Finding the unknown number

Question

In a fraction, twice the numerator is 2 more than the denominator. If 3 is added to the numerator and to the denominator, the new fraction formed is 2/3. Find the original number.

Answer 1

In a fraction, twice the numerator is 2 more than the denominator so let's write the fraction as n/(2n-2).

 

If 3 is added to the numerator and to the denominator, the new fraction formed is 2/3. 

Now we have (n+3)/(2n+1) = 2/3.  This is a proportion and we can cross-multiply; we get 4n+2 = 3n+9

and we solve for n=7.  Original fraction must be 7/12.

 

We check by adding 3 to numerator &denominator; 10/15 does indeed equal 2/3.

 

 

 

Answer 2

The most difficult part of word (story) problems is correctly translating the words and phrases into precise, concise math variables and operations.  It helps to make a "translation dictionary" for yourself:

      twice              means                2x

       is                                            =

      more than                                +

. . .

   (extend with each problem)

 

Let N= numerator

      D = denominator

 

Original number is N/D

 

Translate:

   "twice the numerator is 2 more than the denominator"    means

       2     *       N         =  2      +                   D         [eq1]

   "If 3 is added to the numerator"  means

        3     +                   N            

   "If 3 is added to . . . the denominator"   means

        3       +                        D

   "the new fraction formed is 2/3"     means

        (3+N) / (3+D)           =   2/3                [eq2]

 

From eq1, we have:      D = 2N-2
 

Cross multiplying eq2, we have:

        2(3+D) =3(3+N)

         6+2D = 9 + 3N

 

Substituting for D, we get:

         6 + 2(2N-2) = 9 + 3N

         6 + 4N - 4 = 9 + 3N         [distribute]

          4N + 2 = 9 + 3N            [collect terms]

          N + 2 = 9               [subtract 3N from both sides]

          N = 7                    [subtract 7 from both sides]

      

Now, substitute for N in either equation to solve for D.  Let's use eq1:

   2(7) = 2 + D

    14  =  2 + D

     12 = D

 

The original fraction is:   7/12

 

Checking:

  Is   2(7) = 2+12   ?

         14 = 14   ?yes

  Is   (7+3) / (12+3) =  2/3   ?

          10 / 15   ?

           2/3   ?yes

 

 

Answer 3

Hi Riddhi,

 

A fraction can be expressed as n/d where n is the numerator and d is the denominator. We are told that twice the numerator is two more than the denominator, which translates to

 

2n = d + 2,

 

and this can be written as

 

2n - d = 2

 

We are also told that if three is added to both the numerator and denominator, then the new fraction is 2/3. This means that

 

(n + 3)/(d + 3) = 2/3

 

We can cross multiply this last equation to get

 

3(n + 3) = 2(d + 3)

 

which leads to

 

3n + 9 = 2d + 6

 

which can be reduced to

 

3n -2d = -3

 

 

So, we have two linear equations in two unknowns n, d:

 

2n - d = 2

3n -2d = -3

 

Multiply the top equation by -2. We get

 

-4n + 2d = -4
3n - 2d = -3

 

Now, add these equations and the d terms cancel and we get

 

-n = -7,

 

or n = 7

 

We can now plug n = 7 into either of the original equations to solve for d. I choose:

 

2n - d = 2

 

or

 

d = 2n - 2 = 2(7) - 2 = 14 - 2 = 12

 

So, the original fraction is 7/12

 

===== CHECK ====

 

Twice the numerator is two more than the denominator: 2(7) = 12 + 2 which implies that 14 = 14. Yes.

 

If three is added to the numerator and to the denominator, you get 2/3. (7 + 3)/(12 + 3) = 10/15 = 2/3. Yes

 

So, again, the answer is 7/12