SOLUTION: The ratio of numerator to denominator is 2 to 3. If both are increased by 2, the fraction becomes 3/4. What is original fraction? Which of following systems can be used to solve?

Question 984487: The ratio of numerator to denominator is 2 to 3. If both are increased by 2, the fraction becomes 3/4. What is original fraction? Which of following systems can be used to solve?
N+4=3 and d+5=4
or
3n-2d=0 and 4n+2=3d+2
or
3n=2d and 4n+8=3d+6
How do you figure this out? Lost!
Found 2 solutions by MathLover1, Boreal:
Answer by MathLover1(20849) (Show Source): You can put this solution on YOUR website!
let's numerator be and denominator
given: the ratio of numerator to denominator is to
so, we have
.......eq.1
if both are increased by , the fraction becomes
.......eq.2
to figure out hat is original fraction, solve this system:
.......eq.1
.......eq.2
----------------------------------------
start with .......eq.1 and solve for
....substitute it in eq.2

.......eq.2 and solve for

......both sides multiply by
....simplify

go back to ,substitute for and solve for

so, the numerator is and denominator is
original fraction is:

check:
.......eq.1

and
.......eq.2

Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
x/y=2/3
(x+2)/(y+2)=3/4
cross-multiply both
3x=2y; 3x-2y=0
4x+8=3y+6
4x-3y=-2
;;
Multiply the first equation by 4 and the second equation by (-3) and add it to the first
12-8y=0
-12x+9y= 6
y=6
x=4
(x+2)=6
(y+2)=8; 6/8=3/4
;
3n-2d=0
4n-3d=0, rearranging
multiply top by 3 and bottom by (-2)
9n-6d=0
-8n+6d=0
n=0
d=0
Indeed, that is the only way that 4n+2=3d+2
===================================
3n=2d; 4n-3d=-2, rearranging. Do it by substitution
n=(2/3)d; 4n=(8/3)d
(8/3)d-(9/3)d=-2
(-1/3)d=-2
d=6, multiplying both sides by (-3)
n=4 substituting d into the first and solving for 3n
4(4)+8=3(6)+6
24=24 check
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