SOLUTION: If 2 is subtracted from a numerator of a fraction and 3 is added to the denominator, it reduces to 1/4. If 5 is added to the numerator and 2 is subtracted from the denominator, it

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: If 2 is subtracted from a numerator of a fraction and 3 is added to the denominator, it reduces to 1/4. If 5 is added to the numerator and 2 is subtracted from the denominator, it       Log On


   



Question 1171699: If 2 is subtracted from a numerator of a fraction and 3 is added to the denominator, it reduces to 1/4. If 5 is added to the numerator and 2 is subtracted from the denominator, it reduces to 3. Find the fraction.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

n = original numerator
d = original denominator
n and d are whole numbers. To make things simple, we'll assume they are positive as well.

Let's focus on the part where it says "If 2 is subtracted from a numerator of a fraction and 3 is added to the denominator, it reduces to 1/4"

2 is subtracted from n getting n-2
3 is added to d getting d+3

The fraction n/d turns into (n-2)/(d+3).
We're told this new fraction reduces to 1/4.
We can say (n-2)/(d+3) = 1/4

Cross multiply and solve for d.
(n-2)/(d+3) = 1/4
4(n-2) = 1(d+3)
4n-8 = d+3
d+3 = 4n-8
d = 4n-8-3
d = 4n-11
This will be useful later.

Next we'll focus on the part that says "If 5 is added to the numerator and 2 is subtracted from the denominator, it reduces to 3".

Go back to the original numerator and original denominator (n and d). Add 5 to the original numerator getting n+5. Subtract 2 from the original denominator to get d-2. The fraction (n+5)/(d-2) reduces to 3 meaning we can say:

(n+5)/(d-2) = 3
(n+5)/(4n-11-2) = 3 .... plug in d = 4n-11
(n+5)/(4n-13) = 3
n+5 = 3(4n-13)
n+5 = 12n-39
5+39 = 12n-n
44 = 11n
11n = 44
n = 44/11
n = 4 is the original numerator

That value of n is plugged in like so
d = 4n-11
d = 4*4-11
d = 16-11
d = 5 is the original denominator

The original fraction is n/d = 4/5.

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Let's check our answers.

If n = 4 and d = 5, then,
(n-2)/(d+3) = (4-2)/(5+3)
(n-2)/(d+3) = 2/8
(n-2)/(d+3) = (1*2)/(4*2)
(n-2)/(d+3) = 1/4
This confirms the first part where it reads "If 2 is subtracted from a numerator of a fraction and 3 is added to the denominator, it reduces to 1/4"

Let's check the other part:
(n+5)/(d-2) = (4+5)/(5-2)
(n+5)/(d-2) = 9/3
(n+5)/(d-2) = 3
That part is confirmed as well.

Both parts of the instructions are confirmed.

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Answer: 4/5