SOLUTION: the numerator of a fraction is 1 less than the denominator. when both numerator and denominator are increased by 2, the fraction is increased 1/12`find the original fraction

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: the numerator of a fraction is 1 less than the denominator. when both numerator and denominator are increased by 2, the fraction is increased 1/12`find the original fraction       Log On


   

Question 1192891: the numerator of a fraction is 1 less than the denominator.
when both numerator and denominator are increased by 2,
the fraction is increased 1/12`find the original fraction
Found 3 solutions by Edwin McCravy, greenestamps, josgarithmetic:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
You must have copied this problem wrong, or your teacher or the
the textbook writer made a mistake, for there is no solution. 
However, if 1/12 is changed to 1/9, there will be a solution.  

I will work a problem exactly like yours, changing 1/12 to 1/9. 
The problem I will work is:

The numerator of a POSITIVE PROPER FRACTION REDUCED TO
LOWEST TERMS is 1 less than the denominator. When both numerator
and denominator are increased by 2, the fraction is increased by
1/9. Find the original fraction.

Let n = the denominator

The numerator of a fraction is 1 less than the denominator.
 
Then the numerator = n-1

matrix%281%2C2%2COriginal%2CFraction%29%22%22=%22%22%28n-1%29%2Fn

when both numerator and denominator are increased by 2,
 

matrix%281%2C2%2CIncreased%2CFraction%29+=+%28%28n-1%29%2B2%29%2F%28n%2B2%29

the fraction is increased by 1/9.
matrix%281%2C2%2CIncreased%2CFraction%29%22%22=%22%22+matrix%281%2C2%2COriginal%2CFraction%29%22%22%2B%22%22expr%281%2F9%29matrix%281%2C2%2COriginal%2CFraction%29

%28%28n-1%29%2B2%29%2F%28n%2B2%29%22%22=%22%22%28n-1%29%2Fn%22%22%2B%22%22expr%281%2F9%29expr%28%28n-1%29%2Fn%29

%28n-1%2B2%29%2F%28n%2B2%29%22%22=%22%22%28n-1%29%2Fn%22%22%2B%22%22%28n-1%29%2F%289n%29

%28n%2B1%29%2F%28n%2B2%29%22%22=%22%22%28n-1%29%2Fn%22%22%2B%22%22%28n-1%29%2F%289n%29

Multiply through by the LCD = 9n(n+2)

9n%28n%2B1%29%22%22=%22%229%28n%2B2%29%28n-1%29%22%22%2B%22%22%28n%2B2%29%28n-1%29

9n%5E2%2B9n%22%22=%22%229%28n%5E2%2Bn-2%29%22%22%2B%22%22n%5E2%2Bn-2

9n%5E2%2B9n%22%22=%22%229n%5E2%2B9n-18%29%22%22%2B%22%22n%5E2%2Bn-2

9n%5E2%2B9n%22%22=%22%2210n%5E2%2B10n-20%29

-n%5E2-n%2B20%22%22=%22%220

n%5E2%2Bn-20%22%22=%22%220

%28n%2B5%29%28n-4%29%22%22=%22%220

n+5=0;   n-4=0
  n=-5;    n=4

Ignore the negative answer.

denominator = n = 4
numerator = n-1 = 4-1 = 3

original fraction = 3%2F4

Checking:

The new fraction is %283%2B2%29%2F%284%2B2%29%22%22=%22%225%2F6

1%2F9 of 3%2F4 is %281%2F9%29%283%2F4%29%22%22=%22%223%2F36%22%22=%22%221%2F12

If we add 3%2F4%2B1%2F12%22%22=%22%229%2F12%2B1%2F12%22%22=%22%2210%2F12%22%22=%22%225%2F6.

So the correct answer to the problem I solved is 3%2F4.

Be sure to inform your teacher that the problem as stated 
(with 1/12) has no solution.

Edwin

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Your statement of the problem is not clear. When the problem says the fraction is "increased by 1/12", it is not clear whether that is an increase by a FACTOR of 1/12 or an increase of an ADDED 1/12.

The other tutor interpreted the meaning to be an increase by a factor of 1/12, which makes the problem have no solution.

The problem has a solution if the increase is an added 1/12.

Let the denominator of the original fraction be x; then the two fractions are (x-1)/x and (x+1)/(x+2).

The second fraction is 1/12 MORE than the first:

%28x%2B1%29%2F%28x%2B2%29-%28x-1%29%2Fx=1%2F12
%28x%2B1%29%28x%29-%28x-1%29%28x%2B2%29=%28x%28x%2B2%29%29%2F12
%28x%5E2%2Bx%29-%28x%5E2%2Bx-2%29=%28x%5E2%2B2x%29%2F12
2=%28x%5E2%2B2x%29%2F12
x%5E2%2B2x=24
x%5E2%2B2x-24=0
%28x%2B6%29%28x-4%29=0

x=-6 or x=4

If we choose the positive solution, then the original fraction is 3/4 and the new fraction is 5/6, and 5/6 - 3/4 = 10/12 - 9/12 = 1/12; the conditions of the problem are satisfied.

If we choose the negative solution, the the original fraction is (-7)/(-6) and the new fraction is (-5)/(-4); and (-5)/(-4) - (-7)/(-6) = 5/4 - 7/6 = 15/12 = 14/12 = 1/12; again the conditions of the problem are satisfied.

So the original fraction could be either 3/4 or (-7)/(-6). However, the format of the second "solution" is not standard, so....

ANSWER: The original fraction was 3/4

Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
----------------------------------
the fraction is increased 1/12`
----------------------------------

What's that? Increased how? "1/12" assumed to be missing "by".

Original Fraction Expression, n%2Fd with n=d-1, so fraction may start as %28d-1%29%2Fd.

When both n and d increased by 2,
%28d-1%2B2%29%2F%28d%2B2%29=%28d-1%29%2Fd%2B%281%2F12%29%28%28d-1%29%2Fd%29.
Solve this for d.