Why doesn’t a/b=(a+x)/(b+x)?

Question

Conceptually if you are increasing both numbers a and b by an equal amount I don’t understand why the fraction would not also keep the same value. I would appreciate the clarification.

Answer 1

Here's a refutation:

2/4 = .5

(2+3)/4+3) = 5/7 = .714

.714 >.5

Answer 2

a/b = m/1

(a/b)+p = m+p

but not

(a+p)/(b+p) = (m+p)/(1+p)

any action or operation like +,-,*,/ on one side of (=) sign,

should also be the other side,then only resulting

ratio will remain same

if you divide one side values, the values of the other side

quotient will be always = 1

Answer 3

ab = (a + x) / (b + x)

Cross-multiply:

a(b + x) = b(a + x)

ab + ax = ab + bx

ax = bx

True only if a = b or x = 0



So ab = (a + x) / (b + x) is sometimes true, but not always. For example:

7/7 = (7 + 5) / (7 + 5): TRUE because a = b

3/4 = (3 + 0) / (4 + 0): TRUE because x = 0

5/7 = (5 + 2) / (7 + 2), i.e. 5/7 = 7/9: FALSE



Note that b can't be zero and b + x can't be zero.

Answer 4

increasing by addition is NOT the same as increasing by multiplication



  2 + 3 = 5  while  2 X 3 = 6 . . why .. because multiplication is repeated addition.    2X3 means 2 + 2 + 2  =  6

==================================== 

similarly division decreases something but not in the same way subtraction does . . . 

  [a + (1/2)a] / [b + (1/2)b]  does equal a/b

 similarly  [ (a + xa)/(b + xb)]  =  a/b

  ...  these are examples of proportionally increasing both the numerator and the denominator,,, 



I suppose that the answer to your question comes from ancient math.  5000 years ago they had no rules,  Repeated observations by many people over time led to the accepted rules of algebra.



    let's try your proposition with real numbers

    1/2   =  (1 + 3)/(2+3) ???

. . . .right side becomes  4/5  

 so does 1/2  =  4/5?  NO so  this cannot be a rule of math,

 . . . try it with any numbers that you want (except adding 0) and it does not form an equality.  That is why this is not allowed in math.

Answer 5

In order to keep the ratio the same, you must *multiply* (or divide) both numbers by the same value.



In other words:

a/b = ak/(bk), where k is some non-zero constant.



In fact, when you are reducing fractions, you are looking for that common factor between the numerator and denominator. You then divide each by that number and you get an equivalent reduced fraction.



Example:

8/12 = 2/3



Here you divide top and bottom by 4. Or stated in reverse, if you had the fraction 2/3, you could get an equivalent fraction of 8/12 by multiplying top and bottom by 4.



Here's one more way to think about it.



You have a fraction:

a/b



The only thing you can multiply that by and have it still equal the same value is 1.

a/b * 1 = a/b



So let's convert 1 into an equivalent fraction of k/k (assuming k isn't 0):

a/b * k/k = a/b

(ak)/(bk) = a/b



See, you have to *multiply* top and bottom by the same value to keep the fraction the same. You can't just *add* the same amount.

Answer 6

nononoo. a/b is a proportion. Therefore anything that you add on the top and bottom has to be scaled to that proportion for it to be equal. The only way this could be true is if a = b. 

Answer 7

When doing sums or differences, this works.



When doing products or quotients, you need to increase by the same percentage to be considered "the same amount."  For example, with a/b , if you increase both a and b by 27%, the fraction will still have the same value.

Answer 8

Try a couple of numbers:- 



a/b = > 5/10 = 1/2 ( by cancelling down ). 

 

Then

(5 + 2) / (10 + 2) = 7/12  ( This is neither half nor cancels down). 



NB My answer is NOT a proof, but just a numerical verification. 



hope that helps!!!!

Answer 9

Take a simple case:

a = 0 , b = 1 ;  and x any integer.

E.g.,  (0 + 2) / (1 + 2) = 2/3 , which is not = 0 .



Why?

(Again, by example:)

If you divide a pizza into three parts and  take one, that is clearly not the same as dividing it into five parts and taking three.

Answer 10

I decided to visualize this geometrically with an example (see photo) and it really clicked! Thank you for your quick responses. The reason I had this question is bc a ratio of T/T0 only works when the units are in kelvin (unit: 273+C) and not in Celsius (C) 

which perplexed me! Turns out this is the reason why

Answer 11

because "a" units of "1/b" is not the same as "a+x" units of "1/(b+x)



Proper fractions are portions of a whole, so a/b represents "a" units of "1/b".



But, think for a moment:



suppose



a/b = (a+x)/(b+x)

then ab + ax = ab + bx

==> ax = bx

This is true only if b= 0 or a = b

so there are two cases where a/b = (a+x)/(b+x)

Answer 12

Basically, you are looking at the ratio of two numbers. if you add numbers, they must be in the same ratio. 

Example:

9/3 =3

Now if you add numbers they must be in that same rario of 3/1.

Example:

(9 +3)/(3+1) = 12/4 = 3 same answer as 9/3.

Answer 13

In any fraction we increase denominator and numerator same then the fraction have not same properties.

Answer 14

Just try plugging in some values, and you'll see it doesn't always hold.



For example, if a = 2, b = 3 and x = 4, we have



2/3 ≠ (2+4) / (3+4) = 6/7



Your intuition is correct if you're MULTIPLYING both a and b by x (instead of ADDING x to both a and b) like as follows.:

a/b = (a×x) / (b×x)



2/3 = (2×4) / (3×4) = 8/12 = 2/3

Answer 15

Because it's a proportion.

Think of it this way, 3/4 does not equal 4/5. 

(you'd get 4/5 by letting x = 1)

Answer 16

 a/b = (a × x)/(b × x)

 a/b ≠ (a + x)/(b + x)

Answer 17

If you MULTIPLY a and b by the same number the fraction will stay the same. Addition is different from multiplication so there's no reason to expect the same thing with addition.

Answer 18

a/b = (a + x)/(b + y)



a.(b + y) = b.(a + x)



ab + ay = ab + bx



ay = bx



y = (b/a).x





So, you can see that if you add x to the numerator, i.e. the top of the fraction, you must add (b/a).x to the denominator, i.e. the bottom of the faction.





Example (false):



= 18/6 = 3 → if you add 2 at the top and at the bottom



= (18 + 2)/(6+ 2)



= 20/8



= 5/2



= 2.5 ≠ 3





Example (true):



= 18/6 = 3 → you can see that: a = 18 and you can see that: b = 6



Now, if you add 2 to the numerator (so x = 2), then you must add (b/a).x to the denominator:



= (b/a).x



= (6/18) * 2



= 12/18



= 2/3 ← this is y value





Now, restart:



= 18/6 → you add 2 to the top, and you add (2/3) to the bottom



= [18 + 2] / [6 + (2/3)]



= [18 + 2] / [(18/3) + (2/3)]



= [18 + 2] / [(18 + 2)/3]



= [20] / [(20)/3]



= 20 / (20/3) → to divide by (20/3) is similar to multiply by (3/20)



= 20 * (3/20)



= 60/20



= 3

Answer 19

Suppose a/b=(a+x)/(b+x), where a=/=b=/=0 & x=/=0. Then

a(b+x)=b(a+x)

=>

ab+ax=ab+bx

=>

ax=bx

=>

a=b

The result has a conflict with the property of a, b.

Thus, the assumption of a/b=(a+x)/(b+x) is false.

Answer 20

1/9 is not the same as 2/10, is it?

Answer 21

Remember that fractions are the same as ratios or division.



1/2 = 2/4 = 3/6 = 4/8 = 5/10 



These fractions are equal because in each case the numerator is half of the denominator. As we go along the list we add 1 to the numerator. But we add 2 to the denominator. That's how the denominator stays double the numerator.



We can do the same with any other fraction:

3/7 = 6/14 = 9/21 = 12/28 etc.

Answer 22

Why should it stay the same? Is 1/2 equal to 2/3?





If you add numbers to the numerator and denominator that are in the same proportions, then it remains equal.

In 1/2, the denominator is 2 times the numerator. so what you add needs to be in the same proportion, like 5 and 10

(1+5) / (2 + 10) = 6/12 = 1/2.

Answer 23

on one side you five dollars. the other side you put 6 dollars.



in what way does five dollars equal six dollars?



you explain that first. and that'll be a million dollars for the tuturing lesson.

Answer 24

a/b = (a+x)/(b+x)

a(b+x) = b(a+x)

ab + ax = ab + bx

ax = bx

ax - bx = 0

x(a-b) = 0

x = 0 or a-b = 0

x = 0 or a = b



either x is zero or a equals b



You can add to top and bottom if what you add is in the same ratio as the original. 



a/b = c/d = (a+c)/(b+d) = r



a = br, c = dr

(a+c)/(b+d) 

= (br+dr)/(b+d)

= r(b+d)/(b+d)

= r



replace a with ua, b with ub, c with vc and d with vd and the result still holds. 



Thus a linear combination of the numerators of equivalent ratios over the same linear combination of their denominators maintains the same ratio.

Answer 25

9+1   10                    9

----- = -- = 2.5  not     --  = 3

3+1    4                     3

Answer 26

If we let x = a, then



a+x     a+a        2a        a

----- = ------- = ------- ≠ ------

b+x     b+a      a+b        b

Answer 27

a and b have a certain ratio that is a/b. If we increase the denominator b by a certain amount, then to maintain the same ratio we must increase the numerator proportionally by that ratio. Thus if we just increase them by the same absolute amount they will not be in the same proportion.

Answer 28

It would work if you are multiplying both sides by the same value.  here's an example:



1/2 = (3 * 1) / (3 * 2)



Since we multiplied the fraction by 3/3, which reduces to 1 and anything multiplied by 1 has no change in value.



If you try adding 3 to both sides:



1/2 = (3 + 1) / (3 + 2) = 4/5



That doesn't work since 0.5 isn't equal to 0.8.

Answer 29

Just plug in some numbers and you'll see it's wrong



let a = 1 , b = 3, x = 5



Test

a/b = (a + x )/ (b + x)



1/3 = (1 + 5) / (3 + 5)

1/3 = 6/8

1/3 = 3/4     False



Therefor

a/b ≠ (a + x )/ (b + x)

Answer 30

For Gods sake, if you don't know this stuff, then you aren't born or made for this.  Just don't bother trying to understand.  Google the answers so you can pass it.

Answer 31

just do a/b xb+x that is my way

Answer 32

a/b = ax/bx  

The value of the fraction will be same if a common number is multiplied and not added



example  2/5 = (2*3)/(5*3) = 6/15  

However, if we add , say 2, then (2+2)/(2+5) = 4/7 , which is not equal to 2/5