the numerator and denominator of a fraction are in the ratio of 2:3 if 6 is subtracted from the numerator the result is a fraction that has value 2/3 of the ori

Question

the numerator and denominator of a fraction are in the ratio of 2:3 if 6 is subtracted from the numerator the result is a fraction that has value 2/3 of the original fraction the numerator of the original fraction is

 

in my book solution is

let 2x is numerator

     3x is denominator

(2x-6)/3x=(2/3)*2x/3x

              =x=9

 

my confusion is if ratio is 2:3 so

let

numerator x

denominator y

x/y=2:3

how that ratio multiplied to x               

Answer 1

The book is pointing out a very important rule:

   If two values, a and b, have the ratio c:d, then there is a number x such that a=cx and b=dx.

   If two values, a and b, have the ratio c/d, then there is  number x such that a=cx and b=dx.

 

  This is why we can reduce 2x/3x to 2/3.

 

     Examples are: 1/3 = 2/6  (x=2),   2/5 = 6/15 (x=3),   5/8 = 25/40 (x=5)

 

The problem gives the first ratio as 2/3 (c/d).  This means that there is a number x such that a=2x and b=3x where a is the numerator and b is the denominator of the fraction (a/b).

 

So, the original fraction is 2x/3x (ratio 2/3).

The modified fraction is (2x-6)/3x, which is 2/3 of the original fraction [note: "of" usually means multiply]

      (2x-6)/3x = (2/3)(2x/3x)    "modified is 2/3 of original fraction"

      (2x-6)=(2/3)(2x)                [multiply both sides by 2x]

      6x-18 = 4x                       [multiply both sides by 3]

       2x = 18                         [subtract 4x from both sides; add 18 to both sides]

         x = 9                     [divide both sides by 2]

 

The original fraction is 2x/3x = 2(9)/3(9) = 18/27, it has the ratio 2/3.

Answer 2

You are on the right track, Amey. 

 

You can express this as:

 

x/y = 2/3   because 2:3 is the same as 2/3

 

Now, cross-multiply.

 

3x = 2y

 

y = 3/2 x

 

Using the info given in the problem, we know that:

 

(x - 6)/y = 2/3 (x/y)

 

Cross-multiplication gives us:

 

3xy - 18y = 2xy

xy - 18y = 0

y(x - 18) = 0

 

Since y = 3/2 x, we can substitute to get:

 

3/2 x (x - 18) = 0

 

Set both factors equal to 0 and solve for x.

 

We can disregard the 3/2 x term, though, because if x = 0, then y = 0. However, y is our denominator, and division by 0 is undefined. 

 

So, we have x - 18 = 0

 

x = 18 = numerator

Answer 3

Hi Amey,

 

Let numerator be 2x

denominator = 3x

Original fraction=2x/3x

 

now 6 is subtracted from numerator so ratio 2x-6/3x

which is equal to 2/3 of original fraction.so

2x-6/3x = 2/3 {2x/3x}

 

(2x-6)*3 = 2(2x)

6x-18=4x

2x=18

x=9

Original fraction= 2x/3x =18/27

 

Answer 4

Correct - 2x / 3x is the original fraction.

If x = 9, the fraction is (2)(9)/(3)(9), or 18/27.