can I change 21x34 to 20x35? why or why not?

Question

can I change 21x34 to 20x35? why or why not?

Answer 1

Hi Jamie,

Try this algebra version of your question!

Let x=20 and then x+15=35. Then 20*35 is also (x)(x+15) or x² +15x.

Then x+1 =21 and x+14=34. then 21*34 is (x+1)(x+14) or x²+15x+14.

You can immediately see that the second product has an extra term of 14 that makes it not equal to the first product!

Answer 2

Let's take your numbers and generalize them.

 

You're wanting to know if 

 

(x*y) = (x-1)*(y+1)

 

So let's simplify:

 

xy = xy +x - y - 1

 

As you can see, the two definitely aren't equal. 

 

But can we make them equal?

 

Subtract xy from both sides.

 

0 = x - y - 1

y = x - 1

 

Now let's try it:

 

x = 15

y = 14

 

(15*14)=(15-1)*(14+1)

(15*14)=(14)*(15)

 

HEY HEY WHAT DO YOU KNOW! It does work only in one particular situation.

 

It's only when "y=x-1" that this works, and the reason is obvious from the above: it just swaps the same two numbers around, which is simply the commutative property of multiplication!

Answer 3

Multiplication of numbers has to do with the product of all the factors of each number! in this case, 21 and 34.

So,

21X34= (3)(7)X(2)(17), in other words, by factoring 21 and 34.

If you factor the other pair, 20X35 you will end up with the different pairs of factors,

20X35= (2)(2)(5)X(5)(7).

 

 

 

Addition, however, works if you change the numbers. Since, the number 21 is reduced by one to get to 20 and 34 is increased by one to get to 35; the total sum, therefore, will be

21+34 =20+35

 

 

For MULTIPLICATION,

 

just for your information: if you rearrange the factors of 21X34

 

from (3)(7)X(2)(17)= 

to      (3)(17)X(2)(7)  = 51X14

 OR

to      (3)(2)X(17)(7) = 6X119

 

the product will remain the same, which is 714. This is because the factors that made up the numbers did NOT change.

 

Answer 4

I liked the previous answers.

 

I will just explain that 20 x 34 is different than 21 x 34 because 21 x 34 is one more 34 than 20 x 34.  Then I wonder if you still have a question.

 

 

Answer 5

No you cannot.  This is because 21 × 34      and     20 × 35          have different products. 

 

21 × 34 = 714

 

20 × 35 = 700

 

 

You would think that increasing one factor and decreasing the other factor by the same amount would keep their products the same, but as you can see, it did not.