi all,
First time here and I'm learning calculus for 1st time.
This problem doesn't seem like calculus to me, but it is a test question in a calulus textbook.

Question:
(2/3)(21) (11/14)=?

3rd step of problem is:
(2)(21)(11)
----------- =
(3)(1)(14)

What I can't figure out is WHY the textbook crosses out the:2,3,21,and 14??
Leaving 11/1 .
I agree with the answer. I need to understand the cancellation.

Thanks a ton,
Robert

#### Subhotosh Khan

##### Super Moderator
Staff member
Hi all,
First time here and I'm learning calculus for 1st time.
This problem doesn't seem like calculus to me, but it is a test question in a calulus textbook.

Question:
(2/3)(21) (11/14)=?

3rd step of problem is:
(2)(21)(11)
----------- =
(3)(1)(14)

What I can't figure out is WHY the textbook crosses out the:2,3,21,and 14??
Leaving 11/1 .
I agree with the answer. I need to understand the cancellation.

Thanks a ton,
Robert
=$$\displaystyle \frac{2 * 21 * 11}{3 * 1 * 14}$$ ...................................factorize

=$$\displaystyle \frac{2 * 7 * 3 * 11}{3 * 1 * 7 * 2}$$

=$$\displaystyle \frac{\cancel{2 * 7} * 3 * 11}{3 * 1 * \cancel{7 * 2}}$$ ...........................cancel common terms

=$$\displaystyle \frac{ 3 * 11}{3 * 1 }$$ ...........................cancel common terms

=$$\displaystyle \frac{ \cancel{3} * 11}{\cancel{3} * 1 }$$

=$$\displaystyle \frac{11}{1}$$

Is it clear now........

#### Dr.Peterson

##### Elite Member
Hi all,
First time here and I'm learning calculus for 1st time.
This problem doesn't seem like calculus to me, but it is a test question in a calulus textbook.

Question:
(2/3)(21) (11/14)=?

3rd step of problem is:
(2)(21)(11)
----------- =
(3)(1)(14)

What I can't figure out is WHY the textbook crosses out the:2,3,21,and 14??
Leaving 11/1 .
I agree with the answer. I need to understand the cancellation.

Thanks a ton,
Robert
Can you show an image of what the book does?

I would do something in addition to just crossing those off, at least if I wanted to communicate to someone else what I was doing.

Either I'd first factor,

$\frac{(2)(21)(11)}{(3)(1)(14)}$=$\frac{(2)(3\cdot 7)(11)}{(3)(1)(2\cdot 7)}$=$= \frac{(\cancel{2})(\cancel{3}\cdot \cancel{7})(11)}{(\cancel{3})(1)(\cancel{2}\cdot \cancel{7})}$=$\frac{(11)}{(1)} = 11$​

or I'd cancel partially, writing the quotients:

$\frac{(2)(21)(11)}{(3)(1)(14)} =\frac{(\cancel{2})(\overset{7}{\cancel{21}})(11)}{(\cancel{3})(1)(\underset{7}{\cancel{14}})} = \frac{(\cancel{7})(11)}{(1)(\cancel{7})} = \frac{(11)}{(1)} = 11$​

They are probably thinking one of those.

#### RobertB1985

##### New member
Breaking down the 21 into 3x7, and the 14 into 2x7, seems like the long way of getting to the same place. But using 42 as a common denom and reducing afterwards isn't quick either. Final analysis: factoring may not be quicker, but it's simplier. It did clear it up. Thank you!

#### RobertB1985

##### New member
If they had factored the 21 and 14 in the step list, I wouldn't have bothered you all. Sometimes teachers have been teaching so long that they've forgotten what its like to be a student.

#### pka

##### Elite Member
Hi all,
First time here and I'm learning calculus for 1st time.
This problem doesn't seem like calculus to me, but it is a test question in a calulus textbook.
Question: (2/3)(21) (11/14)=?
$$\left(\dfrac{2}{3}\right)(21)\left(\dfrac{11}{14}\right)=\dfrac{2\cdot 3\cdot 7\cdot 11}{2\cdot 3\cdot 7}=11$$

#### JeffM

##### Elite Member
If they had factored the 21 and 14 in the step list, I wouldn't have bothered you all. Sometimes teachers have been teaching so long that they've forgotten what its like to be a student.
Amen brother

BUT, the other thing is that students will not say “Wait a minute; I did not get that.”

The first time I meet with someone I tutor, I give a little speech. It goes, “My job is to explain clearly. Your job is to tell me out loud when I fail.” At least high school students seem to like the idea of correcting the tutor.

#### Jomo

##### Elite Member
Let's see, they crossed out 2*21 in the numerator and 3*14 in the denominator. You do not see easily why they cross out, fine!! Just check to see if it is correct to cancel out like they did. It is only correct if 2*21 = 3*14. Ah, yes! They are equal. Slow down and think how can the author be correct, if in fact the author is correct.