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?

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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) About Me  (Show Source):
You can put this solution on YOUR website!
let's numerator be n and denominator d
given: the ratio of numerator to denominator is 2 to 3
so, we have
n%2Fd=2%2F3.......eq.1
if both are increased by 2, the fraction becomes 3%2F4
%28n%2B2%29%2F%28d%2B2%29=3%2F4.......eq.2
to figure out hat is original fraction, solve this system:
n%2Fd=2%2F3.......eq.1
%28n%2B2%29%2F%28d%2B2%29=3%2F4.......eq.2
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start with n%2Fd=2%2F3.......eq.1 and solve for n
n=%282%2F3%29d....substitute it in eq.2

%28%282%2F3%29d%2B2%29%2F%28d%2B2%29=3%2F4.......eq.2 and solve for d

%28%282%2F3%29d%2B2%29=%283%2F4%29%28d%2B2%29
2d%2F3%2B6%2F3=3d%2F4%2B6%2F4......both sides multiply by 12
12%282d%2F3%29%2B12%286%2F3%29=12%283d%2F4%29%2B12%286%2F4%29....simplify

4%2A2d%2B4%2A6=3%2A3d%2B3%2A6
8d%2B24=9d%2B18
24-18=9d-8d
highlight%28d=6%29
go back to n=%282%2F3%29d,substitute 6%29 for d and solve for n
n=%282%2F3%296
n=%282%2Fcross%283%29%29cross%286%292
n=2%2A2
highlight%28n=4%29

so, the numerator is n=4 and denominator is d=6
original fraction is: highlight%284%2F6%29

check:
n%2Fd=2%2F3.......eq.1
4%2F6=2%2F3
cross%284%292%2Fcross%286%293=2%2F3
2%2F3=2%2F3
and
%28n%2B2%29%2F%28d%2B2%29=3%2F4.......eq.2
%284%2B2%29%2F%286%2B2%29=3%2F4
6%2F8=3%2F4
cross%286%293%2Fcross%288%294=3%2F4
3%2F4=3%2F4

Answer by Boreal(15235) About Me  (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
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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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